Note: The following paper was printed in the quarterly journal Academic
Questions, published by the National Association of Scholars, which holds
Student Evaluations in a Calculus Course
by Ralph A. Raimi
Towards the end of each semester the Dean's office sends
me a packet of forms, a questionnaire, by which all my students are
encouraged to judge me and the courses I'm teaching. To keep the
results anonymous I am not myself permitted to supervise the
answering, but must leave it to a student to collect the papers and
take them directly from the classroom (from which I have absented
myself) to the University's statistical office. The results are tabulated
this way and that, after the manner of the Social Sciences, and, well
after the semester's end, sent to my Department Chairman and to me.
Among the numerical results are an "Average Grade for
the Professor" and "Average Grade for the Course." These are interes-
ting, of course, to any professor, and the University provides us with
corresponding average data for courses similar to our own so that we can
compare ourselves "as others see us"; but more interesting than these
grades are the raw comments that students are invited to write out (also
anonymously) on a second sheet of paper at the same time. These few
sentences (optional; not every student does this part) appear beneath
three printed headings: A. Comment on the strengths and weaknesses of
the structure and content of the course; B. Comment on the strengths
and weaknesses of the instructor; C. What changes in the course and in
the teacher's methods and manner of instruction would you suggest?
This part is sent back to the professor alone, not to his
Chairman, and not for tabulation and comparison. Here is a selection of
some comments that have come back to me in the last year or two, all
from students in elementary calculus, Math 141, 142, or 143. The
selection, while [sic], is not random.
1. Calculus is a useless course which I will never use
later in life. It is one of the worst courses taught in college.
I believe it's only use is to weed people out.
2. Prof. Raimi provided interesting visual interpretations
of what we were actually doing in math and often referred to past
material when presenting new topics that tied into the new material.
3. This instructor was very unclear, and he continued to
lose the class, day in and day out.
4. Instructor is very good. He knows the material
backwards and forwards...
5. The instructor was not clear, and often confused by his
own work. Also he made many errors in his own calculations.
6. Doesn't relate to us at all, quite boring. But, he
explains things well and works out problems well --- easy to understand.
7. I am appalled and thoroughly disgusted with the math
dept in the 140 series. This is supposed to be a prestigious math &
science school and I feel that those in the math dept care only about
students in the upper level courses. Raimi's attitude was atrocious and
absolutely inexcusable. When I went to him for help and to discuss why
I wasn't understanding something, he called me stupid. Any prof. who
dares to call a student stupid doesn't deserve to be at this university.
I have absolutely no respect for the man. Raimi's attitudes and manners
need immediate attention, not to mention his disasterous teaching. The
course itself follows the book exactly. By having Raimi as a prof., I, as
did others, pretty much taught myself the course. I don't pay all this
money to teach myself a course.
8. He speaks well and makes the subject clear to non-
math majors. He should take more control of the class when a student
asks too many questions.
9. Why the hell is the final worth 40% of the course?
10. As the professor pointed out during the semester, he
wears too much black all the time. He should wear somewhat lighter
clothes so that the atmosphere of the room is made brighter. Somedays
it looks like there may be a funeral going on which makes his presen-
tation a little depressing. But overall, it's no big deal.
11. Completely understands material. Good dresser.
Nice blackboard technique.
12. He was very unclear. Knew the material but couldn't
explain in students terms. Expected all students to have a lot of previous
knowledge. Made people afraid to ask questions. Didn't do examples
from the homework but rather difficult theorems.
13. Tended to go off on tangents --- examples in class
were intirely different from exams.
14. The instructor was himself confused at times. He
would go through the work very quickly, erasing everything before
anyone could get a chance to understand him. Lately, though, class has
improved. He's much more organized.
15. Prof. Raimi applied the material we learned to
practical problems --- something a lot of profs fail to do. Text could
That the text could be better is beyond denial. We change
calculus books every few years, always hoping for a better one. What
we really hope for is a book the students can understand without effort,
though this is not the way we put it in committee.
