Third Edition

Bruce Reznick

University of Illinois

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*Copyright © Bruce
Reznick , 1985, 1994, 1999.*

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form, or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright holder. If you are reading this, you have his permission to download and print a personal copy. If you wish to distribute multiple copies, please contact the copyright holder. You normally will be granted such permission, provided that (i) you use the entire text, without editing and including this paragraph, (ii) the distribution is without charge, and (iii) you let the copyright holder know how you are using the text.

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*"The secret to education is respecting the pupil."*

Ralph Waldo Emerson

I am a professor of mathematics, not a professor of education, and I have not studied pedagogy in an organized fashion. This is a practical essay on teaching, anchored by a few underlying observations.

** Remember that your students are just like you. **You have
probably been a student for most of your sentient life, so you know
more about teaching than you may think. Try to recall what you didn't
like as a student and don't do it as a teacher. After your first
semester, you will find a new respect for your old teachers!
** Remember also that your students are not at all like
you**. You are interested enough in mathematics to pursue an
advanced degree; your first students are likely taking a glorified
high school course. As my mother said to me just before I taught
my first class: "Be kind to your students. Math is much easier for you than
it is for them." ** Your students are
not stupid**. Be patient with them. Try to remember how it felt
to take your least favorite required course and keep in mind that
many of your students will feel that way towards *your* course,
no matter how well you teach. Some will always look forward to a
math class in the way they look forward to a dental appointment.
** There is more in the world than this**. Mistakes
*will* happen.
Nobody's perfect, ever; let alone the first time. (It's the first
time you're teaching this course and it's also probably the first
time your students are taking it.) As a graduate student, you can
only allocate a limited amount of time and energy to your role as
a teacher, and you must use these resources wisely. **
Knowledge cannot be given, it must be taken**. Your students
have also allocated a certain amount of time and energy to your
class. Your job is to direct their efforts in the most fruitful
directions, and trick
them, if necessary, into doing the work that needs to be done. A
proficient practitioner of educational judo can get a class to
teach itself. ** You are not alone**. You do not need
Zorn's Lemma to see that every teacher has been a first-time
teacher. When in doubt, ask a more experienced colleague. Almost
any imaginable problem has occurred before. Recovery can be
made from almost any disaster, but you need to act responsibly.
For example, if you are unable to attend a scheduled class
meeting, you must get in
touch with someone in your department office, so the class can be
covered. Finally, everything you read here is subject to an
important local constraint: **In case of conflict, always
follow the specific instructions of your department.**

Teaching can be extremely rewarding, but that is not why you have
earned an assistantship. Your department has a lot of teaching to do.
** Your primary job as a teacher is to convey as much of the
course material as you can to as many students as
possible**. There is no obvious optimal method. Any reasonable
technique for increasing the transmission of knowledge (facts,
theorems, techniques, examples, algorithms, perspectives, wisdom) is, by
definition, a good teaching technique. Your own enjoyment of teaching
may be important to you, but is secondary to the task at hand. By
accepting a stipend, you have become a professional. In
time, you will become a pro.

A good teacher must be productively manipulative. Almost every
college student has the inherent ability to understand algebra,
trigonometry and even calculus. The barriers to learning are
psychological, environment and linguistic. **A good teacher will
persuade students that they can learn the material**. Motivation
to learn mathematics can be found everywhere, and even the most
recalcitrant students want to pass. I have found in a wide range of
courses and student populations that, if you take your students
seriously, they will respond in kind. **Present yourself as an
alpine guide to your class of climbers, rather than as part of the
mountain**. Create an atmosphere of common purpose rather than
a battleground of wills. Encourage your students. Make them feel
that, with time and effort, they *can* succeed. It's easier than
you may think. Most people want to please those around them and your
students have had a dozen years' experience pleasing their teachers
(at least to their faces). Use this to your advantage. **Take
your work personally**. Make it clear to your students that
*you* want them to learn, that it matters to *you* whether
they understand your presentations, finish the homework, pass the
exams. Make the class a unique, individualized experience, and not
just another soul-less exercise at the Big U. If you really care about
your students' outcome and are even moderately successful in
transmitting the material, then you will find teaching to be very
rewarding. Students remember and appreciate good teachers. One can
bask in the compliments of a satisfied customer for years to come.

