Math 220: CALCULUS I
COURSE: Math 220, Calculus I, 4 credits, Fall 1999
INSTRUCTOR: Dr. Larry Krajewski
Office: MC 526
Phone: (608) 796-3658 (office)
(608) 782-1648 (home)[call before 10 p.m.]
Office hours: 3M,10:30-11:30W, 10F & by appointment
PREREQUISITES:C or betterin Math180 or acceptable placement score
NOTE: A grade of C or better in Math 220 is a prerequisite for Math 221
TEXTBOOK: Calculus from Graphical, Numerical, and Symbolic Points of View, by Arnold Ostebee and Paul Zorn, Saunders, 1997.
LIBRARY How to Study Calculus, by Joseph Mazur
RESERVE: Is Your Math Ready for Calculus?, by Walter J. Gleason
How to Ace Calculus, Adams et al, W. H. Freeman, 1998.
TUTORING: Available in the Learning Center
FINAL EXAM: Tuesday, December 14, 12:50 - 2:50 p.m.
GOALS To help students:
1. learn to value mathematics;
2. learn to reason mathematically;
3. learn to communicate mathematically;
4. become confident in their mathematical ability; and
5. become problem solvers and problem posers.
Core (General Education) Skill Objectives
1. Thinking Skills:
(a) Students will use reasoned standards in solving problems and presenting arguments.
2. Communication Skills: Students will
(a) read with comprehension and the ability to analyze and evaluate.
(b) listen with an open mind and respond with respect
(c) access information and communicate using current technology.
3. Life Value Skills:
(a) Students will analyze, evaluate and respond to ethical issues from an informed personal value system.
4. Cultural skills: Students will
(a) understand culture as an evolving set of world views with diverse historical roots that provides a framework for guiding, expressing, and
interpreting human behavior.
(b) demonstrate knowledge of the signs and symbols of another culture
(c) participate in activity that broadens their customary way of thinking.
5. Aesthetic Skills:
(a) Students will develop an aesthetic sensitivity.
General Education course Objectives:
1. Thinking Skills: Students will
(a) gain a better understanding of the concept of a function;
(b) use graphs to estimate related values, relative rates, extreme values, limits, derivatives;
(c) develop a concept of limit;
(d) understand the derivative;
(e) apply concepts and techniques to calculus to analyze functions and find relative rates, extreme values;
(f) use numerical methods to evaluate derivatives and use calculators and computers efficiently as tools;
(g) model problems from geometry and other disciplines using calculus
2. Communication Skills: Students will
(a) write clear and logical explanations for problem solutions;
(b) participate in group work
3. Life Value Skills: Students will
(a) develop an appreciation for the intellectual honesty of deductive reasoning
(b) understand the need to do one's own work
4. Cultural Skills: Students will
(a) develop an appreciation of the history of Calculus and the role it has played in mathematics and in other disciplines.
(b) learn to use the symbolic notation correctly and appropriately
5. Aesthetic Skills: Students will
(a) develop an appreciation for the beauty of deductive reasoning
(b) develop an appreciation for mathematical elegance.
A. Review and Preview (Chapter One and Appendix A)
Functions are to calculus as numbers are to algebra. We will review some of the important properties and characteristics of functions
and preview some of the topics to follow.
B. The Derivative (Chapter Two)
One of the two major concepts of calculus is introduced. Limits and continuity are discussed.
C. Derivatives of Elementary Functions (Chapter Three)
Various techniques are studied.
D. Applications of the Derivative (Chapter Four)
Differential equations, numerical approximations, related rates and parametric equations are studied.
E. (If time permits) The Integral (Chapter Five)
The second big idea of calculus is introduced.
I view calculus as an introduction to pure mathematics and as a foundation for applications. Concepts, not techniques, are the heart of the course. We will move among graphical, numerical and algebraic representations of central concepts. This way I hope you will gain a better, deeper and more useful understanding of calculus ideas. We shall exploit technology (Maple V and graphics calculators) whenever appropriate. This emphasis on concepts does not mean we shall ignore symbol manipulations or hand computation. Some manipulative practice and skill build and support conceptual understanding. Hand computation illustrates concepts concretely, builds symbol sense (the algebraic counterpart of numerical intuition) and an ability to estimate and can give one a sense of mastery over the material.
