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%TCIDATA{Created=Fri Dec 29 17:40:37 1995}
%TCIDATA{LastRevised=Thu Jun 13 17:05:28 1996}
%TCIDATA{Language=American English}

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\author{%TCIMACRO{\TeXButton{Large}{\Large} }
%BeginExpansion
\Large%
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\textbf{Jonathan Lewin\thanks{
The author of this article has no business connections of any kind with TCI
Software Research, the developers of \emph{Scientific WorkPlace\/ }and has
no financial interest in the success of this product in the marketplace. The
writing of this article was not commissioned by TCI Software Research and
the author receives no commission nor any other remuneration for it.}}}
\title{%TCIMACRO{\TeXButton{LARGE}{\LARGE} }
%BeginExpansion
\LARGE%
%EndExpansion
{\textbf{Building An Information Superhighway Between Student and Teacher}} }
\date{{Kennesaw State University, Marietta, GA jlewin@kscmail.kennesaw.edu}}
\maketitle

\begin{abstract}
In today's academic environment, many students are denied the sort of
quality student-teacher interaction that was taken for granted when their
courses were designed. As a result of this lack of communication, academic
standards have suffered, particularly in disciplines such as mathematics.

This article suggests a way of opening up a line of communication between
student and teacher using \emph{Scientific WorkPlace,\/ }a spectacular new
scientific word processing system and front end for symbolic manipulators
such as \emph{Maple}\/ and \emph{Mathematica.}\/ Our intention is to enable
students to submit questions, homework and tests to their instructors and to
receive assignments, handouts and lecture notes --- all by e-mail or by a
simple operation in a computer lab.

This article also displays some of the features of \emph{Scientific
WorkPlace\/ }as a front end for Maple and Mathematica and reveals the
simplicity with which mathematical operations can be performed in \emph{%
Scientific WorkPlace.}\newpage\ 
\end{abstract}

\tableofcontents

\section{The Communication Dilemma}

The reality of today's college environment, particularly in a commuter
school like the one in which I teach, is that many students maintain their
own households and support themselves by working long hours at their jobs.
To accommodate the needs of such students, many institutions have been
compelled, in recent years, to reduce sharply the number of lectures that
are given in each course and to make each lecture correspondingly longer.
These longer lectures impose a grave burden on students who simply cannot
stay alert for so much time. Furthermore, it is quite common for students in
this kind of course to arrive on campus for their classes and then to leave
as soon as soon as they can to return to their places of employment. This
makes it hard for student and teacher to engage in the one-on-one dialogue
that students may need in order to study mathematics successfully.

Unfortunately, the classical format of a typical college or university
mathematics program is designed for the traditional full-time student who
spends most of his/her time on campus and has ample access to the teacher.
We need to face the fact that one of the root causes of student failure is
our own failure to recognize that our classical teaching technique was
designed to fit an environment that no longer exists at many schools.

The purpose of this article is to suggest a solution to this communications
dilemma. I shall suggest a way of using modern technology to give our
students the individual care they need even when we cannot meet them face to
face. Students will be able to talk to us, to receive lecture notes, to
question us, to submit assignments to us and even to be tested by us when
they are not in our physical presence. In other words, I shall suggest an
efficient way in which students of science and mathematics can engage in
what has come to be called \textbf{distance learning.}

\section{A Software Solution of The Problem}

\subsection{Desired Capability of The Software}

For the establishment of an adequate line of communication between teacher
and student, we need two basic ingredients:

\begin{itemize}
\item  Both teacher and student need to have access to a personal computer.
Ideally, these computers should be networked or, failing that, they should
be connected with e-mail. If neither a network nor e-mail are available then
students and their instructors can still send diskettes to each other.

\item  Both teacher and student should have access to special software that
enables almost effortless reading and writing of complex mathematical
material at the computer display and also provides effortless symbolic
manipulation of such material. In addition, the documents should be stored
as plain text files that can be sent back and forth easily by e-mail.
\end{itemize}

It is the second of these two ingredients that has, until now, been most
illusive. None of the standard word processing products that I have seen is
sufficiently versatile or easy to use to serve as a medium of communication.
However, I am glad to report that this void has been filled recently by a
product known as \emph{Scientific WorkPlace.}

\subsection{\emph{Scientific Word\/} And \emph{Scientific WorkPlace}.}

\emph{Scientific Word\/} is a strikingly new kind of word processing utility
that works by front ending the sophisticated \LaTeX \ typesetting system. It
is quite unlike any other word processing product on the market and there I
believe that it is the word processor of choice for almost all kinds of
scientific and mathematical writing. \emph{Scientific Word}\/ is also sold
bundled together with Maple\footnote{%
Maple V is a product of Waterloo Maple Software.} V,\emph{\ }Release 3 or
with a link to Mathematica\footnote{%
Mathematica is a Product of Wolfram Research Inc.} and this bundle is known
as \emph{Scientific WorkPlace.}\/ In \emph{Scientific WorkPlace,}\/ the word
processing features of \emph{Scientific Word}\/ act as an extremely
user-friendly front end for Maple or Mathematica and, in this way, \emph{%
Scientific WorkPlace\/ }provides the ultimately user-friendly symbolic
manipulator and graphing utility.

From the perspective of working with students, it is not the ability of 
\emph{Scientific WorkPlace}\/ to produce professional quality output that is
so important. Rather, it is the fact that \emph{Scientific WorkPlace\/ }is
so easy to learn and that document preparation is so fast. It takes me only
two meetings with my classes in the computer lab before the overwhelming
majority of my students are comfortable with the process of using \emph{%
Scientific WorkPlace\/ }to read and write documents and to perform
mathematical manipulations.

Part of the essence of \emph{Scientific WorkPlace\/} is that it is so simple
to use. The user does not have to be concerned about such things as tab
stops, fonts, margins, hyphenation, pagination, space to be left between
lines and other similar time-consuming visual activity that is,
unfortunately, the bread and butter of writing with other word processing
systems. Instead, \emph{Scientific WorkPlace\/ }is based on a principle
known as \textbf{logical design }that is described in Section \ref
{logical-design}.

\subsection{Making The Software Available}

On our campus we have \emph{Scientific WorkPlace\/} installed in faculty
offices and in several computer labs to which students have access. All of
these computers are hooked onto a network and all students and faculty have
e-mail accounts.