Please meet now two caricatures of imaginary teaching assistants.
Morrie assumes that his
students are miniature versions of himself: brilliant (of course),
deeply committed to learning the foundations and fine points of
mathematics, and able to assimilate material like a vacuum cleaner.
He doesn't bother to prepare his lectures, because the course is so
trivial that one reading of the text is enough, and besides, his
students should see how a *mathematician* approaches an
unfamiliar situation. He cuts off stupid
questions in class that waste his time and is forever annoyed that
his stupid and lazy students don't make the minimal effort needed to
understand him. He writes a very easy test, but half the class
flunks. Morrie wonders how these idiots will ever find gainful
employment after school. He should wonder how he'll support himself
after they take away his assistantship. Lester, on the other hand,
can't imagine how the poor children will survive without his constant
attention. He has given out his *home* phone number and is
available for consultation at all times. He gives two review
sessions before each test and
encourages students to ask for a make-up test if they aren't ready or
have a bad day. He takes each student's imperfection on an exam as a
personal failure. In fact, he is afraid to give zero credit for
absent work, fearing its effect on his students' delicate
self-esteem. Unfortunately, Lester neglects his own studies and
will also be looking for another job next year. These are extreme,
but not unknown, cases. They come in various intensities and
genders and a little bit of them lives in every teacher, Morrie or
Les.

The rest of this essay is divided into four sections -- preparing,
lecturing, grading and copying.
Ignore any suggestions that strike you as obvious or ridiculous, or so
peculiar that you can't imagine yourself using them. You will soon develop
your own teaching style. To paraphrase Tolstoy,
**good teachers are good in their own way, but bad teachers are
alike: they are unable or unwilling to communicate with their
students. **

I happily acknowledge the suggestions of many friends and colleagues. In particular, Barry Cipra, Nancy Diamond, Dar-Veig Ho, Diem Kratzke, Tom Kratzke, Ray McEachin, Mark Meyerson, Beverly Michael, Larry Riddle and Korin Spongberg were helpful in preparing the first edition and sending me comments after it appeared. Judy Holdener critiqued the second edition and gave me valuable information about the role of teaching assistants in computer labs. Robin Sahner has probably read twenty drafts of both editions, and often saved me from blatant typos and subtle conceptual errors. Remaining mistakes are of course my full responsibility.

This guide was originally written for use in graduate student orientation at the University of Illinois. The generous response it elicited here encouraged me to seek a wider audience. The first edition was produced in 1985 through the auspices of John Martindale at Random House/Birkhauser for distribution as part of their textbook series, and was requested for use by more than sixty graduate programs. In 1993, George Duda at HarperCollins asked me to prepare a second edition and provided many useful suggestions of new topics. In 1997, the rights reverted to me. Tori Corkery has been instrumental in transforming this manuscript into html-ready form. I thank all of these fine people for their encouragement and cooperation.

*"Mere proof won't convince me."*

Caption to a cartoon by
James Thurber

When you teach mathematics, you have an enormous advantage over the
teachers in most other
subjects: you can make statements in complete confidence of their
truth. The value of a definite integral does not depend on political
or religious beliefs. Consequently, the mathematics teacher has a
special obligation to prepare classes carefully. **You must know
what you are talking about, and you must know what you don't
know.** It is far better to admit your ignorance to a class
than to feign knowledge. The old cliche that you never learn a subject
until you teach it happens to be a true cliche. After admitting "I
don't know" (which will happen), you should prepare a full explanation
for the next period. This special effort will show your students that
you are responsive to their needs and will work for them, and this, in
turn, will encourage them to work for you.

Your department will give you a syllabus that should tell you what to
cover in each lecture. **Take the syllabus seriously! ** In
their nervousness, most beginning teachers talk too fast, cover too
much and do not leave time for questions. Some examples must be left
for the students to work out on their own. It is advisable to
rehearse your presentations (at least in the first week) and write
down the material you will put on the board. **Work out all
calculations in detail and make sure you know the correct
definitions.** Be reasonable. There is no mathematical reason why
you can't have *x* as a constant and *c* as a variable in the
same problem, but you will confuse most people seeing calculus for the
first time, so it's a bad choice of notation.
If you are running a recitation section, be
prepared to present the solution to *any* homework problem, **using
the methods you expect from your students.** Emphasize the algebra of
computations and not the arithmetic. The biggest waste of class time
comes from fumbling through examples with errors, false starts, etc.
If you can't do the work, you make it easy for students to rationalize
*their* not doing the work. Students should be actively involved
in the class presentation. There is nothing unethical about asking
them for suggestions when you already know what you want to
do. Another old and true cliche is that mathematics is not a spectator
sport.

Arthur C. Clarke has written that a smoothly running advanced technology would be indistinguishable from magic. Formal mathematical proofs, stripped of tangible intuitive underpinnings fall into the same category. You should always prepare some sort of informal explanation for the mathematics in your presentations, even if, as in the case of trig identities, all you can say is "It turns out that...."

