
How we teach calculus
Posted:
Feb 9, 2002 5:55 PM


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 preprecalculus 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 "neoreform" 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 allday 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 allday 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 halfsemester chunks and give students many opportunities to take them during the year.
The diagnostic test places many of our students into preprecalculus. What students worry most about is falling behind in their studies.
Every halfsemester 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 halfsemester math courses. That means it is possible to start with our preprecalculus 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 allday 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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