Students too, some of them, would want a calculus book -
-- or professor --- or something --- that would be understandable without
effort. But as Euclid or Archimedes is reputed to have said to Ptolemy
or Alexander --- or was it Laplace who said it to Napoleon? --- "Unlike
the rocky lanes of the common people the
broad; but there is no royal road to geometry." Student # 7 is infuriated
at having had to teach himself calculus 141. If I could believe that he
really did, I'd be quite content with my performance as a teacher. That
is the nature of learning: nobody can pour it into your ears. We have all
"taught ourselves," by reading, by writing out exercises, by discussion
with friends and teachers, by solitary thought, by practice, practice,
A teacher is an assistant in all this, but not the whole
story. And his lectures, at least in mathematics, are not the whole of
what guidance he does give. He chooses the text, establishes the pace of
learning, selects the exercises (graded as to difficulty and logical order),
judges and corrects the results. Student # 7 does not regard these things
as "teaching," but maybe he'll learn. I'm sorry it cost so much money
to get him to his present state of understanding, such as it is.
There is internal evidence in his complaint to show that
he does not in fact learn very much by listening. He heard me call him
stupid, for example, and would probably swear to it in court. But I
didn't say that. I never have called a student stupid. There is no
occasion for such a statement, no pleasure in it, no teaching value, no
profit in reputation, ego-building, or cash.
I can imagine, however, how he got the idea that I did. I
must have admitted to him at some time, whether in class or in my
office, that I know something he does not. This is decidedly not an
egalitarian attitude. Furthermore, as is my custom when a student
approaches me with a question, I probably asked him something in
return, to see what part of his question he was actually able to answer
for himself. I don't usually go as far in this manner as did Socrates,
who affected to be nothing but a midwife in the birth of his students'
ideas. There isn't the time, and today's students sometimes feel per-
secuted when one does not immediately set their doubts at rest. Aca-
deme isn't where it used to be, after all.
In any case, one of his answers must have caused me to
interrupt. I can see the scene now: He wants to know how Problem 21
is done. I look at it and see that it involves a bit of trigonometric
information many of my students don't quite grasp, something I assigned
the problem expressly to straighten them out on. So I ask him a prelimi-
nary question about a phrase in the printed problem. This is important
to do, rather than simply write out an answer, because the key transition
in the solution is easily missed by an unschooled eye and mind. In fact,
the author of the textbook has already written out several examples of the
same sort, which this student could not have understood and still have
come to me with his question.
So he answers the simple question and I go on to the next,
which he answers somewhat less patiently. The third question is going
to be the whole point, of course, the revelation. If he answers it in the
mistaken form I anticipate, I will be able to point out how his answer
precisely misconceives the import of the two questions he has just
answered correctly. Magnificent! He will have learned how he could
have answered his own question without ever coming to me, or to any
other "authority." Mathematics is not a question of authority, he will
see, but of inexorable logic. Perhaps, if I put the third question in just
the right form, he will then and there see how its answer flows out of
what we have just established:
I put the question, and (alas) he does not yet see the point.
He gives the unthinking answer he always has given in the past. Testy,
too; why can't the professor tell him, instead of all this jockeying?
"No!" I say, "You can't mean that! What you mean is..." But does #
7 hear the rest of my sentence, or read what I'm writing on the black-
board? Not a bit of it. He hears me say, "You're stupid." He also
observes that I am evading his question, and trying to get him to do
mathematics instead. He didn't come to me for mathematics, you see;
he came for the answer to Problem 21. And he gets called stupid,
Other students come for the answer to Problem 21 and
succeed in getting it. Maybe they have more patience. But that patience
gives out when the examination comes around and Problem 22 turns up,
not 21. Then they say (# 9), "Why the hell is the final worth 40% of
the course?" Or (# 13), "Examples in class were intirely different from
Apart from the comments of praise, which I included
above to prove that I am not condemning, or condemned by, an entire
generation, there are two other currents of thought represented in the
quoted remarks. One of them is a never-failing surprise to me, and that
is the annoyance some students feel when I make a mistake at the black-
board. How can it be that in the same class there are students who say I
lecture clearly, in an organized manner, with a "good blackboard techni-
que," ( # 4, 6, 8, 11) while others in the same classes (# 3, 5, 12, 13,
14) are confused by my disorganization and numerical errors?
The difference, I have discovered from much observation
and questioning of students, has mostly to do with what students are
trying to get out of the class. Those who take down everything I say
into a notebook are getting a set of lecture notes out of the class, while
those who watch and try to understand as they see and hear get some-
thing totally different, even if they sometimes do not understand. The
one who comes to class for lecture notes never understands: every
stenographer knows that to type a letter is not the same as to read it.