Morrie's rationalization for not preparing his lectures -- that it's
good for students to see how a *mathematician* approaches an
unfamiliar problem -- is commonly heard from new TAs and from
world-renowned mathematicians. It is self-serving rubbish from
anybody. There is no reason why you cannot explain your approach as
you present a prepared solution. Unless you are teaching "*Thought
styles of the smart and tenured*", your students are more
interested in competency in their subject than in understanding how
you tick. Most importantly, your behavior in the classroom presents
a model for your students: your hours together constitute a small
and limited intellectual utopian society. If you bluff and blunder
your way through the class, you are validating your student's
efforts to bluff and blunder *their* way through the class.
Such is not the path to knowledge. Never forget that your students'
lack of expertise in mathematics does not make them fools; they will
spot a slacker and act accordingly. It is very hard to regain the
respect of your students when you have lost it through preventable
incompetence.

Nevertheless, it is a waste of time to prepare *too* carefully.
Your presentations are for the moment, not for posterity. **Even
the most brilliant lecture will leave many students befuddled. Do not
take this personally**. Leave room in your preparation for
questions and unplanned tangents. Many people need time to
assimilate new material. If you are uncomfortable in English,
communication must take precedence over spontaneity, but you should
never ignore questions.

You should spend the first weeks of the term familiarizing yourself with the entire course. You must learn the particular notations of the text, even if they seem arbitrary or ungainly, since your students will be asking you questions in that language. Mathematics in a consistent set of notations is confusing enough. If you are running a recitation section, make sure you know what the lecturer has covered, especially where it diverges from the text. You should always have a rough idea of what you will be doing in the next few classes. Every topic lives in four tenses: "soon we shall consider...", "now we consider...", "now we understand..." and "you are responsible for...". If you announce a major topic in advance of covering it, you will enhance the sense of accomplishment upon its completion.

**There are many unexpected resources for the preparation of your
classes. Many texts have a teacher's edition**. There are
literally hundreds of textbooks in print for any course you are
likely to teach as a TA. These provide an excellent source for the
preparation of examples and test questions. Your department probably
has a collection of unused complimentary textbooks sent by
publishers. Shop around. Don't rely too heavily on a single
alternative; an alert student might catch on. There are also
relevant films and videotapes. These can break up the monotony of the
long semester and are a good way to reward the serious students who
show up in class the day before a holiday. Prowl the Web for
additional enrichment ideas.

*"No dark sarcasm in the classroom."*

Pink Floyd

So now that you have prepared for the class, you must face your students. According to pollsters, people are more afraid of public speaking than anything else. Two ways to reduce this fear are to be confident in your knowledge of the material and to create a congenial class atmosphere. Even if you shake visibly, remember that students are much more scared of a trembling teacher than vice versa. It does get easier with time.

Much of your job in the classroom will be specified for you by your
department, but there are a few virtually absolute rules. **Arrive
at your classroom as early as possible, and on time under all
circumstances**. The interval before the bell can be put to
productive use. You can individually return corrected work (and
thereby match names and faces) and you can also talk to your students
in a more relaxed way. If you establish yourself as a human being
before the bell, then it is easier for you to assume the role of
instructor once class starts. Promptness reinforces the notion that
the time between the bells is special: teacher and student must behave
differently than in ordinary life. Once the bell rings, you should
start immediately on organization material (test dates, assignments,
etc.) and old business. Review the work of the previous day and ask
for questions. **Do not cater to late arrivals by waiting for
their attention**. Time is precious. One minute a day, three
times a week for a semester, adds up to a full class period. You'll
need that period by the end of the term.

The same rules of courtesy apply to the end of the class. **Stop
on time or, on rare occasions, one or two minutes late**. By
the time the bell rings, the student has already stopped listening,
and will not give your presentation the attention it deserves. Avoid
the temptation to start a new topic late in the hour. You will almost
certainly have to repeat everything the next time. Do not be afraid
to stop lecturing a few minutes early, if you can think of no
constructive use of the time and there are no questions, but don't do
it often. **If possible, stay for a few minutes after class to
answer short questions**. This is an efficient alternative to a
short office visit for all concerned and reinforces the desired
impression that you care about your students.

The mechanics of lecturing are straightforward, but hard to implement
consistently. Write in a
large, distinct and legible hand, so people in the back can see. (My
former students should please stop
snickering now.) If your chalk squeaks, break it in two and write with the
broken edge. Stand
alternately on either side of your writing, à la Vanna White on
*Wheel of Fortune*, so that students on both sides of the
room can read your work, and pause long enough that it can be copied.
Depending on the
geometry of your classroom, the students in the back row may be unable
to see as much as the
bottom third of the board. Ask. **Speak clearly and slowly enough
to be understandable**. Try to vary your pitch and tone, so the
class will know when you're making an
important point and so they won't be lulled to sleep by your monotone.
Students with strong regional or international accents should be
especially assertive in confirming their comprehensibility.
Urge your students to let
you know if they can't see or hear you. Establish eye contact with
every student. (Harassment tip:
the eyes are *above* the neck, even on students dressed for the beach.)
Eye contact keeps the class
alert and provides you with honest and immediate feedback. No matter
how good you are, some
students will be thinking about their next or their last date; don't let one
distracted face throw you off stride.