In addition, some of our students and faculty have chosen to acquire their
own copies of \emph{Scientific WorkPlace}\/ which they have installed on
their computers at home. This is facilitated by the existence of a low
priced student edition of \emph{Scientific WorkPlace\/ }and by the generous
discount that we receive, as holders of a site license, for individual
copies of the software.

\subsection{Using \emph{Scientific WorkPlace\/ }to Communicate with Students}

As I prepare my courses, I write up sets of lecture notes in \emph{%
Scientific WorkPlace.\/ }I do so, not merely because I want these notes to
be available to my students, but because the process of writing my notes in 
\emph{Scientific WorkPlace\/ }is the easiest and fastest way for me to make
them. One of the logical drives seen by my office computer is drive P: which
belongs to our network. Anything I place in drive P: is visible in read-only
form to our students in the computer labs. So, for example, my students in
Mathematics 202 know that they can find material in the path 
\[
\begin{tabular}{l}
P:\TEXTsymbol{\backslash}files\TEXTsymbol{\backslash}math\TEXTsymbol{%
\backslash}lewin\TEXTsymbol{\backslash}202
\end{tabular}
\]
which is constantly being updated and expanded.

One of the documents in this directory is ask202.tex. When students come to
see me in my office I open this document in \emph{Scientific WorkPlace.\/ }I
find it worth while to write both the question that the student is asking me
and my response to the question directly into the \emph{Scientific
WorkPlace\/ }document on my screen. The typing is so easy that it is
actually easier for me to answer this way than to write with pencil and
paper. When I am done, I save the document. From this moment on, any of my
students can see that this question has been asked and they can read my
response. I have found that many of my students are grateful to see the
answers to questions that they themselves could not ask, either because of
lack of opportunity or because they were unwilling to reveal their ignorance
of the material.

Unfortunately, the academic computing department at my institution has not
(yet) provided a really simple method for students to submit material to me.
They cannot place their work in drive P: for me to pick up because, as I
have said, the P: drive is read-only from any student operated computer. My
suggestion to academic computing (which has, so far, produced nothing much
more than a puzzled stare) is that they supply a new logical drive Q: that
would be \emph{write-only}\/ to students. An added requirement of such a
write-only drive is that a student should not be able to overwrite a file
that is already placed there. This would prevent a student from accidentally
erasing anther student's work.

In the meantime, I teach my students how to place their SWP document into
their e-mail and to send me an e-mail letter that looks something like the
following:

\bigskip\ 

\begin{quote}
\texttt{Dear Dr. Lewin. Here is my homework etc. etc.}

\texttt{\rule{1.47in}{0.01in}cut here \rule{1.47in}{0.01in}}
\end{quote}

\texttt{\%\% This document created by Scientific Word (R) Version 2.5 \%\%
Starting shell: article}

\texttt{\TEXTsymbol{\backslash}documentclass[12pt,thmsa]\{article\}}

\texttt{\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%%
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\texttt{\%TCIDATA\{TCIstyle=article/art4.lat,lart,article\}}

\texttt{\%TCIDATA\{Created=Sun Nov 26 15:26:00 1995\}
\%TCIDATA\{LastRevised=Sun Nov 26 15:26:00 1995\}}

\texttt{\TEXTsymbol{\backslash}input tcilatex \TEXTsymbol{\backslash}%
begin\{document\}}

\texttt{\TEXTsymbol{\backslash}author\{Bertie Wooster\} \TEXTsymbol{%
\backslash}title\{A Question on Series\} \TEXTsymbol{\backslash}%
date\{November 4, 1995\} \TEXTsymbol{\backslash}maketitle}

\texttt{Question: Discuss the convergence or divergence of the series 
\TEXTsymbol{\backslash}[ \TEXTsymbol{\backslash}sum \TEXTsymbol{\backslash}%
frac\{\TEXTsymbol{\backslash}left( 2n\TEXTsymbol{\backslash}right) !\}\{4%
\symbol{94}n\TEXTsymbol{\backslash}left( n!\TEXTsymbol{\backslash}right) 
\symbol{94}2\}. \TEXTsymbol{\backslash}] We write \TEXTsymbol{\backslash}[
a\_n=\TEXTsymbol{\backslash}frac\{\TEXTsymbol{\backslash}left( 2n\TEXTsymbol{%
\backslash}right) !\}\{4\symbol{94}n\TEXTsymbol{\backslash}left( n!%
\TEXTsymbol{\backslash}right) \symbol{94}2\}. \TEXTsymbol{\backslash}] Then 
\TEXTsymbol{\backslash}begin\{eqnarray*\} \TEXTsymbol{\backslash}lim%
\TEXTsymbol{\backslash}limits\_\{n\TEXTsymbol{\backslash}rightarrow 
\TEXTsymbol{\backslash}infty \}n\TEXTsymbol{\backslash}left( 1-\TEXTsymbol{%
\backslash}frac\{a\_\{n+1\}\}\{a\_n\}\TEXTsymbol{\backslash}right) \&=\&%
\TEXTsymbol{\backslash}lim\TEXTsymbol{\backslash}limits\_\{n\TEXTsymbol{%
\backslash}rightarrow \TEXTsymbol{\backslash}infty \}n\TEXTsymbol{\backslash}%
left( 1-\TEXTsymbol{\backslash}frac\{\TEXTsymbol{\backslash}frac\{%
\TEXTsymbol{\backslash}left( 2\TEXTsymbol{\backslash}left( n+1\TEXTsymbol{%
\backslash}right) \TEXTsymbol{\backslash}right) !\}\{4\symbol{94}\{%
\TEXTsymbol{\backslash}left( n+1\TEXTsymbol{\backslash}right) \}\TEXTsymbol{%
\backslash}left( \TEXTsymbol{\backslash}left( n+1\TEXTsymbol{\backslash}%
right) !\TEXTsymbol{\backslash}right) \symbol{94}2\}\}\{\TEXTsymbol{%
\backslash}frac\{\TEXTsymbol{\backslash}left( 2n\TEXTsymbol{\backslash}%
right) !\}\{4\symbol{94}n\TEXTsymbol{\backslash}left( n!\TEXTsymbol{%
\backslash}right) \symbol{94}2\}\}\TEXTsymbol{\backslash}right) \TEXTsymbol{%
\backslash}\TEXTsymbol{\backslash} \&\&\TEXTsymbol{\backslash}mathstrut}

