. . . . . From a mathematics instructor at another university:
"I have just finished reading your article [T.U.L.] in the Notices of the AMS. [We reprint as an appendix to the present article the appendix from T.U.L., Academic Orientation [A.O.], with mild editing. Item #7 from A.O., omitted in the August issue, has been restored and enhanced.] I absolutely agree [with] every word. I taught some undergraduate courses [here] getting extremely good student evaluations. First I was very surprised, then the next semester the ratings were even higher ... and I still didn't understand. Now in the light of your article it makes sense: I taught like this were a high school."
. . . . . From "The sum of mediocrity," an article on pre-college education by Pat Wingert, in the December 2, 1996 issue of Newsweek:
"We expect less from our students, and they meet our expectations."
----------
One thing that should be happening during the first semester of the freshman year is that the students start to figure out how to learn on their own, and get beyond skimming the surface of the subject. This is the main academic adjustment that most students must make when they get to college. We should therefore insist that they do it. This could involve running our calculus courses in a way that is very different from what they are used to from high school. A simple illustration of this is insisting that the students read the textbook, both for concept and examples; we will get back to this later.
At the beginning of the Spring '97 semester, I was talking to a student from my Fall '96 Calculus II (Physical Science and Engineering) course. I had gotten to know her through visits to my office hours for help in getting started on some of the assigned problems. For instance, at the beginning of the course she was unable to make sense of the problem: show that when n is a multiple of 4, the sum 1 + 2 + 3 + ... + n is an even number. Later, she could not figure out how to deal with a sequence defined by a simple two-step recursion. After failing Exam 1, she told me about being panicked during the exam, and she sought my advice. I told her to engage a more secure comprehension of the material, so that it wouldn't get lost under the strain of examination conditions. Her performance on subsequent exams was satisfactory, and her course grade was a C. (I would describe the exams as pretty straightforward, but very thorough.) This is just one example of evidence that the course is being run at a reasonable level for our students.
I felt that my course had helped her substantially, and I sought confirmation from her. What she told me first I'd heard before: she didn't like what I was doing in the beginning. After the semester was over, she said, she had looked back and realized that the way I conducted the course forced her to learn how to learn. Moreover, I was astonished to hear, mine was the only one of her courses that did! [A likely explanation for this is given, more or less, in T.U.L. Untenured faculty are afraid of incurring complaints about their teaching that could hurt their careers. Tenured faculty have lost their will to stand up to student indifference and resistance; besides, doing so can also cost in salary raises, given the way teaching is often evaluated.]
In a way, she was a good student. Before continuing with this, I want
to describe the backdrop. The students in the course were almost
all freshmen
with A.P. credit for Calculus I. The review of the most pertinent material
from Calculus I
was left to the students in the first homework assignment. The handling of
integer variables,
material that would become relevant later for sequences and series, was
treated. An important
thing presented in the
first
week of lectures was the mathematical usage of "if/then" and
"only if",---a majority of the students seem to think at first that
"if" means "if
and only
if"---so they could read the textbook correctly. A good way to confront the
issue
is by showing the two sentences from conversational English:
. . . . . a) If it stops raining, I'll go to the store.
. . . . . b) If I win the lottery, I'll buy a new car.
These have parallel structure but different connotation. It is
not hard to convince
the students that we cannot afford such ambiguity in mathematical writing.
The assignment for
the next week consisted of three problems designed for "consciousness-raising"
(in a course where the students had
been told to expect 8 hours of work per week outside of class). One of
these was the problem about
1 + 2 + 3 + . . . + n mentioned above.
A better one, I
think, was
. . . . . . . . . . True or false, and explain fully (i.e., verify or give a
counterexample):
. . . . . a) f(x) is a rational function only if the antiderivatives
of f(x) are rational.
. . . . . b) If f(x) is a rational function, f'(x) is a
rational function.
The answers to these are familiar from Calculus I
experience, but
few of the students had ever thought about the processes of differentiation and
integration
in the large. To guide them, I went through the verification that the
sum of two rational
functions is a rational function. Also, I explained in lecture what is meant
by a
counterexample. They were instructed
to write up their solutions carefully. Most of the students discover that they
can do such problems if they persevere. While it seems hard to squeeze them
in,
problems of similar depth should be given as the course progresses.