Always ask for questions. **Never say that a question is
stupid**! Every student
who doesn't understand a point (and is brave enough to ask) represents
many others who are afraid
to raise their hand. **You may despair of your students' ignorance
but remember that your
job is to teach these people. ** A curt reaction which suggests
"You are an idiot, get out
of my life" will both reduce the number of questions and stifle the
learning process. Many people
are turned off to mathematics for life by just one such arrogant
teacher. Repeat questions to
make sure everyone has heard them. When you answer a question, be
patient and informative,
and maintain eye contact to see if you've gotten through. It is
proper to defer an answer to later
in the hour or week, if new material will address the issue. The
total absence of student
questions, despite your sincere requests, may be a sign that your
class is lost. When you get
bogged down (it will happen), call "time-out", move ahead in your
presentation and pick up the
topic in the next class. I do not like to interrogate class members.
This is college, not high
school, and if students dread coming to class, they will simply not
show up. Perhaps you are a more skillful interrogator than me.

Be sparing in use of words such as "clear", "trivial" and "easy" when referring to mathematical ideas. These provide little reinforcement to the good student and devastate the poor student. You will probably find yourself in greatest sympathy with your best students, since they are most similar to yourself. Do not skew the class solely to their interests. A "C" student pays the same tuition as an "A" student, and often will eventually contribute more to your department's development fund.

**Never be afraid to repeat yourself**. Nobody pays full
attention all the time.
Some ideas need to be heard a few times in order to sink in. Students
are always reassured when
they hear something they already understand. Never be afraid to
repeat yourself.

After a dozen or more years of schooling, the student establishes a direct link from eye to hand, bypassing the brain. You should never intentionally write something incorrect on the board, even when you say it is wrong. Someone will be distracted and copy it down. Label the source of examples when they come from the text. This encourages students to study from the text as well as the class notes.

New teachers often wonder how they can make their lectures more
interesting. The best way is
for you to **show an interest in the subject yourself!** Don't just copy
down the book's examples,
change the numbers around. If you can calculate quickly, ask students
to provide the values of
some less essential parameters in your examples. Look for
opportunities to pursue student
questions, especially if they are in the direction you already want to
go. If you have a particularly
lively discussion one day about one specific mathematical object, be
sure to return to that object
again whenever feasible.

Showmanship in education is only worthwhile when it increases
attendance or helps maintain
class interest. If you like to tell jokes, tell jokes. **Avoid
saying anything which might
possibly offend anyone, and never make one of your students the object
of humor**. This
is not political correctness; this is common courtesy. Ethnic and
sexual jokes are always
inappropriate. Remember also that you are now an authority figure, it
is inappropriate to smirk
about how dumb Mr. X's mistake was or how hard the next test will be.
**If you don't like
to tell jokes, please don't tell jokes**. Nobody expects a math
class to sound like a
stand-up routine. Take my derivative, please.

The diversity of courses taught by TAs makes it hard to give specific
advice about class content.
Follow your syllabus and use it as a guide to pacing. Ask for help
from your colleagues if you
are having trouble keeping up or filling the hour. Present solutions
in the straightforward way
you expect your students to find them. Emphasize the unity, and
reasonableness of the material,
and de-emphasize the tricks and shortcuts. (A technique is just a
trick you've seen before.) The
level of rigor in a text may distress you, but do not raise it
unilaterally. **Explanations are
always preferable to proofs**. The appreciation of proofs is a
sophisticated taste, and probably
not one of the principal goals of your course. The most important
part of a college education is
learning how to think independently. Most of the low-level
mathematics you are teaching is
valuable as a tool in other disciplines, rather than an end in
itself. Teach your students how to **read **
mathematics.

Many departments now teach some beginning mathematics courses in computer laboratories, in which the roles of the instructors vary tremendously. You will probably be more comfortable with the mathematical component of the course than the computer component. You must become proficient in both and incorporate them into a unified course. Avoid the common arrogance of the mathematician who feels that anything non-mathematical is trivial. Do not underestimate the effort needed to understand the relevant computer work. The level of informality in the lab may make it hard to enforce the bells; you should find useful activities to keep the better students occupied for the full period.

