Now See 'Ems. NO No See 'Ems.
Brian Winkel, Professor
Editor - PRIMUS and Cryptologia
Department of Mathematical Sciences
United States Military Academy
West Point NY 10996 USA
845-938-3200
Brian-Winkel@usma.edu
http://www.dean.usma.edu/math/people/winkel/
To teach visualization skills we need activities in which visualization plays a role. We present four such activities we have used. (1) Fully describe what you would see on one mountain (paraboloid, say) if your eye was on another mountain - issues of dot and cross product, tangent planes, normals to the surface, gradients, regions of integration, surface area, and programming can play major roles in describing what the eye can see. Students must first attempt to visualize and describe what they expect to see and then precisely determine and describe what they can see and how much they can see. (2) Describe the kinematics (motion) of a point, e.g., a point on a piston or the tip of a lever mechanism subjected to spring-like influence on its motion. (3) Explore the parameter space when trying to determine the best fitting parameters to a nonlinear data analysis activity. (4) Examine data points (each student gets a unique input frequency for a circuit which turns out to be a low-hi-pass filter). We plot the resulting output gain as a function of input frequency using class data and attempt to predict the nature of the device from the data points before using Laplace Transform techniques to "nail" the gain response curve.
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Basically I have used computers (specifically computer algebra systems - Maple and Mathematica) to enable students to see images and to do the necessary mathematics to permit them to discover shapes, surfaces, parameters, motions, etc.
The FIRST activity in the abstract is written up in an earlier form at the web site of NSF funded development of Complex, Technology Based Problems in Calculus:
http://www.rose-hulman.edu/Class/CalculusProbs/Problems/OVERVIEW/OVERVIEW.html
and has been published:
Winkel, B. J. 1997. In Plane View: An Exercise in Visualization. International Journal of Mathematical Education in Science and Technology. 28(4): 599-607.
The SECOND activity, can be illustrated from problems at the same site: http://www.rose-hulman.edu/Class/CalculusProbs/Problems/DRAGSTIK/DRAGSTIK.html.
The THIRD activity would be illustrated by forming a direct sum of squares of actual data and predicted data from a model to estimate the parameters, e.g., suppose we have data set data = {(x1,y1), (x2, y2), . . . , (xn, yn)} and we have a function, say f(x) = a Exp(b*x) where the parameters to estimate are a and b. We form the sum of squares from the differences (squared) between the observed data, yi, and the model predicted value f(xi) = a Exp(b*xi), i.e. ss(a,b) = Sum((yi - f(xi)^2,{i,1,n}) = (y1 - a Exp(x1*b))^2 + (y2 - a Exp(x2*b))^2 + . . . + (yn - a Exp(xn*b))^2 and we plot the surface z = ss(a,b) in order visualize the best parameters and to examine other issues such as the possibility there are several best candidates, albeit not in this case.
The FOURTH activity is one which we recently used in our Engineering Mathematics class at West Point, but whose technique we have used many times. Here we give each student a unique data point (in this case the frequency of an input driver function to a system of differential equations the students build to model a specific circuit, a circuit the class knows nothing about, nor does the class understand what the circuit might be used for) and we ask them to compute the gain,. i.e. the amplitude of the output signal divided by the amplitude of the input frequency signal. We then take the individual data reports and plot gain vs. input frequency to see what the situation really is about. From such data collecting we can make conjectures about the nature of the object we are studying, even before a more formal analysis with Laplace Transforms.
If time merited it (and it probably would not, if restricted to the usual ten minute time) I would talk about another visualization, the addition of a high speed highway to a region and how it alters the "neighborhood" of a shopping mall, i.e. the change in the set of points from which you could get to the mall in, say, one hour or less. Again, I have done this with students many times, have placed it on the web at
http://www.rose-hulman.edu/Class/CalculusProbs/Problems/MALLED/MALLED.html
and published the paper at:
Winkel, B. J. 2001. Getting to the Mall: An Activity for Problem-Solving. International Journal of Mathematical Education in Science and Technology. 32(6): 921-931.
Finally (but not really finally, because I could go on and on with illustrations!) consider the following problem:
"The claim is that by slicing parallel sections of equal altitudes from a sphere we get identical surface areas of these sections. The problem is posed in terms of equity of crust distribution for French bread." How about the same question with volume, instead of surface area? This to can be found on the web at:
http://www.rose-hulman.edu/Class/CalculusProbs/Problems/BREADCUT/BREADCUT.html
There you have it. I shall now sit tight, humming the words of Duke Ellington, "Do Nothing Til You Hear From Me". . Me being You in this case. Take care.