To return to the student in my office, I asked her to evaluate 1 + 2 + 3 + . . . + n in order, for n = 1, 2, 3, 4, and emphasized the parity of the answers. That already triggered something in her head, and she was prepared to persist with the problem. A bit later, she reported something very "unusual": as she spent more and more time with it, she saw her understanding start to grow. I assured her that this was very normal. The expectation that the answer is either there or not there is one of the many misconceptions that freshmen have about mathematics.
I have been teaching that Calculus II course every fall semester since 1993. In '94, I changed my attitude towards my role as instructor [I should thank my colleague W. Stephen Wilson, who was Chair at the time, for pushing me in this direction.], substantially increasing the amount of thought during the preparation and energy into the delivery of the lectures, and additional effort in printing up handouts to supplement the lectures as needed. But I also expected the students to match it with increased learning; the aspiration had become command of the material. Though the results were good, there was still a lot of grumbling in the class. Indeed, the libelous review of that course in the student course guide led to the shutting down of that publication. My aspirations have not changed much since 1994, though my understanding of the educational issues has.
One of the students in the 1994 course, who also reported being unhappy at first, wrote of the lectures, "He made the material very easy to understand, if and only if you were doing the work necessary to keep up with the class." I think he was trying to convey the message that a student who was not keeping up would be unable to see that the material was being explained in a clear and helpful manner. It reflects poorly on the current state of affairs that I started to wonder whether that was fair! With the support of my department chair, I decided that it was. I was encouraged further by the dean of the college, now provost (whose academic credentials are in the humanities, by the way).
I remember vividly when, during the second or third week of the 1993 course, a student came to see me, feeling that he was hopelessly lost. I probed with a few questions, after which both of us could see that he was very close to understanding the material. He, like so many other high school graduates, had been trained to absorb mathematics in tiny controlled doses, which are to be memorized and later regurgitated. It is no wonder that the suggestion of learning concept often gets perceived by students as irrelevant theoretical digression, rather than the means to better comprehension that it is.
It struck me just a couple of weeks before Fall '95 began that there was no clear way for the students to figure out what it was, so different from their high school experience, that I wanted them to do. Some students make the transition instinctively [a strong student reported that it took her a couple of weeks to get into stride], while others simply blame the instructor for teaching poorly. This led me to do some serious orientation for my own students that year. It was a nice coincidence that during the first week of classes, I crossed paths with a senior who had taken Calculus I from me in his freshman year. I started telling him about my ideas on academic orientation. At one point he said to me, "You have to remember that they are freshmen, and that they don't realize that whatever they think of their instructor, they'll be learning most of the material on their own anyway." I recall that I came back with Why don't they know that?! If the high schools and older students aren't communicating that, then it has to come from us.
I then pressed hard for academic orientation in the university. For obvious reasons, it is better that the message have the ostensible endorsement of the university: of the math department and the academic deans. I ended up giving a presentation on mathematics during Orientation Week, 1996, a first. At Hopkins, it comes down to communicating to the entering students what most sophomores, and virtually all juniors and seniors, know about education in college. This involves making the new rules explicit (see the appendix Or-2 Academic Orientation [A.O.]), and debunking the misconceptions that so many of the freshmen bring to college. Appended as Or-1 is a compilation of such misconceptions. The first five items were discussed first to help soften the impact of the potentially startling A.O., which was circulated about half-way through the presentation.
I am convinced that the level at which my course is now conducted is about right (for its audience). If we assume that this is correct, it is hard to justify on educational grounds running the course at a lower level. My university has 900 to 1000 entering students each year, who are ambitious (at least in the abstract) and "bright" (mean SAT Math score a little over 700). Most of them did not have to work hard in high school; here lies a big part of the problem. One of my 1996 students, who was used to having to work hard, reported what another student had said: "I'm so pissed off! I got a C-. I couldn't believe it. I never worked in high school, and I always got A's!" And that was after orientation! Nobody said that our task is an easy one.