There are at least five distinct media through which a student learns: the textbook, the classroom, the homework, the exams and out-of-class work with other students. Each medium is best suited for a different sort of learning. The most brilliant lecture series makes for a superficial or incomplete textbook, and the best textbook, if read aloud, would be ponderously dull. The text is best as a reference, used to learn material which requires reflection and careful study, and for detailed calculation and lengthy and precise definitions. The best use of the lecture is for real-time interaction with students, answering questions and presenting examples which show how things "really" work. Handy hints and suggestive analogies which would be awkward in print are ideal for the lecture. Lectures are most effective when the students already have made some attempt to learn the material. I encourage my students to read the appropriate text sections both before and after the lecture. Some even do it.

Homework is essential for the assimilation of skills. However,
problems should not be due before
similar work has been done in class. Exams are not often considered a
tool for learning. **The
effort your class makes to study for your exams provides you an
opportunity to focus attention on
the crucial parts of the syllabus**. For this reason, I think it's
foolish to be coy about the material you plan to test. Out-of-class
interaction is largely beyond your control, but
it is often the way students learn the most. I distribute a voluntary
phone and e-mail list; some students will be understandably
reluctant to give out this information, and you should never force
them. At my school, I can arrange for
newsgroups specific to each class. I make these unmoderated, and
encourage class members to post.

I explicitly encourage students to work together on their homework, but this may violate the rules of your department. Check first! Assuming it is OK, I encourage the students to discuss the homework on the newsgroup before it is due. I also post any e-mail correspondence I have had with students, after anonymizing the source of the questions.

3. Grading

*
"Be courteous, impartial and firm, and so compel respect from all."*

General instructions to umpires, Official Baseball Rules (9.05).

In the ideal academic setting, there would be no testing and grading, just class discussions and the correction of assignments. Students would not need certification and would care enough about the content of their courses to make the stimulus of grades unnecessary. We are not in the ideal. Virtually every American university agrees on the need for some hour exams during the term and a final at the end. Writing, administering and grading these tests provide some of the most difficult and least rewarding aspects of the teaching profession.

You will probably be teaching one section of a multi-section
course. and will be expected to
provide a final course grade for each student. **It is essential that
your grading schemes be
consistent with those of your colleagues**. The components of a
final grade usually consist of homework and/or quizzes, hour-long
exams and a final. This section represents the current state of
evolution of my personal taste as an autonomous instructor and is far
from pedagogical dogma. As before, if what you read here contradicts
the instructions you have been given, *ignore this material*!

Many teachers use weekly quizzes as a substitute for graded daily homework. I think this is a bad idea, but I am in a distinct minority. I think that quizzes absorb class time and merely replicate the exams in miniature. They tend to distort the entire class period in which they are given. There are few situations in the "real world" in which mathematics must be applied quickly and without reflection and without recourse to reference books or knowledgeable colleagues. Homework assignments more accurately simulate the circumstances in which students will use their mathematical knowledge. To be sure, graded homework assignments are more time-consuming for instructors than quizzes.

I assign homework to be due most class periods for a lower-level class, and weekly for an upper-level class. Not all problems are graded, but the students know in advance which ones will be graded. (Ungraded problems are used as the basis for some exam questions.) Students get a chance to test their skills without a time constraint and are encouraged to work steadily throughout the semester. Daily homework gives me a rapid and accurate idea of what my students have learned. I grade on a fairly crude numerical scale of 3 or 4 points per assignment, which minimizes hair-splitting discussions of partial credit. I drop the lowest few homework scores in computing homework percentage for the course, which minimizes arguments about whether a student should be excused from a particular assignment due to outside activities.

When the homework is due, I distribute worked solutions to all problems assigned. This fills many needs at once: it "extends the hour" by limiting the amount of class time I need to spend on routine homework problems; the homework grader can save time by referring the erring student to the worked solutions; I can justifiably refuse to accept late homework for credit (the bane of every grader and a bad study habit); by the end of the semester, students have a supplement of several hundred worked problems; finally, the fresh preparation of homework problems keeps the instructor's attention on the material. (You'll know what I'm talking about after teaching the same class for the third or fourth time.) It is possible that your department limits the number of handouts you can give your class. Check! Instructors sometimes make homework solutions available at the local campus copy shop or post solutions outside their office. Neither of these has the same immediacy as the scheme given here.

Homework assignments ought to require at least an hour's work per
class for the average student,
beyond reading in the text and reviewing class notes. I usually give
homework on the fly, waiting
for the end of the class period to put the next assignment on the
board. This is not very popular,
but allows me to tailor the assignments to the actual material
covered. You should probably
stick to problems from the text until you are an experienced
instructor. I try to return homework
by the next class period, so both the students and instructor get
rapid feedback on what has been
mastered. **Students will work together**. Many do their most
productive learning in
groups. If you encourage this, maybe some weaker students will copy a
better student's
homework without fully understanding it. (I confess I did this in college
physics.) If you forbid
collaboration, many students will still work together and you will
have created a large group of
scofflaws. In my opinion, this hurts the social contract of the class
even more than the inequity
of a student getting unearned credit through copying. Your department
may forbid collaboration,
however.