\noindent \texttt{\ \TEXTsymbol{\backslash}\TEXTsymbol{\backslash} \&=\&%
\TEXTsymbol{\backslash}lim\TEXTsymbol{\backslash}limits\_\{n\TEXTsymbol{%
\backslash}rightarrow \TEXTsymbol{\backslash}infty \}\TEXTsymbol{\backslash}%
left( \TEXTsymbol{\backslash}frac n\{2\TEXTsymbol{\backslash}left( n+1%
\TEXTsymbol{\backslash}right) \}\TEXTsymbol{\backslash}right) =\TEXTsymbol{%
\backslash}frac 12. \TEXTsymbol{\backslash}end\{eqnarray*\} It therefore
follows from Raabe's test that \$\TEXTsymbol{\backslash}sum \TEXTsymbol{%
\backslash}frac\{\TEXTsymbol{\backslash}left( 2n\TEXTsymbol{\backslash}%
right) !\}\{\% 4\symbol{94}n\TEXTsymbol{\backslash}left( n!\TEXTsymbol{%
\backslash}right) \symbol{94}2\}\$ diverges.}

\texttt{\TEXTsymbol{\backslash}end\{document\}}

\texttt{\rule{1.47in}{0.01in}cut here \rule{1.47in}{0.01in}}

\texttt{I hope you approve of the way I wrote this. I tried hard. Yours
sincerely. Bertie.}

\bigskip\ 

After I receive Bertie's e-mail letter, all I have to do is copy his \emph{%
Scientific WorkPlace\/ }document to my clipboard and then paste it into a
file that I might call bwooster.tex. I can then open his document in \emph{%
Scientific WorkPlace,}\/ write my comments and then e-mail it back to him.

Thus, using \emph{Scientific WorkPlace,\/ }students can send and receive
mathematical documents very easily even if they are restricted to a text
only type of e-mail. Students who are unable to meet with the professor
personally can still submit questions, homework assignments, projects and
tests in the form of \emph{Scientific WorkPlace\/} documents. By using the
network or e-mail to answer many of the questions that must today be
answered during their office ours, professors are able to enhance the level
of contact they have with students and to make much more efficient use of
their office hours. Instead of simply waiting in their offices for students
who may or may not turn up, professors will be able to work at the computer
screen communicating with their students. Even a student who is out of town
at the time of a given lecture can receive a perfect set of lecture notes by
providing his/her e-mail address. This is the sense in which \emph{%
Scientific WorkPlace\/} can help us to set up an ``information
superhighway'' between student and teacher.

\subsection{Some Added Benefits}

One of the most difficult lessons to convey to students of mathematics is
that true learning does not take place unless the student has learned to
write mathematics in such a way that the sentences are complete, meaningful
and readable. All too often, college students are victims of the poor
teaching (often coupled with multiple choice testing) that they experienced
in their prior schooling. All too often, their work consists of a bunch of
meaningless symbols that are dotted aimlessly around the page together with
an ``answer'' that is gift wrapped in a big box.

In order for a mathematics course to be truly successful, we have to break
this habit. In my opinion, the most fundamental principle of mathematics
teaching is that the process of learning mathematics must be seen as being
identical to the process of learning to teach mathematics. Whatever
mathematics a student writes should be presented as if the student were
teaching it to whomever will be reading it. In becoming a good teacher, a
student is becoming a good teacher to him or her self.

I mention all this because I have noticed, among my own students, that the
moment they start writing their mathematics on the computer screen they
start taking pride in the appearance of their work and in the quality of
their mathematical writing. As their writing improves, some of their
lifelong bad habits begin to disappear and they become better problem
solvers. \textbf{I believe that the learning of mathematical word processing
should be part and parcel of the training that students receive in a
university or college mathematics curriculum.}

\subsection{Other Goals}

The availability of \emph{Scientific WorkPlace}\/ will encourage teachers to
write up their lecture notes and to make them available to their students.
The advantages of compiling such lecture notes are as follows:

\begin{itemize}
\item  Teachers who prepare polished lecture notes may give more organized
and polished lectures.

\item  Students who are ill or out of town will still have the course
materials at their fingertips.

\item  Students who lack note taking skills will still have a proper set of
lecture notes.

\item  Courses may come to depend less on expensive (and often worthless)
textbooks.

\item  Teachers will be encouraged to make their lecture notes available to
other teachers in their departments. In my own department, for example, we
have (in addition to the P: drive) a logical drive H: that is visible in
faculty offices but not visible at machines operated by students. We can
place lecture notes, examination papers and other material on this drive,
even when we don't want students to see it. Eventually, a bank of material
will begin to accumulate. I believe that this facility will enable the
members of a department to communicate more closely with each other. It can
serve as a basis for faculty discussion and it can be used to guide new
faculty and part time faculty.
\end{itemize}

\section{\emph{Scientific WorkPlace}\/ as A Word Processor}

In this section I shall provide a brief discussion of the way in which I
believe a mathematics course can benefit from the word processing features
of \emph{Scientific WorkPlace.}\/ In the next section I shall discuss the
capabilities of \emph{Scientific WorkPlace\/ }as a friendly front end for
Maple or Mathematica.

\subsection{Getting The Students Started}

Every one of my classes begins with a couple of sessions in the computer lab
where I instruct my students in the rudiments of \emph{Scientific WorkPlace.}%
\/ I make no attempt to teach them everything that \emph{Scientific WorkPlace%
}\/ can do. Instead, I teach them exactly what they need to know: how to
open a document, how to preview it or print it, how to write in a document
that has already been opened and how to perform mathematical operations
using Maple or Mathematica.

After these initial sessions, the students can use the lab computers at any
time when they are not being used by another class. I encourage those
students who have their own computers to acquire the student version of 
\emph{Scientific WorkPlace\/} if they can afford to do so. Many of my
students have done this.

To simplify their lives even more, I have placed a document called
labdoc.tex in the P: drive and instructed students to open this document and
save it with another name on a diskette. If they forget to save the file
onto a diskette they receive an error message because the P: drive is
read-only to them. This document labdoc.tex has been set up with
instructions that allow them to get going with their typing without having
to make any formatting or style decisions before they begin. If they decide
to print the document they have written it will be perfect. If they e-mail
the \emph{Scientific WorkPlace\/ }document to me then I can open it and read
it on my own machine.