Because some students can't, most--around 85%--of our students were never asked to learn mathematics in high school by reading from a textbook. They grew accustomed to picking up the material from classroom presentation alone, even in A.P. courses. But in fact, the students who attend a good college are capable, with some exertion, of reading the book; we therefore want them to do it. For some, it's a struggle. Even the stronger students will encounter things in the book that they can't, or just don't, figure out. However, there are numerous ways they can have this straightened out, viz., lecture, section meeting, TA office hours, help room, professor's office hours, discussion with classmates, . . . . Given the advent of extended help room hours, I find even more outrageous the suggestion that the lectures be aimed at those who want to skimp on their effort in learning the material.
When college students say they want a good teacher, some want a good educator, one who will help them to make it through the material. They accept that inherent to mathematics is the need to decide what to do to solve a problem, to make distinctions and choices, to reject ideas that are fallacious, and to persist when one's first attempt doesn't work. This flexibility [Intellectual flexibility is a notion that makes sense in all disciplines. As such, it is wise to talk about it when we explain to educators in non-scientific fields what we want from our students in mathematics courses.] is often absent from high school experience, where mathematics is taught largely by repetition. As such, they were trained to learn the subject inflexibly. Many students want it to stay that way in college; they want to be helped around the material, in effect, bailed out by the instructor from having to understand it. Pandering to the latter group is slowly but surely eroding the quality of mathematics education in American colleges and universities, and even abroad.
An interesting thing I learned in 1996, when the course was divided into two lectures, with the other run in tandem with mine by a remarkably skillful assistant professor, was that the freshmen here will praise an instructor who runs the course at a high level provided the lectures are well-organized, focused, and "self-contained". However, it does not follow that they learn better; his students and mine performed comparably on exams (cf. misconception #13). All too many students want to come into class cold, expecting to get "taught" by the professor, from the ground up. [That would serve to keep the level of the course down, unless the students are expected to pick up a lot more as they read the textbook later. Students here are capable of learning the easier things in the course largely by themselves, and I remind them of that.]
What is the point of the instructor's commitment of time and effort to attempt to supply teaching that will be judged to be "better" when it is not getting matched by demonstrated better learning? I would go further. No style of teaching mathematics can substitute for insisting that the students pick up their share of the work, unless one is willing to compromise standards. We should stop seeking panaceas that place the burden on the competent instructor; I do not believe in the "Fountain of Education," and it is time to stop looking for it!
How can we fulfill our role as educator? The main theme is that one must aim for the determination of the appropriate level for the course, one that matches the level of the students who take the course in one's own university. Of course, I do not mean here the level that we see when the students follow their high school instincts, thinking that they won't be reading the textbook, or equating learning to memorizing a list of formulas, or declaring that spending three or four hours on homework is a lot of work. We must then have the conviction, and ideally the backing of the department and the administration, to hold to that level. However, in doing this we also put a greater obligation upon ourselves, for the threshold requirements [This notion is mentioned in misconception #13. I know no algorithm for determining where the threshold is for a given level of aspirations and a given student body.] for the instructor, in giving a course at a higher level, are correspondingly higher.
Academic orientation is necessary, of course. The students must be told how things are going to be different from high school. This is more likely to succeed if there is a web of support that will block the students from resisting "because their professor has these crazy ideas." As reasonable as we may find the statements of A.O., most freshmen are shocked by them; they even wonder if it's for real. It doesn't help matters if some of our colleagues teach calculus as though it were "grade 13 of high school," an easy way to endear oneself with freshmen.
On the other hand, many students have heard in high school such things as "Don't just memorize. Learn concept." However, they found that they could score well on tests by ignoring this advice and behaving as they were advised not to. That must stop in college. Another big problem is that many freshmen wrongly believe that adjustment is needed only for students other than (weaker than) themselves.
