Note: The following paper was printed in the quarterly journal Academic

Questions, published by the National Association of Scholars, which holds

the copyright.



            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

be better.


          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 Royal Highway is smooth and

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:  Eureka!  No further explanation re-



          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

being understood.


          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

facilitate learning.


          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                                                  

University of Rochester

4 May 1988