Interactive On-line Exercises of Basic Mathematical Functions

H. Nishizawa
Department of Electrical Engineering, Toyota National College of
Technology, Eisei-cho 2-1, Toyota 471-8525, Japan

In this paper, a system is described for serving on-line exercises of mathematical functions for high-school students. The system basically depends on WWW technology and uses WWW browsers running on students' computers for displaying questions, answers, and explanations. An exercise page has a graph of a mathematical function and requests a student to fill the text field with a mathematical expression appropriate for the graph. When the student clicks on the "evaluate" button, the expression is sent to the system server and compared with the answer expression. If the expression is not equal to the answer, graphs of both expressions, and the comment on how different his expression to the answer are displayed. Comparing the graphs and reading the comment, the student is able to continue his guessing work until he gets the correct expression. The exercises have been developed for students who have difficulty in understanding mathematical functions in the author´s algebra course, who seem to lack graphical images of the functions they learned in the classes. They usually try to remember the written rules for converting a mathematical function to its graph and reverting it. Such attempts sometimes succeed in simple cases like linear functions but fails in quadratic functions or more complicated functions because the rules for converting those functions become too complicated just to remember. What they lack is rich experiences of handling real graphs. Although recent development of electrical worksheets and symbolic computing program like Mathematica made it easy and quick for a student to draw graphs, it is only half the way to the understanding. Because the drawing is automated and includes usually few guessing works, the student does not necessarily have ideas about, for example, the effect of a coefficient value on the shape or the position of a graph. The only rigid knowledge he has is how he operates the program. The on-line exercises provide with the other half. The first thing a student does in the exercise is a guess which tries to describe the given graph correctly. There is no instructions of expressing the graph nor hints before the first guess. After the first guess is done, then comes the hint or comment which help him to make the next guess. This makes a kind of experiment on the desktop. A series of the experiments tell him the behavior of graphs in detail and the relation with the expressions. Adding to that, because the evaluation are done symbolically using Mathematica functions, wide variety of expressions are allowed, which is easier to accept for usual students. The exercise provides the students with opportunities to learn from their mistakes and to build their own theory about mathematical functions.