Tests. You have done your best to explain the material and provide
illuminating examples and
have patiently collected the homework. **Some of your students will
inevitably disappoint you on their
test papers with egregious lapses of knowledge**. This is a fact of life.
Some students only care about their grade in your course, which
they regard as an annoying
obstacle in their academic path. **Students always study the material
on which they think they
will be tested**. You can use this leverage to great advantage by
testing the most important
parts of the course. Let your students know what you consider the
core knowledge, base the bulk
of the homework on it, and then write boring and straightforward exam
questions. **Deep
creative thought should not be necessary for the construction or
solution of an examination in a
basic mathematics course**. I do not like to see difficult
questions on marginal topics or
"trick" questions on exams. These send an unduly cynical message
about education. **You should not use your exams to prove to your
students that you are smarter than them.** They will concede the point.

Many students for their part become pre-law majors before an exam, looking for loopholes in the syllabus. It is both merciful and efficient to tell a class that you have no intention of testing them on a particularly tricky offshoot of the main discussion. Students should learn to hit the fastball before worrying about the curve. You want them to concentrate their limited studying time on the most important topics. On the other hand, you shouldn't give away the store. It is a good idea to distribute review questions before a test, especially if you have taught the course before. This minimizes the unfair advantage of students who have access to fraternity and sorority files of previous tests. I have given up on review sessions unless the class is in an unusually panicky mood before an exam. These perpetuate the false idea that passing the exams (rather than learning the material) is the main purpose of the course. One final thought, be merciful: an exam should be a dipstick into the crankcase of a student knowledge and not just a shaft.

**In writing a test, you should first identify the types of problems
you expect your students to
solve, and then write problems to fit. ** A good test should take a
good ("B") student about
forty-five minutes of a fifty-minute class period to complete, leaving
some time to check the
answers. **Always take the test yourself. ** Some problems take
longer to write out than
you expect. If your test takes *you* forty-five minutes, then it
is too long. You will want problems
with a range of difficulties, with an "A"-caliber question to challenge
the best students. However any
problem which is only solved by one or two students is necessarily a
bad problem for that
particular class. Not only will it frustrate most of the students, but
it will ruin the scores of those
whose poor work habits make them spend most of their time on it. Make
the first question
relatively easy, to reduce the frequency of choking. You should
always indicate the point value of each problem, and balance your
allocation of points over the
entire corpus to be tested. I think it is a good idea for instructors
of the same course to
collaborate in writing tests, especially if the exams will be given
simultaneously. It is prudent
however to assume that any two sections will have friends in common,
so you do not want to
duplicate the same problems without some variation in the numerical
parameters.

Multi-part questions should be used with extreme caution. A foolish
error on the first part of
such a question destroys the testing value of the rest of the
question, unless you are willing to give
full credit for a correct answer in (b), based on incorrect information
from (a). This takes a lot of
time to grade. Cute story questions are also tempting, especially to a
new instructor.
Unfortunately, student literacy is often so low that many students
cannot find the mathematical
problem embedded within. **Prepare your tests carefully. ** It is
no fun to deal with the
consequences of a botched question, and you are obligated not to make
the students suffer from your mistake. Even when you have written a fair
and careful exam, you
may be nervous before you administer it. This is natural. You may
also be made uncomfortable
by the accumulated stress in a room of test-takers. Keep in mind that
the reaction of a class
during and immediately after an exam may be a poor reflection of their
true performance. I have
seen students grimace and tug at their hair during an exam, leave in a
funk, and turn in a perfect
paper.

Your course will have overall grading philosophy. Here's mine. I am
generous in partial credit
towards those who know how to do a problem, but make silly errors or
miss relatively minor
points. Care in working is useful, but most exams force students to
work at a reckless pace and
most students will be using silicon-based helpers when they later
apply mathematics. **An exam
should be graded problem-by-problem to standardize partial
credit. ** Correct a few papers
first without assigning point values in order to get a sense of how
partial credit will be needed.
Make sure you grade the method, not just the answer. Don't write exam
questions in which the
correct answer is easy to guess. Mark clearly the first place where a
student errs. Avoid
correcting solutions in full detail; I ask my students to redo their
incorrect problems informally as
a tool for learning. **A successful exam is one which accurately
measures the abilities of the
students, not necessarily one which maps them neatly onto a set of
five letters. ** Consistency
in grading is essential when teaching a multi-section class; do not
make unilateral policy decisions.
**Students deserve privacy regarding their grades; return the exams
individually. ** You will
probably know some student better than others, and it is not
unreasonable to "root" for an
especially hard-working student. It is definitely not okay to grade
students differently on the same
work, based on your estimate of what they "really" knew. Finally, it is worth
remembering that ungraded
papers do not become easier to grade while marinating on your desk.
Get the grading done as
quickly as feasible. Your students really appreciate a rapid turnaround.