To put the preceding into effect, we must be free to give the students what they need, not what they say they want. But if we do so, we risk lowering our ratings in the course evaluations, [Actually, my own ratings went up from '93 to '94.] for students who retain their belief that they are entitled to do well without exerting themselves are not going to be happy. This places us in the ironic situation that we might be penalized just for doing our job conscientiously. [I know of some (many?) past students (including, of course, the one mentioned at the beginning of the article) came to realize how much they benefitted from my course only after the course was over, e.g., while they were taking Calculus III. "The world rewards the appearance of merit oftener than merit itself." -- F. de La Rochefoucauld]
I'll summarize what I have been doing toward upgrading the freshmen's expectations. [What one can do in the classroom is based in part on one's nature; it is unreasonable for ourselves or others to regarded us as teaching machines. Also, it may be relevant here that I had control of the entire course.] When I gave a forceful presentation in my course in 1995, the Department had circulated J. Martino's Survival Guide to all students taking a large lecture course, and that reinforced my message nicely. In Fall 1996, my presentation was on the Orientation Week program, and that "implied" backing by the University. In Fall 1997, my previous experience enabled me to carry out efficient major orientation for my own course. I appealed to something I had ascertained was mentioned by deans of both the College of Arts & Sciences and the Engineering School in their addresses to the freshmen: the amount of work outside of class expected in a college-level course (cf. Academic Orientation #3). After reminding the students of that during the first lecture, I told them, "I don't want to hear any griping about the workload in this course unless you are consistently putting in more that 8 hours a week. And if you are not highly talented, you may decide that you want to put in more." I feel that I've told the students this year everything I might want to say in the way of orientation.
Above all, the students should get the message that we intend to help them learn the material, but we are not going to bail them out if they don't. The exams in the course must uphold the level announced for the course. They must be made up so that they (de facto) penalize students who insist on operating on the basis of high school notions that we have declared inappropriate. The problems should be taken from the heart of the material, not from the surface (as students learn to expect in high school). No practice exams to suggest programming. I tell them that doing new problems of sufficient difficulty with books closed comes closest to simulating the exam situation. In lecture this year, I asserted before the first exam, "If you take all of the homework problems, examples done in lecture and the book, problems from past years' exams [available to the students], the problems on the exam will be different from all of them; but they can be done by the same methods."
The messages of A.O. must be reinforced throughout the semester, and frequently
in the first part of the course. Here's a sample of what my class heard or
read this year:
--- Talent and background will make some difference, but you are
going to have to work in order to succeed, both in this course and in
pursuing your career goals. If you choose to shoot for less, you do
so at your own risk.
--- The goal is to reduce your dependence on the instructor.
--- Think about it after class. You should know by now that you don't
have
to understand it here.
--- It's impossible for me to explain that to you. Some things
you must try to
figure out for yourself.
--- Mathematics is about concept, attitude and control.
--- The purpose of the exam questions is to determine the extent of your
command of the material. Though you should be getting the correct answer to
the problems
if you have good command, it is not the main point. After all, if I
only wanted to see the right answers, I'd just do the problems myself!
A strategic point: we shouldn't overlook the power of negative
reinforcement.
In Calculus II, the following lines of "reasoning" are all too common:
. . . . . a) Determine whether the series \sum_{n = 1}^\infty 1/n
converges. Well, 1/n goes to 0, so the series converges.
. . . . . b) Compute the limit, as n goes to \infty of
(1 + 1/n)^n.
Well,
1 + 1/n goes to 1, and 1 to any power is 1. The limit is 1.
These are not just silly mistakes; they are fundamental errors that
show disrespect for the
methods of the course and for the instructor. (Why didn't we teach them the
"easy"
way to do it?) Indeed, it is essential to make the
student see
that these are wrong. [Students sometimes object to the idea of
showing them
that a way of "thinking" is wrong
(rather than programming them with the right way to do the problems). I'm
quite sure
that every math instructor has the experience that students, having been shown
the right way, make such mistakes
anyway.]
I term (a) the ultimate sin, and (b) the penultimate; it is
announced that committing the
penultimate sin in a problem
gets an automatic zero credit, and the ultimate sin gets negative
credit.
These sins occur with surprisingly low frequency in my course; the class
performs better now with infinite series.
I should mention that the immediate reason for my writing "Teaching at the University Level" was that I felt the calculus reform movement was gaining too much momentum. The notion that students at universities like Harvard and Stanford needed reformed calculus was out of line with my own observations. Moreover, it is unlikely that today's college students are less intelligent than their predecessors; rather, something has happened to their sense of learning. While the transition from high school to college has always been a hurdle for students, today's students find themselves in a weaker position to deal with it. Since I also refuse to believe that they have been irreparably damaged, trying to repair the damage makes far more sense than pretending that the subject needs "genetic engineering". In particular, we don't have to wait for the difficult underlying problem with pre-college mathematics education to get resolved before we can start to remedy the situation in the colleges.