Some of you may have had mathematics courses that were based on the transmission, or absorption, view of teaching and learning. In this view, students passively "absorb" mathematical structures invented by others and recorded in texts or known by authoritative adults. Teaching consists of transmitting sets of established facts, skills, and concepts to students. I do not accept this view. I am a constructivist. Constructivists believe that knowledge is actively created or invented by the person, not passively received from the environment. No one true reality exists, only individual interpretations of the world. These interpretations are shaped by experience and social interactions. Thus, learning mathematics should be thought of as a process, of adapting to and organizing one’s quantitative world, not discovering preexisting ideas imposed by others.
Consequently, I have three goals when I teach. The first is to help you develop mathematical structures that are more complex, abstract, and powerful than the ones you currently possess so that you will be capable of solving a wide variety of meaningful problems. The second is to help you become autonomous and self-motivated in your mathematical activities. You will not "get" mathematics from me but from your own explorations, thinking, reflecting, and participation in discussions. As independent students you will see your responsibility is to make sense of, and communicate about, mathematics. Hopefully you will see mathematics as an open-ended, creative activity and not a rigid collection of recipes. And the last is to help you become a skeptical student who looks for evidence, example, counterexample and proof, not simply because school exercises demand it, but because of an internalized compulsion to know and to understand.
Assessment is a continuous process. The constant flow of feedback and dialogue between students and teacher helps each better define how successful they have been in the exploration of a subject. In trying to align my assessment procedures with my philosophy of learning and teaching I have come up with the following plan for this semester.
You must come to class regularly. If you miss a class you are expected to find out what happened. Unexcused absences may lower your grade. One cannot be an active learner if one is not present in class. Reading someone else’s notes is not comparable to participating in class.
You will do some of the work in this class in cooperative learning groups. You will be working together on class activities. Working
well within a group is an important skill that is essential for many of the jobs for which our graduates apply. The objective for group
work in this course is two-fold:
1. to give you moral support while you are solving problems, and
2. to develop skills in working effectively as part of a team.
• Pass a test on differentiation rules with at least 80% accuracy
• Complete group assignments
• Complete a group project
• Complete individual assignments
• Pass exams
I suggest a lot of homework, and assume that:
• You are working on calculus problems regularly, almost daily. In particular, I expect that you will do some work outside of class between every class.
• A 4-credit class requires 8 to 12 hours of study time each week. It is most effective if your study time is distributed over the whole week, and not concentrated only on the weekends.
• You are studying individually and working regularly with your group.
• you are responsible for your own learning, and so you will ask questions - in your group, and in class discussion - about problems that have you stumped.
• You can't really be stumped until you've given the problem a good try.
• I do not need to "collect and grade" homework in order to motivate you
• You cannot completely understand a new concept until you have applied it to solving problems.
Keep a Problem Notebook
You may maintain a notebook with your work on the assigned problems.. You must do all of the assigned problems to get credit. The notebook may be collected unannounced several times during the semester so bring it to class. [Worth 50 points and reduces the weight of one exam from 100 points to 50 points.] Let me know during the first week of classes if you select this option.
Four chapter exams 60%
Graded assignments 10%
Final exam 15%
Differentiation test 0%_____
Cheating will not be tolerated. ANY INDIVIDUAL WORK THAT IS TURNED IN TO BE GRADED (with the exception of the project) MUST REPRESENT YOUR OWN EFFORT AND NO ONE ELSE'S. Please refer to the student conduct code for a discussion of cheating and plagiarism. First offense—F for the assignment. Second offense—F for the course.
NOTES TO THE STUDENT
Begin by orienting yourself. Read How to Study Calculus and How to Use This Book: Notes for Students on pages xix - xxi of your text. Get a rough feel for what we are trying to accomplish by perusing the chapter headings. Read the notes written to you by previous classes. (I asked the students to write these during the last week of classes.)