If I place a homework assignment on the P: drive for my students then all
they have to do is open the \emph{Scientific WorkPlace\/} document insert
their solutions directly into it. If they decide to print their work, the
printed copy will be automatically hyphenated, paginated and formatted and
will have the appearance of a publication quality document. Alternatively
they can e-mail back the \emph{Scientific WorkPlace\/} document for me to
read on screen; in which case, I can e-mail it back with corrections and
comments. Note that even complicated documents that contain binary pictures
can be sent over e-mail, even if the e-mail system is primitive and allows
only text files. The reason for this is that \emph{Scientific WorkPlace\/}
can store documents in a ``wrapped'' form in which the entire document,
pictures and all, is a plain text file

At worst, if network drives and e-mail are not available, then the teacher
can make the file available on diskette and the students can return the
diskette with their completed homework. This is still vastly superior to
handing in homework that has been written with pencil and paper.

I have found that even those of my students who are not particularly
computer or keyboard literate can become comfortable with the use of \emph{%
Scientific WorkPlace.\/} After a little practice with \emph{Scientific
WorkPlace,}\/ they too are able to write complicated mathematical
expressions effortlessly the computer screen.

\subsection{Typing up A \emph{Scientific WorkPlace\/ }Document}

Although it is not the primary purpose of this article to extol the virtues
of \emph{Scientific WorkPlace,\/ }I need to say something about the way in
which the process of typing in \emph{Scientific WorkPlace\/ }differs from
the process of typing in other word processing systems. This difference
plays a very important role in the ability of students to write up their
work quickly and effortlessly in \emph{Scientific WorkPlace.}

\emph{Scientific WorkPlace\/ }operates very differently from other word
processors. Unlike all of the standard word processors (such as Describe,
Wordperfect, Wordpro and Microsoft Word) that are designed as ``wysiwyg''
(what you see is what you get) products, \emph{Scientific WorkPlace\/ }is
based on an entirely different philosophy that is called\label%
{logical-design} \textbf{logical design.}\footnote{%
I find it interesting that the developers of \emph{Scientific WorkPlace\/ }%
(TCI Software Research) were also the developers, a decade ago, of the $T^3$%
{}scientific word processing system which was one of the very first wysiwyg
word processing systems and which won an \emph{Editor's Choice}\/ award from
PC Magazine in 1986 for its excellence as a technical word processor. TCI
was one of the first software developers to embrace wysiwyg and now they are
the first to abandon it for something better.}

The idea of logical design is that the author of a document need not also be
responsible for making decisions about the way the document will look when
it is printed. In other words, the author of a document should be just that:
an author - not a secretary. The process of writing up a \emph{Scientific
WorkPlace\/ }document is akin to the process or writing up a pile of
handwritten notes that are to be handed to a secretary for typing. The
secretary will shoulder the duty of producing a manuscript that is fit for
publication. Since \emph{Scientific WorkPlace\/ }takes over the role of
secretary, the author need not bother with such traditional nit picking
issues as margins, tabs, fonts, spacing between sections. hyphenation of
words, pagination of the document etc. Thus the author can focus only on
what is to be written, which makes the process of preparing a document
considerably simpler in \emph{Scientific WorkPlace\/ }than in other systems.

Moreover, the secretarial functions of \emph{Scientific WorkPlace\/ }go far
beyond the duties of a human secretary. \emph{Scientific WorkPlace\/ }%
provides automatic numbering of chapters, sections, theorems, exercises,
mathematical displays etc. It provides automatic cross referencing,
automatic table of contents and automatic index. If changes, additions or
deletions are made in a document, all of these items are updated
automatically.

Even more exciting features are in the pipeline. For example, the next
version of \emph{Scientific WorkPlace\/ }will allow insertion of hypertext
links and will include an on-screen table of contents that will enable the
user to jump to any particular section by a simple mouse click. It will also
provide a history feature that will allow the user to jump back to the place
of origin. Because of these features, the next version of \emph{Scientific
WorkPlace\/ }will be particularly well suited for the production of
documents that are meant to be read on the computer screen.

\subsection{Workshops for Faculty}

In order for the students to benefit properly from a knowledge of \emph{%
Scientific WorkPlace,\/ }it is important that as many as possible of their
courses make use of it. Therefore, it is necessary to convince as many as
possible of the professors in a department to take the trouble to become
competent \emph{Scientific WorkPlace\/ }users. In my own department we
conduct frequent workshops in order to train our faculty to become efficient
users of the system.

As the system becomes accepted as the principal word processor in a given
department, members of the department should start exchanging lecture notes,
homework assignments, test papers and other material that has been written
in \emph{Scientific WorkPlace\/ }documents. If the department computers are
linked by a network then it would be a good idea to provide a special place
in the network for this shared material.

\section{The \emph{Scientific WorkPlace\/ }Front End for Maple And
Mathematica}

Apart from its role as a communications medium between student and
instructor, \emph{Scientific WorkPlace\/ }also serves as a front end for the
symbolic manipulators Maple and Mathematica, and this feature of the
software gives it a second important role in the teaching of many modern
mathematics courses.

In this short article, I can't even begin to describe all of the Maple and
Mathematica operations that are supported by the \emph{Scientific WorkPlace.}%
\/ I shall demonstrate just a few of them in order to whet the appetite for
more and to exhibit their intuitive simplicity.\emph{\ }This simplicity is
the key difference between symbolic manipulations in \emph{Scientific
WorkPlace\/} and symbolic manipulations in any other software product that I
have seen. In order to use either Maple or Mathematica directly, or any of
the other symbolic manipulators, it is necessary to type an instruction in
the native language of the product. Personally, I find the process of
learning these native languages to be quite painful and I find the process
of remembering what I have learned to be almost as painful. Furthermore,
even when the language of a symbolic manipulator has been learned, the
process of writing commands in this language is slow and tedious and prone
to errors. The process of working this way with my students does not appeal
to me.

