As a mathematics instructor the majority of the time spent in the class room is on problem solving at the board. In this setting technology is a supplement, since I have found its useful for students to see the steps worked out in real time in order to internalize the thought process. However, there are two specific examples of how technology can enhance learning even during the problem solving process. The first occured when I was substitute teaching Math 230 Multivariable Calculus. It is near impossible to draw the 3-D shapes and surfaces of revolution encountered so that they are clear. I have used the surface area of revolution notebook from Mathematica to draw clear images such as a torus or sphere



Another example occurs when teaching direction fields and stability of linear systems as part of Math 251: ODE and PDE. Here I have also used Mathematica notebooks. The first picture is of a direction field in a 2D place which illustrates the slope of the solution at every point in space. To draw a million of these tiny arrows on the board takes more time away from learning, but with a computor we can plot the direction field immediately and analyze the solution. The second picture is a phase portrait for a 2 x 2 linear system, it would take a lot of time to draw this at the board, but instantly students can see the arrows leaving the origin indicating an unstable spiral.

















Reflections on Technology

I have noticed that in lower-level math courses (lowwer than 200-level) the problems and diagrams are so simple that most things can be done without technology directly in the classroom, but I always perfer to have a course webpage (link to Angel off my home webpage) to post lecture notes and review materials for exams which are both crucial for the learning process. In the higher level courses with multiple dimensions and more advanced diagrams technology is a good supplement. When a 2D blackboard cannot capture something that is necessary for student understanding. My use of technology during class has consisted only of displaying detailed graphics using Mathematica notebooks for imaging beyond the course webpage and course materials. An example of a bad use of technology is the dependence of students on their calculators and not wanting to learn how to do it by hand. They do not realize that the concepts need to be understood to even be able to discern whether the calculator is giving a reasonable answer. I would like to use more technology, but I have only taught in one technology-enabled classroom at Penn State. I strongly encourage students to print out lecture notes from the course webpage and fill in the examples as we do them in class. I also refer students to review websites and technology features for their use at home to aid in the learning process. The bottom line is that technology will never surpass the importance of the lectures itself or the textbook, but as a supplement it can enhance a lecture by a factor of at least 1.5.

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