The stenographer-student hopes to understand the material later on,
when he "goes over" his notes. If the professor is forever making
mistakes and erasing this to substitute that as he goes along, the
stenographic problem becomes insufferable. It is, on the other hand,
exactly the non-stenographic student who points out the errors when
they turn up, or at least asks the meaning of a momentarily puzzling
formula or argument, and thus reinforces his understanding with each
error the professor makes. The first sort of student is angry, and
carries his anger home in his notebook, while the second sort of
student will hardly remember that the eraser was ever used. And
I might add that the second sort learns more, both in class and
later, and would learn more than the stenographer even if the lecture
had been as perfect as the textbook, complete with plastic overlays
in two colors.
This is not to say that I should cultivate the making of
mistakes. They can waste time and they can confuse the issues. But
there is only one sure cure for errors: the totally written-out lecture, i.e.
the blackboard textbook copied from the manuscript in the professor's
possession, hour by hour and chapter by chapter. This sort of thing was
traditional in European universities in (say) the 19th Century, and often
for fairly good reason, because the subject was advanced (by today's
American undergraduate standards) and printed textbooks practically
nonexistent at that level. But the Calculus 141-143 under discussion here
has a textbook that weighs ten pounds and contains lengthy exposition,
perfectly worked-out examples, exercise sets with printed answers to the
odd-numbered ones, and pictures by airbrush and computer. The book-
store also sells a Student Guide to accompany the textbook, and this
contains even more worked-out examples and more answers to exercises.
Shall the student then attend class only in order to substitute his own
smudges for this glorious and more than complete stock of absolutely
accurate printed information?
I tell them on the first day of class that notes on my
lectures are not worthwhile. I tell them why. I repeat this from time to
time as the semester goes on, and as I notice more and more notebooks
creeping out of hiding, and more and more pencils working as I talk.
Keep a notebook, I say, and put your exercises in it as you go through
the course. Use it at home to record your puzzlements, so you may
remember what to ask in class, or if you visit me in my office. Look
over its earlier pages as the semester goes on, to see how some of your
earlier confusions have evaporated with experience and practice. Bring
it to class too, and open it to enter a phrase that seems obscure to you as
I talk, so you can go to the book later on and find out what you missed.
It will be there. But don't, please don't, make your notebook into a
second text. It is guaranteed not to be as good as the one you bought, I
tell them, while in the meantime you are robbing yourself of what value
a live professor can bring to the classroom.
By the end of the semester almost everyone is writing a
mile a minute. It is part of the student culture to do so; no amount of
cautioning from me will convince more than a few. (# 14 perhaps?) No
wonder some of them complain that they have to "teach themselves" the
course; they haven't given me a chance to help.
The second current is made explicit only in the remarks of
Student # 1 in the list, but it is an undercurrent in some of the others.
"Calculus is a useless course..." says # 1, "quite boring" says # 6
(though he thinks he is saying it of me rather than of calculus),
"...makes his presentation a little depressing.." says # 10 (though he says
he's talking about my black clothing).
I think that if I lectured on The Black Death instead of calculus,
# 10 might find my presentation quite lively, whatever clothing
I wore, and # 1 might never think to call the subject "useless." Nobody
ever thinks an interesting thing to be useless. I wonder about Student #
1; does he find music "useless"? It certainly is useless in the sense he is
using the word, and so is the story of Brutus and Caesar, and nine-tenths
of whatever else goes on in college.
Actually, calculus is very useful in the restricted sense # 1
intended, and every calculus book and professor makes this clear by
numerous examples. # 15 notices this with approval; where was # 1 at
that time? Eighteen-year-olds are not always great judges of such
matters. But though every calculus course and book does point out
practical application where it can, bearing in mind that the audience is
made up of scientific beginners, it does no good to defend calculus on
these grounds in hopes of attracting students to either its beauties or its
uses. The real message of the student who finds calculus useless,
boring, or depressing, and even in many cases of the student who finds
the book tedious or the professor confused, is that the subject is not
What then have we learned by having the students "grade"
the professor and the course? Pretty much the same thing as we learn by
giving a good examination on the subject matter. Furthermore, the
discovery that a lot of students think the teaching can be better is not
necessarily useful. Some years ago we used a questionnaire that con-
tained the following question: Is the textbook for this course (a) Too
easy, (b) Too hard, or (c) About right? In my classes the vote usually
came out pretty evenly split among the three answers. One interpretation
of this result is that two-thirds of the students are dissatisfied with the
book; should we change it then? In which direction?