It is not my intention to condemn the calculus reformers outright. But if we are to get serious about resolving the difficulties our students are having in learning mathematics, we must first address the issues that really matter most, namely the low expectations that many of them hold when they arrive in college and their overestimation of the effort that they are putting in. Only then does it make sense to judge the merits of different methods of instruction . . . for a given group of students.
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Or-1. Misconceptions (and rejoinders)
A. Common misconceptions about high school and college
. . . . . 1. Only a jerk could get less than a B-grade in a course!
If that's true, then nearly half of the freshman in math courses in
previous
years are jerks!
. . . . . 2. I had a good math teacher in high school who taught at my
level.
But most of the freshmen admitted here say they were in the top quarter
(say) of their math class, yet agree that a teacher [in high school] is
supposed to make sure
that even the weakest students in the class learn. Now, whose level was the
course run at?
. . . . . 3. I did well in math, even calculus, in a good high
school.
I'll
have
no trouble with math at Hopkins.
There is a different standard at the college level. The student will have to
put in more effort in order to get a good grade (or equivalently, to learn the
material sufficiently
well by college standards).
. . . . . 4. My AP calculus course in high school was like a calculus
course in college.
The student here is expected to do much more learning outside of the classroom;
see also #3 above. (On the other hand, the Advanced Placement Tests are
college-level exams.)
. . . . . 5. College will be like high school, just a little harder.
(See all of the above)
B. Misconceptions about learning mathematics
. . . . . 6. In a calculus course, theory is irrelevant, for what is
really at
stake is doing the problems. The lectures should be aimed just
at showing you how to do the problems.
We want you to be able to do all problems---not just particular kinds
of
problems---to
which the methods of the course apply. For that level of command, the student
must attain some conceptual understanding and develop judgment. Thus, a
certain amount of theory is very relevant, indeed essential. A student who
has been trained
only to do certain kinds of problems has acquired very limited expertise.
. . . . . 7. The purpose of the classes and assignments is to prepare the
student for the exams.
The real purpose of the classes and homework is to guide you in
achieving the aspiration of the course: command of the material.
If you have command of the material, you should do well on the exams.
On the other hand, some students act as though the exam problems have been
decided in advance, and expect the lectures and assignments to be
leading up to performance on those problems, or ones just like
them. The latter would constitute the avoidance of our goal.
. . . . . 8. The best way to study mathematics is to just memorize
everything
very carefully.
As a colleague in the Physics Department once put it, "You can't memorize
problem-solving!"
Here, problem-solving refers to the
ability to take a problem and attempt
to carry out whatever methods might be relevant to solve it. This is
a skill that grows with experience. (You might keep
in
mind as an analogy that memorizing the dictionary of a foreign language is not
enough to achieve fluency in that language.)
. . . . . 9. Students learn best when everything they have to know is
presented
slowly in the classroom.
If everything the student has to know is presented slowly in the classroom,
the total amount of material in the course will be rather little. Thus,
students actually learn least that way.
. . . . . 10. It is the teacher's job to cover the material.
As covering the material is the role of the textbook, and the
textbook is to be
read by the student, the instructor should be doing something else,
something that helps the
student grasp the material. The instructor's role is
to guide the students in their learning: to reinforce the essential
conceptual points of the subject, and to show the relation between them and
the solving of problems (cf. #6).
. . . . . 11. Since you are supposed to be learning from the book,
there's no
need to go to the lectures.
The lectures, the reading, and the homework should combine to
produce true comprehension of the material. For most students, reading a
math text won't be easy. The lectures should serve to orient the student in
learning the material.
. . . . . 12. A good teacher is one who can eliminate most of the
struggle for the
student, making the material easy to learn.
Of course, it is possible to direct the students toward correct ways
of thinking, but a certain amount
of struggle is inevitable. Experience cannot be taught. Moreover, many
topics are
inherently difficult so they cannot be understood either
passively or quickly. Eliminating the
struggle can only be achieved by excising substance from the course (e.g.,
constricting
the scope of the course, or reducing the means for recognizing where
the methods of the course
apply).
Then, the fraction of the material that remains could
well
be easier to learn, but the student will be acquiring diluted skills.
. . . . . 13. When the students are happy with the instructor's lectures,
they learn the material better.
This statement is wishful thinking. According to the evidence I've seen,
once threshold requirements are met the perceived quality of
the instructor makes little, if any, difference in learning. What makes a
real difference in learning is appropriate effort by the student.
