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<p><strong> MathXpert and Mathematical Education <br>
by Michael Beeson,<br>
author of MathXpert
</strong></p>
<p>The famous mathematician Karl Gauss called mathematics the <em>Queen of Sciences. </em>Physical
science rests on a mathematical basis; engineering rests on physical
science; technology rests on engineering. Without mathematics, your car
and television wouldn't exist, and your house would be dark and cold.
The refrigerator works on principles of thermodynamics that could never
have been understood without mathematics, and the same goes for Diesel
engines, so without mathematics, you would have to eat what could be
grown nearby, and you would have to eat it before it spoiled. Life
would be "nasty, brutish, and short."</p>
<p>Without an understanding of mathematics, a person cannot understand
the principles on which refrigerators, cars, airplanes, television
sets, cell phones, and computers have been designed. Mathematically
uneducated persons may simply take these products of human intelligence
for granted, treating them as part of the environment. This should not
be allowed to happen: every person should have the opportunity to learn
and understand the accumulated technical knowledge of the past.</p>
<p>Mathematical literacy is much rarer in our society than verbal literacy. Two main reasons for this are</p>
<ul>
<li>You have to be very careful in mathematics, because a slight error can throw you completely off track.</li>
<li>Mathematics is cumulative. You must master each part of the subject before moving on to the next part.</li>
</ul>
<p>You must have encountered both of these problems in your attempts to learn mathematics!</p>
<p>I once analyzed the errors my students made on exams in freshman
calculus. Eighty percent were not calculus errors at all, but algebra
or trigonometry errors. That is, these were errors in things the
students were supposed to know before taking calculus at all. Once such
an error had been made, the students could not complete the calculus
problem correctly. Notice how the two difficulties are connected: for
lack of mastery of earlier stages, the students are not <em>able</em> to proceed carefully.</p>
<p>Mastering a subject before proceeding to the next subject is
definitely not encouraged by the present educational system. We find
students at all levels lacking fundamental skills they should have
mastered years ago. Being very careful with your work is not encouraged
by our society, either. Examples are easy to find at the individual,
corporate, and governmental levels. It is hardly surprising, then, that
a subject that <em>requires</em> care and mastery is widely regarded
as difficult and frightening. But the roots of "math anxiety" are to be
found in the two difficulties mentioned above, not in any inherent
difficulty of mathematics itself.</p>
<p>In 1981, I thought that the computer could be harnessed to make a
fundamental attack on these two problems in mathematical education.
Using a computer, a student can proceed at his or her own pace, so that
mastery can be achieved. In a classroom, if a student is a little too
slow, he or she can easily get lost, and understand nothing at all of
the rest of the lecture. When using a computer projector instead of a
blackboard, I learned that about one-third of my class typically lags
at least one full blackboard behind the lecture. These students
complained that the solutions would scroll off the projector screen
before they were finished copying them. In years of lecturing at the
blackboard, I had never realized that many students had their attention
on the previous board instead of on the developing lecture.</p>
<p>Using a computer privately, these students will be able to look at
each screen as long as they desire, and even go back over the problems
again and again. Moreover, the computer can offer solid help with the
necessity of being careful. If the program is properly written, I
thought, it should be <em>impossible to make a mistake</em>. You could, of course, take a less than ideal step, but not a mathematically incorrect step. </p>
<p>If a program is going to help students to achieve mastery, it must
be able to present each subject at the appropriate level. Beginning
algebra students may need a five-step solution to a simple
common-denominator problem such as 1/2 + 1/3. Calculus students must
see common denominators as a one-step operation, perhaps even to be
performed as a small part of the operation <em>simplify</em>. </p>
<p>There were, even at that time, some mathematical programs in
existence, but they had several basic flaws, and I decided in 1985 to
write a new program from scratch. This program became MathXpert. The
name is a contraction of "Mathematical Expert", because the program is
able to show you how to solve most of the problems you would encounter
in algebra, trigonometry, and two semesters of calculus. (The term <em>precalculus</em>
is used only in the US, and includes trigonometry, logarithms,
exponentials, and an introduction to complex numbers, all of which lie
within MathXpert's expertise.) </p>
<p>MathXpert has been designed from the beginning to satisfy these principles: </p>
<ul>
<li><em>Correctness</em>. You cannot make a mathematical mistake.</li>
<li><em>Transparency</em>. You can see and understand every step, not just the final answer.</li>
<li><em>Appropriate Step Size.</em> Detailed steps for beginners, powerful short steps in advanced topics.</li>
<li><em>Real Problem Solving</em>. Not just an electronic page turner, MathXpert can help you solve any problem, including those in your homework assignment.</li>
<li><em>Easy and Accurate Graphing</em>.