In sharp contrast, all of the symbolic manipulations that are performed by
Maple or Mathematica in \emph{Scientific WorkPlace\/ }can be obtained by a
couple of simple mouse clicks that are applied to mathematical expressions
that appear on the screen in exactly the same form in which we would have
written them down on paper. Anyone who knows how to write the mathematical
expressions with pencil and paper automatically knows how to perform
operations on the expressions with Maple or Mathematica in \emph{Scientific
WorkPlace.}\/ It takes me only a few minutes to make my students feel like
expert users. To display this contrast I have included with some of the
following examples a brief description of my attempts to solve the same
problems working directly in the front end that is supplied with Mathematica%
\footnote{%
Although I have ragged Mathematica a little in this section, I must
emphasize that Mathematica is a first class high end symbolic manipulator.
There are many important types of application for which Mathematica is a 
\emph{sine qua non.}\/ No doubt, my failure to produce results working
directly in Mathematica is the result of my own incompetence. The point is,
however, that though I am equally incompetent when I communicate with Maple
or Mathematica inside of \emph{Scientific WorkPlace,\/} I can, nevertheless,
perform any operation I like that way.} Version 2.2.3 for Windows.

\subsection{Matrix Operations}

Typing a matrix is easy. In order to type the matrix 
\[
\begin{array}{rrrrr}
1 & 1 & 3 & -2 & 2 \\ 
0 & 1 & 4 & 2 & 1 \\ 
2 & 3 & 1 & 1 & -3 \\ 
3 & 1 & 1 & -1 & -1 \\ 
2 & 2 & 1 & 6 & -2
\end{array}
, 
\]
all one has to do is click on the matrix button to obtain an empty grid,
select the size and then fill it in, moving around with the mouse or the
arrow keys. To type this matrix in any of the standard symbolic manipulator
front ends one has to type something like

\[
\left[ \left[ 1,2,3,-2,2\right] ,\left[ 0,1,4,2,1\right] ,\left[
2,3,1,1,-3\right] ,\left[ 3,1,1,-1,-1\right] ,\left[ 2,2,1,6,-2\right]
\right] . 
\]
In order to invert this matrix using any of the standard manipulators one
would now need to type some inversion command that I've forgotten. However,
in \emph{Scientific WorkPlace\/ }I can ask Maple or Mathematica to invert
this matrix, simply by raising it to the power of $-1$ yielding 
\[
\left[ 
\begin{array}{rrrrr}
1 & 1 & 3 & -2 & 2 \\ 
0 & 1 & 4 & 2 & 1 \\ 
2 & 3 & 1 & 1 & -3 \\ 
3 & 1 & 1 & -1 & -1 \\ 
2 & 2 & 1 & 6 & -2
\end{array}
\right] ^{-1}, 
\]
opening the Maple or Mathematica menu and clicking on \textsf{simplify, }%
yielding 
\[
\left[ 
\begin{array}{rrrrr}
\frac 8{83} & -\frac{12}{83} & -\frac{17}{83} & \frac{27}{83} & \frac{14}{83}
\\ 
\frac{163}{249} & -\frac{40}{83} & \frac{79}{249} & -\frac{145}{249} & \frac{%
19}{83} \\ 
-\frac{80}{249} & \frac{40}{83} & \frac 4{249} & \frac{62}{249} & -\frac{19}{%
83} \\ 
\frac 2{249} & -\frac 1{83} & -\frac{25}{249} & -\frac{14}{249} & \frac{15}{%
83} \\ 
\frac{51}{83} & -\frac{35}{83} & -\frac{15}{83} & -\frac{25}{83} & \frac{27}{%
83}
\end{array}
\right] 
\]
or clicking on \textsf{evaluate numerically, }yielding 
\[
\left[ 
\begin{array}{ccccc}
9.6386\times 10^{-2} & -.14458 & -.20482 & .3253 & .16867 \\ 
.65462 & -.48193 & .31727 & -.58233 & .22892 \\ 
-.32129 & .48193 & 1.6064\times 10^{-2} & .249 & -.22892 \\ 
8.0321\times 10^{-3} & -1.2048\times 10^{-2} & -.1004 & -5.6225\times 10^{-2}
& .18072 \\ 
.61446 & -.42169 & -.18072 & -.3012 & .3253
\end{array}
\right] . 
\]

\subsection{Some Simple Evaluations}

To evaluate the limit

\[
\lim_{n\rightarrow \infty }\frac{\left( n!\right) ^2e^{2n}}{n^{2n+1}}. 
\]
in \emph{Scientific WorkPlace},\/ one simply has to click on the Maple or
Mathematica menu and then click on the option \textsf{evaluate.} This will
produce the answer: 
\[
\lim_{n\rightarrow \infty }\frac{\left( n!\right) ^2e^{2n}}{n^{2n+1}}=2\pi . 
\]
I tried several times to evaluate this limit using Mathematica directly but
I never did succeed. For my fifth try I typed 
\[
\begin{tabular}{l}
Limit[(((Gamma[n+1])\symbol{94}2)*(Exp[2*n]))/n\symbol{94}(2*n+1),n-%
\TEXTsymbol{>}Infinity]
\end{tabular}
\]
\newline
which yielded several complaints about essential singularities. Then I gave
up. Presumably there is a way to ask Mathematica to find this limit but it
is beyond me and I dread the thought of teaching it to students.

Some other expressions that I evaluated in \emph{Scientific WorkPlace\/} by
clicking on \textsf{simplify }are the following: 
\[
\lim_{x\rightarrow 0}\frac{e-ex/2-\left( 1+x\right) ^{1/x}}{x^2}=-\frac{11}{%
24}e, 
\]
\[
\sum_{n=1}^\infty \frac 1{n^4}=\frac 1{90}\pi ^4, 
\]
\begin{eqnarray*}
\frac{d^3}{dx^3}x^4e^x\cos 5x &=&\allowbreak 24xe^x\cos 5x+36x^2e^x\cos
5x-180x^2e^x\sin 5x \\
&&-\allowbreak 288x^3e^x\cos 5x-120x^3e^x\sin 5x \\
&&-74x^4e^x\cos 5x+\allowbreak 110x^4e^x\sin 5x
\end{eqnarray*}

\subsection{Evaluating An Integral}

To evaluate a simple integral such as 
\[
\int_0^1\sqrt{1-x^2}dx 
\]
in \emph{Scientific WorkPlace,\/ }all one has to do is type the expression,
place the cursor there, click on Maple or Mathematica and \textsf{evaluate.}
This yields 
\[
\int_0^1\sqrt{1-x^2}dx=\frac 14\pi . 
\]
I also asked Mathematica to find this integral directly. After a couple of
tries I finally typed 
\[
\begin{tabular}{l}
Integrate[Sqrt[1-x\symbol{94}2],x,0,1]
\end{tabular}
\]
and I received output that looked something like 
\[
\begin{tabular}{l}
Pi \\ 
--- \\ 
4
\end{tabular}
. 
\]

