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How we teach calculus
Posted:
Feb 9, 2002 5:55 PM
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What follows is a long description of how the Department of Mathematics
at Polytechnic University is teaching the freshman and sophomore
math courses that all science and engineering majors take.
There is almost nothing new in what we do. The important thing is
that we made a commitment to do all of them simultaneously and
to continue doing them well. Once all of the instructors and students
recognized that high standards and expectations were being set for
all of them and were not going to be compromised in any way, everyone
responded accordingly. As a tenured math professor, I had had
no experience working as part of a cohesive team before. Now that
I have, I find it exhilarating.
We also know how little we know, so we welcome feedback, criticisms, and
new ideas.
The following is a list of some of the things we do:
1) We set tough and consistent grading standards. A "C" means
the student understands the material and has acquired the necessary
skills, but makes a few too many mistakes. A "B" means the student
only makes a few minor mistakes. An "A" is reserved only for outstanding
students who can do the most difficult problems.
In particular, we no longer curve any of our grades at all. Students
have a hard time believing this, but we really do this.
2) We separate the grading and teaching responsibilities.
The director of freshman math is responsible for creating the tests
(she does delegate this to other instructors) and computing
all course grades. No instructor sees the exam for his students
before it is given. Although the instructors assist in
grading worksheets and exams, their primary responsibility is teaching
the students and preparing them for the exams. This insures uniformity
of grading standards across different sections of the same course
as well as across different semesters. At the same time it frees
an instructor to put 100% of his effort towards helping each and
every student.
In the past, a student could simply retake a course repeatedly
until he or she stumbled onto an easy instructor. This is no longer
possible.
3) We have put together a highly motivated and skilled team of nontenured
fulltime instructors, who do all of the teaching of our freshman courses.
Most tenured faculty have too many conflicting interests and
responsibilities to teach well. Although there are exceptions, most simply
cannot be depended on to work cohesively as part of a team and fulfill
all of their responsibilities as an instructor.
Our solution is to hire fulltime instructors whose sole responsibility is
teaching well. We make sure that, as long as the instructors fulfill their
responsibilities well, they are provided with the most comfortable working
conditions possible and generous raises. They are viewed as valuable long
term employees of the department and treated accordingly.
We are very, very picky about hiring instructors. Every instructor is
auditioned, to verify that he or she has not only a solid grasp of
mathematics but also solid skills and talent for teaching.
The result is that we have an incredibly hardworking and dedicated team
of instructors who work and party together. In terms of their teaching,
they deliver far more than tenured faculty for the same amount of money.
(It is interesting to note that the Harvard math department has been doing
something like this quietly since the early 80's.)
4) We have carefully designed a diagnostic test that assesses a student's
mathematical skills, ranging from elementary arithmetic to integral
calculus. Every entering freshman and transfer student is asked to take
this test, and we place the students on the basis of the results.
In particular, students who demonstrate weaknesses in arithmetic and
algebra are asked to take our pre-precalculus course. This turned out
to be a fairly large fraction of the freshman class. And a large
fraction of those taking this course fail it the first time.
5) We demand that students continue to demonstrate in every math course
the skills learned in prerequisite courses, ESPECIALLY arithmetic
and algebra. In particular, we demand that a problem be done correctly
from beginning to end and that all answers be fully
simplified. We have become very stingy with granting partial credit.
A student who makes chronic arithmetic or algebra errors in an advanced
courses is heavily penalized.
6) The focus of each course is on teaching students how to use and
do mathematics in useful contexts. The corollary is that students
must have both the understanding to set up a problem properly AND the
computational skills to work out the solution completely without error.
(I call this "neo-reform" calculus.)
To enforce the need for computational skills, students are required to get
an 80% or better on a gateway exam that focuses completely on
computational skills.
The homework, worksheets, and exams are dominated by "word problems".
Students who stumble on EITHER the setup of a problem OR the
computational part are penalized severely, losing more than half the
points.
7) We replaced the one hour recitations by all-day workshops. This is
the one that we stumbled onto and had the most unexpected results.
We had always known that the 1 hour recitations for going over homework
were a farce. We took advantage of the fact that at our school Fridays
are used only for labs and recitations. So a couple of years ago we
tried an experiment. Even though students were still assigned to
1 hour recitation times, we told them that in fact they could show
up to the recitation room any time from 9am to 6pm on Friday.
Once there, they would be given a worksheet with four or five problems
to do. They would be allowed to do them any way they wanted. They were
allowed to consult anything and anyone they wanted to. They were allowed
to stay as long or as little as they wanted. We created worksheets
that contained problems that really went to the heart of the material.
In other words, problems that were very easy, if you really understood
the mathematics AND could do the computations without error, but
impossible otherwise.
We were completely astounded by what happened. Students came in hordes,
spent hours (yes, hours!) in the recitation, and struggled through
the worksheets together and separately.
I know that this is no surprise
to those of you who are better trained or more experienced in mathematics
education, but, remember, most of us college professors have never
experienced anything like this.
The effect of these all-day workshops was so powerful that I could go on
forever about them. Drs. Shah and Van Wagenen have a nice paper
discussing them available at http://www.math.poly.edu/research
We are in the process of implementing them for ALL of our courses.
We call what students do in a workshop "supervised struggling".
The idea was at least partially inspired by the Montessori approach
to education.
We now tell students that lectures are not for learning but for learning
what you need to learn and that workshops are for learning. We
always compare it to learning a skill like carpentry or basketball.
How many people believe that they can learn how to play basketball
by attending lectures, reading a book, and then trying to practice
by oneself without supervision? Why should learning math be
any different?
8) We broke up all of our courses into half-semester chunks and give
students many opportunities to take them during the year.
The diagnostic test places many of our students into pre-precalculus.
What students worry most about is falling behind in their studies.
Every half-semester course is taught not only during each half of the
fall and spring semesters, but also during minisessions that run
in January and May, as well as during the summer sessions. Within
one calendar year, it is possible to take 10 consecutive half-semester
math courses. That means it is possible to start with our pre-precalculus
course in September, fail it once, and still finish not only all freshman
but also all sophomore math courses by the following September!
We also offer all of the minisession and summer freshman courses for $100
per credit, to minimize the financial impact on the students.
Those are just some of the facts about what we do. What is
even more important are the philosophy and values that guide what we do.
I can talk about that, too; again, a lot of that is also familiar to many
on this list, but foreign to too many research mathematicians.
Some of it is discussed in the all-day workshop paper. Also, see
http://www.math.poly.ed/courses/tips.phtml to see what we tell our
students about how to do well. Feel free to cruise around our web site to
see the worksheets and tests that we administer.
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