The computer can make correct graphs instantly, and change them
instantly so you can see how the graph depends on the numbers in the
formula. You are never more than one click away from a relevant graph.</li>
</ul>
<p>MathXpert is to solving a mathematics problem somewhat as a word
processing program is to writing an essay. You still direct the course
of the solution, but you use the computer to get the steps on the
screen. You tell the computer <em>add these fractions and put them over a common denominator</em>,
for example, and it carries out that operation, copying the rest of the
line over onto the next line. MathXpert provides an easy method for you
to accomplish the task of telling the computer what to do, without
having to remember (let alone type) complicated commands or look them
up. A lot of work has gone into the design of the "term selection"
method that lets you accomplish this easily.</p>
<p>Where MathXpert differs from a word processing program is in the
amount of help it can give you in solving your problems. A word
processing program can't write your essay; but MathXpert can solve your
mathematics problem (most of the time, I hope). This power is made
available to you through the <em>Hint</em>, <em>AutoStep™</em>, and <em>AutoFinish™</em> buttons, whose functions are explained in this manual. </p>
<p>I wrote MathXpert as a tool for you to use in learning mathematics.
You will be able to use it to get past the obstacles to doing careful
work in mathematics, and mastering mathematics level by level. However,
no program, no matter how sophisticated, can do the learning for you.
In studying mathematics, you should take it as your ideal to master the
subject, and to do absolutely careful work. In the past, this may have
been too difficult for you, due to the classroom situation and the lack
of immediate feedback when you made mistakes in your homework. Those
days are over: now, using MathXpert, you can see every detail at your
own pace, and never make another mathematical mistake.</p>
<p>Not only the students who have difficulty with mathematics will
benefit from MathXpert, but also the bright ones, who normally are
bored in their mathematics classes, and are forced to sit through day
after day of trivial repetitions of things they have heard already.
These students should fly like birds in the new world that MathXpert
will open up to them. They too can proceed at their own pace, and solve
problems that will challenge them instead of bore them. I expect to
hear from those students about a lot of difficult problems that they
were able to solve with MathXpert, which MathXpert's internal
algorithms could not solve, or could not solve so beautifully.
MathXpert's methods are good, but they are general, and students will
find they are sometimes able to improve on MathXpert's auto-generated
solutions.</p>
<p>MathXpert is not only for students enrolled in classes. It can also
be used for home study. Many mathematics teachers and engineers will be
brushing up their mathematics in the evenings using MathXpert. A lot of
people who say that math was their worst subject in school secretly
wish that hadn't been the case; and some of them will be interested
enough to give it another try, especially when no human being will be
looking over their shoulder to see their mistakes, and especially when
they are guaranteed <em>not to be able to make any mistake. </em></p>
<p>There is another reason why Gauss called mathematics the Queen of
Sciences: she has an austere but compelling beauty. She reveals herself
only gradually, after long and patient attention. If you are careful,
and seek to master the subject, you too will fall under her spell, and
find that you have developed a genuine love of mathematics.</p>
<p>Dr. Michael Beeson<br>
Professor Emeritus of Mathematics<br>
Professor Emeritus of Computer Science<br>
San Jose State University<br>
<a href="mailto:contacthelpwithmath@gmail.com">contacthelpwithmath@gmail.com</a></p>
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