By the end of the term, both you and your students should have a pretty accurate expectation about the final. I like to give a two-hour exam in our three-hour slot. The final should cover the course material in a balanced way, including variations on most of the highlights of the course. It is reasonable to be less generous with partial credit for conceptual errors on final papers, since the material is being tested for a second time.

**Your grading scheme should be consistent and well-publicized from
the start. ** Your
conscience must be your guide, but you must also be consistent with
the standards established in
the course. I prefer grading on a numerical basis throughout,
assigning a letter grade only at the
end. (Students naturally want to know how they are doing; you can give
them a range of grades.)
Numerical grades are more sensitive to the information in the exam
scores and are commonly
used in large classes. In this case, a student with two low "A"'s and
one low "B" will probably have a
high "B" on average, but perceive a low "A" by majority rule. You should
make your students aware
of this possibility. I grade lower level classes as follows: the
homework counts 10%, each of the
three hour exams counts 20% and the final counts 40%. This adds up to
110%, so I drop the
10% with the lowest percentage. This cushions one bad test if the
student has done good
homework. I do not weigh class involvement in the course grade. There
are two escape hatches
in my scheme: any student who gets at least 96% on the final will get
an "A" and any student who
gets at least 75% on the final will pass the course. These make
students feel better, but in practice
rarely lead to a grade that would not otherwise be earned. I announce
that the "A/B", "B/C" splits
will be no higher than 90%, 80%, etc., but might be lowered, depending
on how the class does.
This gives students specific goals to aim for, while retaining the
promise of a curve. You
might be able to keep your grade records on a computer in
your department. I am perhaps
irrationally dubious of the security in such a set-up. Keep a
back-up, in any case.

An amazing statistical fact is that students' scores tend to clump
nicely into the various grades,
except for a few exasperating borderline cases. **The first time you
make up final course
grades, you should show them to at least one more senior instructor
for comments.** I still
stay up nights at the end of the semester wondering whether I've given
the correct grade. **Do
not feel guilty if you flunk a student.** Most students who flunk
have not put in the time and effort necessary to pass. The student
who is working hard and still flunking should be
encouraged, early in the term, to use one of the many support services
your department provides
or to engage a tutor. **It is never appropriate to tutor one of your own
students beyond the usual office hours, particularly for pay.**

**Do not be talked into changing a student's grade by a hard-luck
story.** You would be
surprised at
how many hard-luck stories are not quite ... accurate. At virtually
every university, one bad
grade in one course does not get a student in trouble. Your school
does not want to admit that its
Admissions Office goofed, and so has an elaborate apparatus for
handling students in hot water.
A crucial part of this apparatus is the objective evaluation of a
student's performance from a
professional instructor. That means you. If you feel that a course
grade does not represent the full
story of a student's work, give the grade earned and get in touch with
the appropriate dean.
**Do not be reluctant to pass such problem students on to a more
senior instructor or
administrator in your department.** Teaching provides many
opportunities for self-delusion.
A good grade or a bad grade on a poorly designed test may be close to
meaningless. Morrie and
Lester would undoubtedly have radically different opinions about the
same students. Maybe your
class is stupid, or maybe you gave them two problems too many on their
tests. When your
median score is 93%, let someone look over your tests and grading
policies. Maybe you're the
next Jaime Escalante and your students are brilliant, or maybe you are
not expecting enough of
them. Even experienced instructors stay up nights wondering about
these issues.

*"I was thrown out of NYU for cheating on my metaphysics final. The
professor caught me looking into the soul of the boy sitting next to
me"*

Woody Allen

This last section is a miscellany of canned advice for dealing with
unexpected difficult questions. I
offer no warranty for my suggestions, because individual cases vary so
much. **Again, always
ask more experienced instructors for advice!** Some of the most
useful information comes
from listening to the "war stories" swapped by your department's
grizzled veteran teachers.
Professors can help, too.