I returned to \emph{Scientific WorkPlace\/} and typed the harder integral

\[
\int_0^1\sqrt[3]{1-x^2}dx. 
\]
A click on \textsf{Evaluate }gave 
\[
\int_0^1\sqrt[3]{1-x^2}dx=\frac 2{15}\left( \sqrt{\pi }\right) ^3\frac{\sqrt{%
3}}{\Gamma \left( \frac 23\right) \Gamma \left( \frac 56\right) }. 
\]
(I decided that I didn't want to see the latter expression working directly
in Mathematica.) Clicking on \textsf{Evaluate Numerically }yielded 
\[
\int_0^1\sqrt[3]{1-x^2}dx=.84130926319527255670501144743. 
\]

\subsection{Solving An Ordinary Differential Equation}

I am no expert on differential equations but I thought I should show one
anyway. So I began with the equation 
\begin{equation}
x^2y^{\prime \prime }-xy^{\prime }+y=0.  \label{euler}
\end{equation}
When I placed the cursor in this equation clicked on the Maple option 
\textsf{Solve ODE, }I obtained,

\begin{center}
Exact solution is : $y\left( x\right) =C_1x+C_2x\ln x.$
\end{center}

Since Equation (\ref{euler}) is a simple Euler equation I decided to spice
it up a little. So I tried a bunch of variations. For each of these I
clicked on the \textsf{Solve ODE} option in \emph{Maple.}\/ The results
included Bessel functions, hypergeometric functions and various other
special functions that I have never heard of. Then, quite by accident I
tried the equation 
\begin{equation}
\left( 1-x^2\right) y^{\prime \prime }-xy^{\prime }+y=0  \label{not-special}
\end{equation}
which yielded

\begin{center}
Exact solution is : $y\left( x\right) =C_1x+C_2\sqrt{\left( -1+x^2\right) }.$
\end{center}

I don't know what it is about Equation (\ref{not-special}) that
distinguishes it from equations like 
\[
\left( 1-x^2\right) y^{\prime \prime }+xy^{\prime }+y=0 
\]
whose solutions involve special functions. But this experimentation with 
\emph{Scientific WorkPlace\/} taught me something that I could never have
discovered on my own. Finally, I tried some nonhomogeneous variations of
Equation (\ref{not-special}). Among these, the equation 
\[
\left( 1-x^2\right) y^{\prime \prime }-xy^{\prime }+y=x 
\]
gave me 
\begin{eqnarray*}
y\left( x\right) &=&-\frac 12\frac{-x\sqrt{\left( -1+x^2\right) }-\ln \left(
x+\sqrt{\left( -1+x^2\right) }\right) +\left( \ln \left( x+\sqrt{\left(
-1+x^2\right) }\right) \right) x^2}{\sqrt{\left( -1+x^2\right) }} \\
&&+\allowbreak C_1x+C_2\sqrt{\left( -1+x^2\right) }
\end{eqnarray*}
and the equation 
\[
\left( 1-x^2\right) y^{\prime \prime }-xy^{\prime }+y=x^2 
\]
gave me 
\[
y\left( x\right) =\frac 23-\frac 13x^2+C_1x+C_2\sqrt{\left( -1+x^2\right) }. 
\]
I think that the simplicity of \emph{Scientific WorkPlace\/} makes it an
ideal tool for experimentation. I would have liked to try these equations
directly with Mathematica as well but I simply couldn't see how to make
Mathematica recognize the symbol $y^{\prime \prime }.$ I got as far as
solving the equation $y^{\prime }=3.$ For this purpose I had to type 
\[
\begin{tabular}{l}
DSolve[y'[x]=\thinspace =3,y[x],x]
\end{tabular}
\]
which yielded the output 
\[
\begin{tabular}{l}
\{\{y[x]-\TEXTsymbol{>}3x+C[1]\}\}
\end{tabular}
\]
\emph{Scientific WorkPlace}\/ gives the solution as $y\left( x\right)
=3x+C_1 $ which I kinda prefer.

\subsection{Using The Maple or Mathematica Front End as A Writing Tool}

The combination of the powerful word processing features of \textsl{%
Scientific WorkPlace}\/ and the way it works as a front end to Maple and
Mathematica allows us to write many kinds of documents very rapidly and
simply; much more rapidly than one could possibly write the document with
pencil and paper.

Let us suppose, for example, that we wish to test the series 
\[
\sum \frac{27^n\left( n!\right) ^3}{\left( 3n\right) !} 
\]
for convergence. Typically, one would begin with the words: 
\begin{equation}
\text{\emph{Write} }a_n=\frac{27^n\left( n!\right) ^3}{\left( 3n\right) !}
\label{first-display-series}
\end{equation}
\emph{for each positive integer} $n.$ In order to write Equation, \ref
{first-display-series} there is no need to retype the expression $\frac{%
27^n\left( n!\right) ^3}{\left( 3n\right) !}.$ Since this expression
appeared in the previous line, it can be reproduced at once with a drag and
drop mouse action. The next step in testing this series is to say that for
each $n,$%
\begin{eqnarray}
\frac{a_{n+1}}{a_n} &=&\frac{\frac{27^{\left( n+1\right) }\left( \left(
n+1\right) !\right) ^3}{\left( 3\left( n+1\right) \right) !}}{\frac{%
27^n\left( n!\right) ^3}{\left( 3n\right) !}}=9\frac{\left( n+1\right) ^2}{%
\left( 3n+2\right) \left( 3n+1\right) }  \label{second-display} \\
&&\mathstrut  \nonumber \\
&=&\frac{\left( 3n+3\right) \left( 3n+3\right) }{\left( 3n+2\right) \left(
3n+1\right) }>1.  \nonumber
\end{eqnarray}
and to conclude from the fact that the sequence $\left( a_n\right) $ is
increasing that $\sum a_n$ diverges.

