**1. With a math problem, you either know how to do it or you don't.
**

**
The intended meaning in the above is that in trying to solve a math problem
on an exam, if you
don't see how to do it at once, you will not be able to recognize how to
do it after thinking about it. This
attitude presumes that all problems on an exam will be at the surface of the
material
being tested. While that may be the experience of many students in high
school, there is no
reason to presume it will persist in college. The student may well have to
think and
decide what to do on a problem, and may need to do an involved calculation or
consider alternate approaches.**

**2. In a calculus course, theory is irrelevant, for what's
really at
stake is doing the problems. The lectures should just show the student
how to do the problems.
**

**We want you to be able to do 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!
**

**3. 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.
**

**4. 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.
**

**5. 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. #2).
**

**
6. 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.
**

**7. When the students are happy with the instructor's lectures,
they learn the material better.
**