Cheating can be a profoundly troubling problem. **The truly
dedicated cheater is probably
close to undetectable.** There are students who put more effort
into cheating than they would
need to pass the course honestly. The best vaccine is an atmosphere
of common purpose in which
cheating is perceived as antisocial and destructive -- an offense
against the majority of honest
students and an assault on the integrity of the educational system.
Many cheaters justify their
actions as a response to the indifference of their instructors. Never
give them this excuse. Peer
pressure can be an effective preventive. Remind your students not to
cheat in a friendly
"I-know-you-wouldn't-do-this-to-me" way. **Proctor exams with
care.** Move around the
classroom, including the back if possible, so prospective cheaters
have no protected angles. Look
at the directions of your students' gazes, but do not be too zealous.
Many honest students look
straight ahead when they think. If you are suspicious of a student,
make eye contact until it is
acknowledged. This is usually enough. Minimize the opportunities for
out-of-class cheating by
circumspect treatment of test materials. Write the test at home,
after the last class meeting,
if possible, and personally operate the copier which produces the exams.

When grading a test, be suspicious of non-sequiturs in reasoning, or oddly missing steps in the calculation. Be suspicious also of friends who sit together at an exam and make the same errors. (However, this behavior may have an innocent explanation: students who study together often share misconceptions.) Cheating is a serious charge to make, but you are well within your rights to ask that friends not sit together at the next test, especially if you do it tactfully. If you unambiguously catch a cheater, let someone in authority know immediately. Most universities have rather complicated regulations on cheating. These may strike you as lax. I announce that "nobody has ever cheated in one of my classes . . . twice", with a deliberately ambiguous smile.

Students often complain about the grading of their work. They may point out something the grader missed or claim that they deserve more credit. In either case, they deserve a serious hearing. If you see no merit in their case, tell them so. Otherwise, take the work back and review the grading on your own time, away from their scrutiny. Be reluctant to change your mind on a judgment call. Even as a new teacher, you are a professional with a richer perspective on the course material than your students. But grading mistakes must be acknowledged and corrected, on all relevant papers if warranted. Again, you may want to consult a more experienced instructor for advice. Some unscrupulous students have been known to amend their papers after they have been graded and returned, and demand more credit. This is hard to prove. You should always make a mark on a blank examination page, to discourage after-hours inspiration. You can also photocopy an exam before returning it in hopes of snaring the repeat offender.

Teachers at the University of California are required by state law to
post their office hours. It
seems that an Assemblyman's son could never find his professor. In
any case, this is an excellent
idea. Most departments require their instructors to reserve two or
three hours per week for
answering questions, meeting absent students. etc. Most students are
comfortable with email, and I've found that if you can respond
quickly, you can handle much of the demand for office hours. You should also
make yourself available for other appointments, but setting fixed hours is a
good time-management
technique. Do not expect a lot
of activity until just before a test. Office hours are a good time to
prepare for your classes and
grade homework, and they provide a safety net for students, but they
are an inefficient teaching
medium.
**If two people ask you the same question outside of class,
you are well-advised to
cover it in class.**

Many mathematicians are naturally shy people who find it hard to
maintain the self-image of
authority in the classroom. **Always remember that you have the
superior knowledge and the
power to assign grades.** When you see a student learn as the
result of your efforts, your own
self-confidence as a teacher (and in life) will grow. Mine did. This
brings me to the issue of
unruly students. It is hard to give general advice; books have been
written on the subject. I start
with polite requests and try to get the rest of the class on my side.
If necessary, you can ask the
offending party to leave. **If you have trouble keeping the attention
of the class, silence is the
best weapon.** Stand at the front, look annoyed (this part is easy),
and say nothing, Do not
speak until you have regained their attention. It will seem like
forever, but fifteen seconds is
usually all it takes.

Gilbert Highet noted that a teacher must be friendly without becoming a friend. However relaxed
your classroom atmosphere, you must retain a proper distance from your students. You are in a
power relationship with them, you are not equals. You cannot let personal relationships interfere
with professional responsibilities. **Never date a student!** Not only is this unprofessional,
it is almost certainly considered sexual harassment by your university. Avoid even the
suggestion of an inappropriate personal interest in your students' lives. A prominently displayed
photograph (authentic or not) of you in a couple will discourage inappropriate student interest in
your life. In cases of sudden true love, I counsel patience or a change of section.

As a new teacher, you will be bombarded with advice, much of it contradictory and confusing.
You cannot possibly accept it all. **Teaching becomes easier with experience.** You will
discover that many obvious ideas fail and many dubious ideas work. Before long, you will be
giving advice to new instructors and swapping your own war stories. I thank you for reading this
far -- I think I'll let you out a few screens early.

6. About the Author

**Bruce Reznick** is a Professor of Mathematics at the University
of Illinois at
Urbana-Champaign. He received his degrees from Caltech (BS, 1973) and
Stanford (Ph.D., 1976). He has been a Sloan Foundation Fellow and
Chairman of the Problems Committee for the Putnam Competition. His
research ranges
promiscuously among number theory, algebra, analysis, geometry and
combinatorics.
He roots for the Chicago Cubs, but has forgotten why.
The author was supported in part by the National Science Foundation.

*Last modified: August 23, 1999*