In having us administer these questionnaires the University
intends us a service: By learning of our students' dissatisfactions we
might be induced, it thinks, to improve our performance. Perhaps we
may, though I have as yet seen no scientific evidence to this effect. I
have, on the other hand, seen evidence showing that our grading of
students does improve their performance. The symmetry of the idea of
having students grade professors just as professors grade students is
illusory, and obscures the essential asymmetry in the student-teacher
relationship, that we are better judges of them than they can be of us.
And that it is more important that we judge them than that they judge us.
And more healthy.
There is no harm in our knowing what students think of
our performance, as there is no harm in any form of knowledge. There
may, however, be some harm in gathering this not very useful
knowledge in quite the way we do.
Most teachers are bad teachers. It may be that in a
singularly good college one will find a lot of good teachers, but even
then college is not all of life, and not all learning takes place within ivied
walls. We learn first from our parents, later from relatives, lovers,
employers, children; we learn from newspapers, from housepainters and
automobile repairmen, from doctors, policemen, Senators, social
workers, from trashy novels and good ones, from movies, tax forms,
symphonies, advertisements. I repeat, then, that most of our teachers
will inevitably be bad teachers. Many are ignorant of what it should be
their business to know, some will be inarticulate, some will be liars.
Some will have little time for us, nor will they take the trouble to
administer questionnaires, much less be guided in their later behavior by
our opinions of their performance. Yet we must learn from them all.
There is no other way.
Somehow this lesson must be conveyed to our college students.
Do you want to learn? Then it is for you to dig it out. If you find
a book that is well written you are in luck. If you find an articulate
and knowledgeable professor who cares to see that you understand, you
have found a treasure above rubies. But you must expect mostly to find the
obscure, the garbled, the false, the irrelevant, and the indifferent. It
is there your desire to learn will be put to the test, for if you cavil and
complain --- quite correctly, no doubt --- that you have found only the
obscure, the garbled, the false, the irrelevant, and the indifferent, you
will end by learning nothing except complaint.
There are those, even in college, to whom this lesson
comes early on. They are the ones who get something interesting and
valuable out of every course they take, and who, if they have trouble
learning (as some of them do; the people I speak of are not only the
"bright" ones), try harder.
And there are those who never learn how to profit from
bad instruction; these people tend, as they grow hardened in their dis-
satisfactions, to find bad instruction everywhere they go. They become
great judges of teaching flesh. They know just what the failings are in
their professors, that caused them to dislike calculus, Chaucer, or
International Trade and Payments. Later in life they will know just what
the failings are in their employers, that caused their work to be under-
valued, or the failings in their spouses that caused the marriage to end.
The dichotomy between the one who wits to learn and the
one who will not learn is doubtless extreme, and we all partake some-
times of the attitude of the one or the other. Perhaps temperament,
genetically determined, accounts for a great deal here. But until the
science of psychology can demonstrate otherwise we must assume that
soreheads are as much created by education as by birth, and that it
is the duty of every professional teacher to encourage attitudes that
The most important such attitude is self-reliance. It isn't
that we "teach ourselves," as if the world contained no books and
lectures; it is that we ourselves are the ones to blame if we fail to learn.
The world is full of bad teachers, to be sure, but they are teachers, all of
them, if we look at them correctly. How can our students be taught to
look upon the wealth of the world of the mind, mixture that it is of the
true and the false, the beautiful and the ugly, the ordered and the non-
sensical, and then extract from it what we would like to call an education?
Setting them eight times a year to sit in judgment of the
knowledge, the lucidity, the friendliness, or the black clothing of the
professor is not a step in the right direction. Sure, we all judge our
neighbors, and our students are bound to judge us, and even tell their
friends which courses to take or avoid. But this is not the same as what
we are having them do now. We are elevating their judgments to an
unwonted level, focusing their attention on a crotchet of the professor as
if it were equal in importance to the death of Hector, and as if in conse-
quence of that professor's failings they were justified in ignoring the
assignment. I can see no benefit arising from the student evaluation
system sufficient to counterbalance this one misdirection of the student
conscience, this postponement of the day when our students will, if ever,
learn how to learn.
Ralph A. Raimi
4 May 1988