You can learn a little bit about calculus by reading the textbook, but you can learn to use calculus only by practicing it yourself. MATHEMATICS IS NOT A SPECTATOR SPORT! Do the assigned problems (and some that aren’t assigned). Keep a problem notebook. Ask questions. Make comments. Volunteer to do problems on the board when requested.
A graphics calculator is helpful in visualizing some of the graphs but remember that it only gives a window of pixels to view. My philosophy is:
a. Do analytically (with pencil and paper), then support numerically and graphically (with graphics calculator).
b. Do numerically and graphically, then confirm analytically.
c. Do numerically and graphically because other methods are impractical or impossible.
Read the text carefully. Have a pencil and calculator in hand. Study the examples—they are not templates for the exercises but examples of calculus ideas. Study the pictures—they are not illustrations or decorations but an important part of the language of mathematics. An ability to think pictorially—as well as symbolically and numerically—about mathematical ideas may be the most important benefit calculus can offer. Be sensitive to the language.
I will try to challenge you with interesting and relevant problems. The British philosopher and mathematician Bertrand Russell once said, "Most people would rather die than think—and most do!" Thinking is a dynamic process. To be a critical thinker you need to be truthful, open-minded, empathetic, questioning, active, autonomous, rational, self-critical, and flexible. You should be able to analyze and solve problems effectively; generate, test, and organize ideas; form, relate and apply concepts; construct and evaluate arguments; explore issues from multiple perspectives; reason analytically with concepts, relationships, and abstract properties; develop evidence and reason to support views; exchange ideas with others in a systematic fashion; apply knowledge to new situations; and become aware of your own thinking processes in order to monitor and direct it.
Consider forming a study group. Research has shown that students learn better if they work cooperatively in small groups to solve problems and learn to argue convincingly for their approach among conflicting ideas and methods. You will have several opportunities to work together in class on problems. Another bit of research points to the value of working together on homework. In the 1970s, Uri Treisman, a mathematician at the University of California at Berkeley, began an extensive study to determine why students did poorly in calculus. Currently about 40–50% of those who start calculus do not finish. Professor Treisman found that students who did homework in groups were far more likely to do well than students who worked alone. He observed students who worked in groups of three or more. One student would get an answer that was wrong; a second student would find the error and correct it. The process was repeated continually with the result that virtually all the students in the group understood how to approach the problem under discussion correctly. For students who worked alone, misconceptions went unchallenged. This led to a downward spiral of frustration and self-doubt. So I encourage you to work together on homework.
AMERICANS WITH DISABILITIES ACT
If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me and Wayne Wojciechowski, The Americans With Disabilities Act Coordinator (MC 320—796-3085 ) within ten days to discuss your accommodation needs.
Berlinski, David, A Tour of the Calculus, Pantheon, 1995.
Cohen, Marcus, et.al., Student Research Projects in Calculus, Mathematical Association of America, 1991.
Connoly, Paul and Teresa Vilardi, editors, Writing to Learn Mathematics and Science, Teachers College Press, 1989.
Countryman, Joan, Writing to Learn Mathematics, Heinemann, 1992.
Cruse, Allan B. and Milliane Granberg, Lectures on Freshman Calculus, Addison Wesley, 1971.
Davidson, Neil, editor, Cooperative Learning in Mathematics. A Handbook for Teachers, Addison Wesley, 1990.
Dick, Thomas P. and Charles M. Patton, Calculus, Volume I, The Oregon State Calculus Curriculum Project, PWS-Kent, 1991.
Dubinsky, Ed and Keith Schwingendorf, Calculus, Concepts and Computers, Preliminary Version, Volume I, West Publishing, Co., 1992.
Dubinsky, Ed et. al., Readings in Cooperative Learning for Undergraduate Mathematics, MAA Notes, 1997.
Edwards, C.H., Calculus and the Personal Computer, Prentice Hall, 1986.
Ellis, Wade, Jr., et al, Maple V Flight Manual, Brooks'Cole, 1992.
Finney, Thomas, et.al., Calculus: A Graphing Approach, Preliminary Edition, