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\begin{document}
\paragraph{}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\title{A Guide to Writing in Mathematics Classes}
%\date{\today}
%\maketitle

\begin{center}
{\LARGE {\bf A Guide to Writing in Mathematics Classes}}
\end{center}
\vspace{1cm}

%{\bf Table of Contents}
%\begin{enumerate}
%     \item Why Should You Have To Write Papers In A Math Class?
%     \item How is Mathematical Writing Different?
%     \item Following the Checklist
%     \item Good Phrases to Use in Math Papers
%\end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Why Should You Have To Write Papers In A Math Class?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     For most of your life so far, the only kind of writing you've done in
math classes has been on homeworks and tests, and for most of your life
you've explained your work to people that know more mathematics than you do
(that is, to your teachers). But soon, this will change.

     Since you are in this calculus class, you know far more mathematics
than the average American has ever learned---indeed, you know more
mathematics than most college graduates remember. With each additional
mathematics course you take, you further distance yourself from the average
person on the street. You may feel like the mathematics you can do is
simple and obvious (doesn't everybody know what a function is?), but you
can be sure that other people find it bewildering. It becomes
increasingly important, therefore, that you can explain what you're doing
to others who might be interested: your parents, your boss, the media.

     Nor are mathematics and writing far-removed from one another.
Professional mathematicians spend most of their time writing: communicating
with colleagues, applying for grants, publishing papers, writing memos and
syllabi. Writing well is extremely important to mathematicians, since poor
writers have a hard time getting published, getting attention from the
Deans, and obtaining funding. It is ironic but true that most mathematicians
spend more time writing than they spend doing math.

     But most of all, one of the simplest reasons for writing in a math
class is that writing helps you to learn mathematics better. By explaining a
difficult concept to other people, you end up explaining it to yourself.

%------------------------------------------------------------------------
\begin{quote} 
\textit{Every year, we buy ten cases of paper at \$$35$
each; and every year we sell them for about \$$1$ million each. Writing
well is very important to us.}\\
--- Bill Browning, President of Applied Mathematics,
Inc. 
\end{quote}
%------------------------------------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{How is Mathematical Writing Different from What You've Done So 
Far?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     A good mathematical essay has a fairly standard format. We tend to
start solving a problem by first explaining what the problem is, often
trying to convince others that it's an interesting or worthwhile problem to
solve. On your homeworks, you've usually just said, ``9(a)'' and then
plunged ahead; but in your formal writing, you'll have to take much
greater pains.

     After stating the problem, we usually state the answer immediately.
Then we move on to showing how we arrived at this answer. Sometimes we even
state the answer right along with the problem. It's uncommon to read a
math paper in which the answer is left for the very end. Explaining the
solution and then the answer is usually reserved for cases where the
solution technique is even more interesting than the answer, or when the
writers want to leave the readers in suspense. But if the solution is messy
or boring, then it's typically best to hook the readers with the answer
before they get bogged down in details.

     It is important to explain as many of your mathematical derivations as
possible in English words and complete sentences.  For example:

          To solve for $x$ when $3x^2 - 21x + 30 = 0$,
          we used the quadratic formula, and found that either
	  $x =5$ or $x=2$:
\vspace{.2cm}
\[
x \ \ = \ \ \frac{21 \pm \sqrt{21^2-4 \cdot 3 \cdot 30}}{2 \cdot 3} \ \ 
= \ \ \frac{21 \pm \sqrt{441-360}}{6} \ \
= \ \ \frac{21 \pm 9}{6} \ \
= \ \ \frac{30}{6} \mbox{ or } \frac{12}{6} \ \
= \ \ 5 \mbox{ or } 2.
\]
\vspace{.2cm}
          Thus either $x = 5$ or $x = 2$.


     Math is difficult enough that the writing around it should be simple.
``Beautiful'' math papers are the ones that are the easiest to read: clear
explanations, uncluttered expositions on the page, well-organized
presentation. For that reason, mathematical writing is not a creative
endeavor the same way that, say, poetry is. You shouldn't be spending a
lot of time looking for the perfect word, but rather should be developing
the most clear exposition. Unlike students in humanities, mathematicians
don't have to worry about over-using ``trite'' phrases in mathematics. In
fact, Section~\ref{sec:phrases} contains a list of trite but useful
phrases that you may want to use in your mathematical papers.
     This guide, together with your checklist, should serve as a reference
while you write. If you can master these basic areas, your writing may not
be spectacular, but it should be clear and easy to read---which is the
goal of mathematical writing, after all.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Following the Checklist}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


When you turn in your writing assignment, you should use a paper clip to
attach the checklist to the front. You should use both the checklist and
this booklet as a guide while you write because you will be graded
directly on the criteria outlined on the checklist. What follows here is a
more detailed explanation of the criteria that will be used for grading
your papers.

%%%============================================================
\vspace{.5cm}
\noindent
{\bf 1. Clearly restate the problem to be solved.}
\vspace{.5cm}


Do not assume that the reader knows what you're talking about. You don't
have to restate every detail, but you should explain enough so that someone
who's never seen the assignment can read your paper and understand what's
going on without any further explanation from you. Outline the problem
carefully.

%%%============================================================
\vspace{.5cm}
\noindent
{\bf 2. State the answer in a complete sentence which stands on its own.}
\vspace{.5cm}


If you can avoid variables in your answer, do so.  If not, remind the
reader what they stand for. If your answer is at the end of the paper and
you've made any significant assumptions, restate them. Do not assume
that the reader has actually read every word and remembers it all (do you?).

%%%============================================================
\vspace{.5cm}
\noindent
{\bf 3. Clearly state the assumptions which underlie the formulas.}
\vspace{.5cm}


For example, what physical assumptions do you have to make? (No friction?
No air resistance? That something is lying on its side, or far away from
everything else?) Sometimes things are so straightforward that there are no
assumptions, but not often.

%%%============================================================
\vspace{.5cm}
\noindent
{\bf 4. Provide a paragraph which explains how the problem will be
approached.}
\vspace{.5cm}


It's not polite to plunge into mathematics without first warning your
reader. Carefully outline the steps you're going to take, giving some
explanation of why you're taking that approach. It's nice to refer back to
this paragraph once you're deep in the thick of your calculations.

%%%============================================================
\vspace{.5cm}
\noindent
{\bf 5. Clearly label diagrams, tables, graphs, or other visual
representations of the math (if these are indeed used).}
\vspace{.5cm}


     Even more than in literature, in math a picture is worth a thousand
words, especially if it's well labeled!
     Label all axes with words if you use a graph. Give diagrams a title
describing what they represent. It should be clear from the picture what
any variables in the diagram represent. Make everything as clear and
self-explanatory as possible.
     Warning: if you decide to draw pictures using a computer program, make
sure to save them in such a way that they can be easily edited.  You will
probably need to go back and change some aspect before completing the
project.


%%%============================================================
\vspace{.5cm}
\noindent
{\bf 6. Define all variables used.}

\begin{enumerate}
\item Even though you have labelled your diagram, you
should still explain in words what your variables represent.

\item If there's a quantity you use only a few times, see if you can
get away with not assigning it a variable. As examples:

\vspace{.5cm}
Example (1):
\noindent
\begin{center}
\vspace{-.5cm}
\begin{tabular}{|p{6in}|}
\hline
{\bf Good:}\newline
We see that the area of the triangle will be one-half of the product of
its height and base that is, the area of the triangle is $(1/2)\times
3\times 4 = 6$ square inches.
\\\hline\hline
{\bf Not as good:}\newline
We see that $A = {1\over 2}hb$, where $A$ stands for the area
of the triangle, $b$ stands for the base of the triangle, and $h$ stands
for
the height of the triangle, and so $A = (1/2)\times 3\times 4 = 6$
square inches.
\\\hline
\end{tabular}
\end{center}


\newpage
Example (2):
\noindent
\begin{center}
\vspace{-.5cm}
\begin{tabular}{|p{6in}|}
\hline
{\bf Good:}\newline
Elementary physics tells us that the velocity of a falling
body is proportional to the amount of time it has already
spent falling. Therefore, the longer it falls, the faster
it goes.
\\\hline\hline
{\bf Not as good:}\newline   
Elementary physics tells us that $vt = g(t-t_0)$,
where $vt$ is the velocity of the falling object at time $t$, $g$
is gravity, and $t$ is the time at which the object
is released. Therefore as $t$ increases, so does $vt$: i.e.,
as time increases, so does velocity.
\\\hline
\end{tabular}
\end{center}


I hope that you'll agree that the first example of each pair is much easier
to read.

\item The more specific you are, the better. State the units of 
measurement.
Try to use words like ``of'', ``from'', ``above''.  For example:

\noindent
\begin{center}
\vspace{-.5cm}
\begin{tabular}{|p{6in}|}
\hline
{\bf Good:}\newline
We get the equation $d = rt$, where $d$ is the distance from
Sam's car to her home (in miles), $r$ is the speed at which
she's traveling (measured in miles per hour), and $t$ is the
number of hours she's been on the road.
\\\hline\hline
{\bf Not as good:}\newline
We get the equation $d = rt$, where $d$ is the distance, $r$
is the rate, and $t$ is the time.
\\\hline
\end{tabular}
\end{center}

Avoid words like ``position'' (height above ground? sitting
down? political situation?) and ``time'' (5 o'clock? January? 3 minutes
since the experiment started?). Never mind that your instructor uses these
words freely; you can too when you pass this class.

\item Variables in text should be italicized to tell them apart from
regular letters.
\end{enumerate}

%%====================================================================
\vspace{.5cm}
\noindent
{\bf 7. Explain how each formula is derived, or where it can be found.}
\vspace{.5cm}


Don't pull formulas out of a hat, and don't use variables which you don't
define. Either derive the formula yourself in the paper, or explain exactly
where you found it (so other people can find it, too).
     Put important or long formulas on a line of their own, and then center
them; it makes them much easier to read:

\noindent
\begin{center}
\begin{tabular}{|p{6in}|}
\hline
{\bf Good:}\newline
The total number of infected cells in a honeycomb with n
layers is   
\[
                  1+2+ . . . +n =n (n +1)/2.
\]
Therefore, there are $100(101)/2 = 5,050$ infected cells in a
honeycomb with 100 layers.
\\\hline\hline
{\bf Not as good:}\newline
The total number of infected cells in a honeycomb with n
layers is $1+2+ . . . +n =n (n +1)/2$. Therefore, there are
$100(101)/2=5,050$ infected cells in a honeycomb with 100
layers.          
\\\hline
\end{tabular}
\end{center}

Most advanced word processors have an equation editor. If you don't have
an equation editor, you may try formulas with tabs and keyboard
characters, or you may wish to write the mathematics in by hand. All of
these are fine options.


%%=================================================================
\vspace{.5cm}
\noindent
{\bf 8. Give acknowledgment where it is due.}
\vspace{.5cm}


It's {\em extremely} important to acknowledge where your inspiration, your
proofreading, and your support came from. In particular, you should cite:
any book you look at, any computational or graphical software which helps
you understand or solve the problem, any person you talk to (your
instructor, or another projects instructor, in this case.)
The more specific you are, the better.

%%=================================================================
\vspace{.5cm}
\noindent
{\bf 9. Use correct spelling, grammar, and punctuation.}

\begin{enumerate}

\item Don't forget, spelling and grammar are just as important in
mathematics papers.  Please spell-check and proofread your work for
grammar mistakes. Better yet, ask a friend to read your
paper.

\item Don't get sloppy with punctuation in sentences which contain
mathematical formulas. Put a period at the end of the computation if
it ends a sentence; use a comma if appropriate. An example
follows.

\noindent
\begin{center}
\begin{tabular}{|p{6in}|}
\hline
If Tweek's caffeine level varies proportionally with
time, we see that
$$C_t = kt,$$
where $C$ is his caffeine level $t$ minutes after 7:35
a.m., and $k$ is a constant of proportionality. We can solve
to show that $k =202$, and therefore his caffeine level by 
11:02 ($t =207$) is
$$C_{207} = (202)(207) = 41,814.$$
In other words, he's mightily buzzed.
\\\hline
\end{tabular}
\end{center}

\vspace{.2cm}
\item Do not substitute mathematical symbols for English words ($=$ and
\# are especially common examples of this). The symbol ``$=$'' is used
only in mathematical formulas---not in sentences:

\noindent
\begin{center}
\begin{tabular}{|p{6in}|}
\hline
{\bf Good:}\newline
We let $V$ stand for the volume of a single mug and $n$
represent the number of mugs. Then the formula for the
total amount of root beer we can pour, $R$, is $R = nV$.
\\\hline\hline
{\bf Not as good:}\newline
We let $V =$ volume of a single mug and $n =$ the \# of mugs.
Then the formula for the total amount of root beer $R = nV$.
\\\hline\hline
{\bf Bad:}\newline
We let $V$ stand for the volume of the mug and $n$ represent   
the number of mugs. Then the formula for the total amount
of root beer we can pour, $R$, is $R$ is $nV$.
\\\hline
\end{tabular}
\end{center}


\vspace{.2cm}
\item Do, however, use equal signs when you state formulas or equations,
because mathematical sentences need subjects and verbs, too.

\noindent
\begin{center}
\begin{tabular}{|p{6in}|}
\hline
{\bf Good:}\newline
Then the formula for the total amount of root beer we can
pour is $R = nV$.
\\\hline\hline
{\bf Not as good:}\newline
Then the formula for the total amount of root beer we can
pour is $nV$.
\\\hline
\end{tabular}
\end{center}


\end{enumerate}

%%%============================================================
\vspace{.5cm}
\noindent
{\bf 10. Use correct mathematics.}
\vspace{.5cm}

This is self-explanatory.

%%%============================================================
\vspace{.5cm}
\noindent
{\bf 11. Completely answered the original question.}
\vspace{.5cm}

So is this.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Good Phrases to Use in Math Papers:}
\label{sec:phrases}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{it}
\begin{itemize}
  \item Therefore {\em (also:} so, hence, accordingly, thus, it follows
	that, we see that, from this we get, then {\em )}
  \item I am assuming that {\em (also:} assuming, where, M stands for; 
	{\em in more formal mathematics:} let, given, M represents {\em )}
  \item show {\em (also:} demonstrate, prove, explain why, find {\em )}
  \item This formula can be found on page 243 of Stewart's 
	Calculus: Concepts and Contexts, single variable, Second edition, 2001. 
  \item If you have any further questions, feel free to contact me or Sam
     	Smart, who helped me develop this formula for you.
  \item While I am very glad to help you this time, you should be advised
	that my usual consultation fee is \$$85$.
  \item (see the formula above ). {\em (also:} (see *), this tells us that
	$\ldots$ {\em )}
  \item if {\em (also:} whenever, provided that, when {\em )}
  \item notice that {\em (also:} note that, notice, recall {\em )}
  \item since {\em (also:} because {\em )}
\end{itemize}
\end{it}


\vspace{1cm}
{\Large {\bf Remember to please proofread the final draft before you
submit it. Please, please. No, really.}}

\vspace{1in}
\underline{\hspace{3in}}\\

This handout was adapted from materials developed by Harel Barzilai,
Annalisa Crannell, and ``Guidelines for Projects'' in Calculus: An Active
Approach with Projects and Appendix B of Student Research Projects in
Calculus.


\end{document}

