Modeling explicitly goes over the various types of graphs, makes them graph
them, linearize them, and write the final equation. This is done repeatedly
with real experiments, and the results are graded with a lot of credit on
the correct graphing procedures. Students also have to describe the
relationships.
I take the point of view that they must do graphs by hand until all of them
can do it correctly. I have never finished the semester with all doing it
correctly, but by the end the vast majority are doing it well.
One big problem in doing graphs is the ability to interpolate the points and
get them in the right place. This relates to numeracy, and when it is low
this is a partial indication of low numeracy. I have considered making a
practice program for them to learn how to interpolate values graphically.
As to the source of graphing problems it is much more than just the
calculators. The textbooks, both science and math, generally present graphs
with straight lines at 45 degrees going through the grid points, and going
through zero. Curves are often distorted and in many cases there is subtle
distortion because the draftsman did not understand what it should be. My
biggest beef here is the temperature graphs for constant heating. Usually
the get the slopes wrong, and often horizonally compress the graph in some
regions without indicating it. Also broke axes are used in scientific
graphs, and students think they can analyze curves with broken axes. The
texts need to have much better graphs. The only text that I know actually
attempts to attack the graphing problem is Minds-on-Physics where they have
students analyze both a perfect, and a more accurate graph. They frequently
have students get data from graphs, and even present maps with tilted axes.
Math books are SOOOOO limited in that they never present tilted axes,
triangles at very odd angles... Also in math students are generally not
required to use scales and to calculate slopes from scaled graphs, so they
just do blocks no matter what the scale is. Indeed math is very deficient
for not doing units consistently and for presenting math analysis as just a
bunch of number without any meaning. They always use X & Y as a map, but
never really draw the distinction between a map and a graph.
So if students had to first learn graphing properly, the graphing
calculators would be a great aid, but instead they go to the calculator a
wee bit too early. They should never use graphing calculators if they have
low numeracy. But computer programs that allow them to easily explore
things like slope and intercept can be of great benefit before they
memoirize the mantra of mX + b. The graphing calculators are both too
limiting in some cases, and too powerful in others. Being able to
continuously vary m or b using a slider and see what they do is a valuable
thing.
John M. Clement
Houston, TX
-----Original Message-----
I don't know about ability to sketch certain algebraic equations, but I have
seen that the words "constant," "linear," and "quadratic" are disconnected
from the graph shapes if the students have to draw them. They can recognize
them if I draw them, which really puzzles me.
Students have difficulty in actually producing plots of data, especially
when they have to choose the axis scales. We force them to do hand plots for
labs the first half of the semesters (plotting d vs t^2, d vs t, <v> vs t,
range vs theta, etc.) just in case they don't have a computer around when
they need to plot something, and to make them think about ratios, scaling,
perspective, linearity, etc. I still fear that the "skill" disappears when
we let them use the computer for their graphing. When I let them use the
computer, I force them to describe the type of curve....[primal screams
heard].