I shall describe now how I typed the material in the display \ref
{second-display}: I began by typing 
\[
\frac{a_{n+1}}{a_n}= 
\]
and, on the right of the $=$ sign, I created a fraction object. Then I used
a mouse drag and drop to paste $\frac{27^n\left( n!\right) ^3}{\left(
3n\right) !}$ both above and below the fraction bar yielding 
\[
\frac{a_{n+1}}{a_n}=\frac{\frac{27^n\left( n!\right) ^3}{\left( 3n\right) !}%
}{\frac{27^n\left( n!\right) ^3}{\left( 3n\right) !}}. 
\]
Next, I highlighted the top and used search-replace to replace $n$ by $%
\left( n+1\right) $ yielding 
\[
\frac{a_{n+1}}{a_n}=\frac{\frac{27^{\left( n+1\right) }\left( \left(
n+1\right) !\right) ^3}{\left( 3\left( n+1\right) \right) !}}{\frac{%
27^n\left( n!\right) ^3}{\left( 3n\right) !}}. 
\]
Then I typed another $=$ sign, and use a mouse drag and drop to produce 
\[
\frac{a_{n+1}}{a_n}=\frac{\frac{27^{\left( n+1\right) }\left( \left(
n+1\right) !\right) ^3}{\left( 3\left( n+1\right) \right) !}}{\frac{%
27^n\left( n!\right) ^3}{\left( 3n\right) !}}=\frac{\frac{27^{\left(
n+1\right) }\left( \left( n+1\right) !\right) ^3}{\left( 3\left( n+1\right)
\right) !}}{\frac{27^n\left( n!\right) ^3}{\left( 3n\right) !}}. 
\]
Finally I highlighted the second of the expressions 
\[
\frac{\frac{27^{\left( n+1\right) }\left( \left( n+1\right) !\right) ^3}{%
\left( 3\left( n+1\right) \right) !}}{\frac{27^n\left( n!\right) ^3}{\left(
3n\right) !}} 
\]
and (holding down the control button), I clicked on \textsf{simplify},
yielding 
\[
\frac{a_{n+1}}{a_n}=\frac{\frac{27^{\left( n+1\right) }\left( \left(
n+1\right) !\right) ^3}{\left( 3\left( n+1\right) \right) !}}{\frac{%
27^n\left( n!\right) ^3}{\left( 3n\right) !}}=9\frac{\left( n+1\right) ^2}{%
\left( 3n+2\right) \left( 3n+1\right) }. 
\]

This entire process can be done in a flash.

\subsection{Using The Maple Define Utility}

I shall now show how to test the series 
\[
\sum \frac{27^n\left( n!\right) ^3}{\left( 3n\right) !} 
\]
for convergence even more rapidly using the Maple define utility. As before
we begin with the definition 
\[
a_n=\frac{27^n\left( n!\right) ^3}{\left( 3n\right) !} 
\]
but this time we inform Maple of the definition. We highlight the above
equation, open the Maple menu and click on \textsf{define}\emph{,} and then
on \textsf{New Definition.} We then type 
\[
\frac{a_{n+1}}{a_n}= 
\]
and copy the expression $\frac{a_{n+1}}{a_n}$ to obtain 
\[
\frac{a_{n+1}}{a_n}=\frac{a_{n+1}}{a_n}, 
\]
then highlight the right side and, holding down control, we click on \textsf{%
Simplify} and obtain 
\[
\frac{a_{n+1}}{a_n}=9\frac{\left( n+1\right) ^2}{\left( 3n+2\right) \left(
3n+1\right) }. 
\]

\subsection{Drawing A Graph}

As with other Maple operations, the drawing of graphs is simple and quick.
So is the act of rotating the graphs and zooming towards them or away from
them. The entire process is simple enough to encourage students to
experiment with a large number of examples, to make them feel like experts
and to provoke them to learn from their experience. I have included just a
few of the many kinds of graph that can be drawn.

To draw the graph 
\[
y=8x^3-6x-1 
\]
I highlighted the right side of the preceding equation, opened the Maple
menu and clicked on \textsf{Plot 2D, Rectangular. }This yielded \FRAME{dtbhFU%
}{2.9992in}{1.9994in}{0pt}{\Qcb{The graph $y=8x^3-6x-1$}}{\Qlb{first-cubic}}{%
Figure }{\special{language "Scientific Word";type "MAPLEPLOT";width
2.9992in;height 1.9994in;depth 0pt;display "USEDEF";plot_snapshots
TRUE;function \TEXUX{$8x^3-6x-1$};linecolor "black";linestyle
1;linethickness 2;pointstyle "point";xmin "-5";xmax "5";xviewmin
"-5";xviewmax "5";yviewmin "-1010";yviewmax "1009";rangeset"X";phi 45;theta
45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name
\TEXUX{$x$};valid_file "T";tempfilename
'C:/SWP25new/docs/DJHULO8S.wmf';tempfile-properties "XP";}} which,
after a bit of zooming, came out as \FRAME{dtbhFU}{2.9992in}{1.9994in}{0pt}{%
\Qcb{The graph $y=8x^3-6x-1$}}{\Qlb{zoomed-cubic}}{Figure }{%
\special{language "Scientific Word";type "MAPLEPLOT";width 2.9992in;height
1.9994in;depth 0pt;display "USEDEF";plot_snapshots TRUE;function
\TEXUX{$8x^3-6x-1$};linecolor "black";linestyle 1;linethickness 2;pointstyle
"point";xmin "-1.06853";xmax "1.283904";xviewmin "-1.069";xviewmax
"1.284";yviewmin "-4.6";yviewmax "8.484";rangeset"X";phi 45;theta
45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name
\TEXUX{$x$};valid_file "T";tempfilename
'C:/SWP25new/docs/DJHUM6PO.wmf';tempfile-properties "XP";}}

