H. Nishizawa

Department of Electrical Engineering, Toyota National College of

Technology, Eisei-cho 2-1, Toyota 471-8525, Japan

E-mail: nisizawa@tctcc.cc.toyota-ct.ac.jp

**Abstract**

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.