The best
thing that a decent teacher can do, in order
to get the students to learn better, is to hold high yet reasonable
expectations of them.
Or-2. Academic Orientation
What follows is what an entering freshman should hear about the academic side
of university life
[in mathematics (and the sciences)].
It is distilled from what I've learned and written concerning the need
for academic
orientation.
The underlying premise, whose truth is very easy to demonstrate, is that most
students who
are admitted
to a university like JHU were being taught in high school well below their
level. The intent here is
to reduce the time it takes for the student to appreciate this and to help
him or her
adjust to the demands of working up to level.
. . . . . 1. You are no longer in high school. The great majority of
you, not
having done so
already, will have to
discard high
school notions of teaching and learning, and replace them by university-level
notions.
This may be difficult, but it must happen sooner or later, so sooner is better.
Our goal
is for more than just getting you to reproduce what was told to you in
the classroom.
. . . . . 2. Expect to have material covered at two to three times the
pace of high
school. Above
that, we aim for greater command of the material, esp. the ability to apply
what you have
learned to new situations (when relevant).
. . . . . 3. Lecture time is at a premium, so must be used efficiently. You cannot be
"taught" everything
in the classroom. It is your responsibility to learn the material.
Most
of this learning
must take place outside the classroom. You should willingly put in
two
hours outside the classroom for each hour of class.
. . . . . 4. The instructor's job is primarily to provide a framework, with
some of
the particulars,
to guide you in doing your learning of the concepts and methods that comprise
the material of the course.
It is not to "program" you with isolated facts and problem types, nor to
monitor your progress.
. . . . . . 5. You are expected to read the textbook for comprehension. It
gives the
detailed
account of the
material of the course. It also contains many examples of problems worked out,
and these
should be used to supplement those you see in the lecture.
The textbook is not
a novel,
so the reading must often be slow-going and careful. However, there is the
clear advantage
that you can read it at your own pace. Use pencil and paper to work through
the material,
and to fill in omitted steps.
. . . . . 6. As for when you engage the textbook, you have the
following
dichotomy:
. . . a) [recommended for most students] Read, for the first time, the
appropriate section(s) of the book
before
the material is presented in lecture. That is, come prepared for class. Then,
the faster-paced
college-style lecture will make more sense.
. . . b) If you haven't looked at the book beforehand, try to pick up what
you can
from the lecture. Though the lecture may seem hard to follow (cf. #2),
absorb
the general idea and/or take
thorough notes,
hoping to sort it out later, while studying from the book outside of
class.
. . . . . 7. It is the student's responsibility to communicate clearly in
writing up
solutions of the questions and problems in homework and exams. The rules of
language still apply in mathematics, and apply even when symbols are used in
formulas, equations, etc. Exams will consist largely of fresh problems that
fall within the material that is being tested.
Or-3. Two Telling Tales
. . . . . 1. Analogy: French in high school and college. I knew that the course French Elements covers about the same material as the first two years of high school French. This is typical of first year college language courses. Also, the semesters are shorter here, and one can calculate that the material is covered approximately three times as rapidly here as in high school.
After looking at the catalog description of French Elements, I called the instructor. I felt sure that there was more to it than just the triple speed. "Yes," she replied. "In our course, we aspire for fluency."
I admit that I had four years of French in high school, but no one ever spoke of fluency. It should be obvious that most of the work must occur outside of class. You can expect something like the tripling of high school pace, a lot of work outside of class, plus aiming for the mathematical analogue of fluency (perhaps command is the correct word), in a calculus course here.
. . . . . 2. Analogy: Martial arts. An 18-year-old enters a tae kwon do studio, walks up to the instructor, and states proudly, "I want to learn how to put my hand through a stack of bricks!"
The instructor thinks a moment, then replies. "Well, that's very difficult, and will take time. First, you must develop self-control and mental discipline. Then ..."
The youth interrupts, "Don't give me that discipline crap! Just teach me how to put my hand through the bricks!"
The instructor walks away shaking his head, as does the would-be student. One of the regulars of the studio, who teaches math in a local high school, steps up to the instructor. "You know, the young man has a point. All you have to do is make the bricks out of softer material, and crack them a little in advance."