To draw the graph 
\[
x^3y-xy^2+x^2+y-4=0 
\]
I simply selected \textsf{Plot 2D Implicit }from the Maple menu and obtained 
\FRAME{dtbpFU}{2.9992in}{1.9995in}{0pt}{\Qcb{The graph $x^3y-xy^2+c^2+y-4=0.$%
}}{\Qlb{implicit-plot}}{Plot }{\special{language "Scientific Word";type
"MAPLEPLOT";width 2.9992in;height 1.9995in;depth 0pt;display
"USEDEF";plot_snapshots TRUE;function
\TEXUX{\EQN{6}{1}{}{}{\RD{\CELL{x^3y-xy^2+x^2+y-4=0}}{1}{}{}{}}};linecolor
"black";linestyle 1;linethickness 2;pointstyle "point";xmin "-5";xmax
"5";ymin "-5";ymax "5";xviewmin "-5.2";xviewmax "5.204";yviewmin
"-5.2";yviewmax "5.204";rangeset"XY";phi 45;theta 45;plottype
12;num-x-gridlines 25;num-y-gridlines 25;plotstyle "wireframe";axesstyle
"normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name
\TEXUX{$y$};valid_file "T";tempfilename
'C:/SWP25new/docs/DKDDNJCQ.wmf';tempfile-properties "XP";}}To draw
the graph 
\[
z=xy\sin x 
\]
\FRAME{dtbhFU}{2.9992in}{1.9994in}{0pt}{\Qcb{The graph $xy\sin x.$}}{\Qlb{%
graph}}{Plot }{\special{language "Scientific Word";type "MAPLEPLOT";width
2.9992in;height 1.9994in;depth 0pt;display "USEDEF";plot_snapshots
TRUE;function \TEXUX{$xy\sin x$};linecolor "black";linestyle 1;linethickness
1;pointstyle "point";xmin "-5.2";xmax "5.204";ymin "-5.2";ymax
"5.204";xviewmin "-5.2";xviewmax "5.204";yviewmin "-5.2";yviewmax
"5.204";zviewmin "-24.93";zviewmax "24.95";viewset"XYZ";phi 45;theta
45;plottype 5;num-x-gridlines 25;num-y-gridlines 25;plotstyle
"wireframe";axesstyle "none";plotshading "Z";xis \TEXUX{x};yis
\TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};valid_file
"T";tempfilename 'C:/SWP25new/docs/DJHUQFYB.wmf';tempfile-properties
"XP";}}To draw this surface I highlighted the right side, opened the Maple
menu and This graph is just one of a wide variety of possible graph types
that are supported by \emph{Scientific WorkPlace}.\/clicked on \textsf{Plot
3D, Rectangular.}

The possibilities are endless. A thickened form of the parametric graph 
\begin{eqnarray*}
x &=&-10\cos t-2\cos \left( 5t\right) +15\sin \left( 2t\right) \\
y &=&-15\cos \left( 2t\right) +10\sin t-2\sin \left( 5t\right) \\
z &=&10\cos \left( 3t\right)
\end{eqnarray*}
came out as \FRAME{dtbhFU}{3.2085in}{2.1395in}{0pt}{\Qcb{A Tube Plot}}{\Qlb{%
tube}}{Plot }{\special{language "Scientific Word";type "MAPLEPLOT";width
3.2085in;height 2.1395in;depth 0pt;display "PICT";maintain-aspect-ratio
TRUE;plot_snapshots TRUE;function \TEXUX{[ \left( -10\cos t-2\cos \left(
5t\right) +15\sin \left( 2t\right) ,-15\cos \left( 2t\right) +10\sin t-2\sin
\left( 5t\right) ,10\cos \left( 3t\right) \right) ]};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";radius
\TEXUX{$1$};xmin "0";xmax "6.3";xviewmin "-22.5";xviewmax "22.46";yviewmin
"-16.75";yviewmax "24.79";zviewmin "-11.42";zviewmax "11.42";rangeset"X";phi
45;theta 45;plottype 13;num-x-gridlines 50;num-y-gridlines 10;plotstyle
"patch";axesstyle "none";plotshading "ZGREYSCALE";xis \TEXUX{t};valid_file
"T";tempfilename 'C:/SWP25new/docs/DJHUQK71.wmf';tempfile-properties
"XP";}}Finally, the M\"{o}bius band 
\[
\left( x,y,z\right) =\left( \left( 1-t\sin \theta \right) \cos 2\theta
,\left( 1-t\sin \theta \right) \sin 2\theta ,t\cos \theta \right) 
\]
came out as \FRAME{dtbhFU}{2.9992in}{1.9994in}{0pt}{\Qcb{Mobius band}}{\Qlb{%
mobius}}{Plot }{\special{language "Scientific Word";type "MAPLEPLOT";width
2.9992in;height 1.9994in;depth 0pt;display "PICT";maintain-aspect-ratio
TRUE;plot_snapshots TRUE;function \TEXUX{[ \left[ \left( 1-t\sin \theta
\right) \cos 2\theta ,\left( 1-t\sin \theta \right) \sin 2\theta ,t\cos
\theta \right] ]};linecolor "black";linestyle 1;linethickness 1;pointstyle
"point";xmin "-0.3";xmax "0.3";ymin "0";ymax "3.14";xviewmin
"-1.327";xviewmax "1.047";yviewmin "-1.265";yviewmax "1.266";zviewmin
"-0.312";zviewmax "0.3122";phi 55;theta 0;plottype 5;constrained
TRUE;num-x-gridlines 20;num-y-gridlines 20;plotstyle "wireframe";axesstyle
"none";plotshading "ZGREYSCALE";lighting 4;xis \TEXUX{t};yis
\TEXUX{v627};var1name \TEXUX{$t$};var2name \TEXUX{$\theta $};valid_file
"T";tempfilename 'C:/SWP25new/docs/DJHUQX5C.wmf';tempfile-properties
"XP";}}

\section{In Conclusion}

The value of \emph{Scientific WorkPlace\/ }as a tool for the teaching of
mathematics is twofold:

\begin{itemize}
\item  \emph{Scientific WorkPlace's}\/ unique word processing capability
makes it ideally suited as a communications medium between teacher and
student.

\item  As a front end to Maple and Mathematica, \emph{Scientific WorkPlace\/ 
}provides simplicity and convenience that is not available with other
products. Students using \emph{Scientific WorkPlace\/ }can be taught rapidly
and simply to perform complicated symbolic operations.
\end{itemize}

I believe that if we take advantage of these unique features of \emph{%
Scientific WorkPlace\/} we can take a big step forward in our quest for
quality mathematics education.

\section{Some Ordinary Differential Equations}

\subsection{Example}

Solve the equation 
\[
x^2y^{\prime \prime }-xy^{\prime }-6y=x\cos \left( \log x\right) 
\]
, Exact solution is : $y\left( x\right) =-\frac 1{16}x^{1+j}-\frac
1{16}x^{1-j}+C_1x^{1+\sqrt{7}}+\allowbreak C_2x^{1-\sqrt{7}}$

\end{document}
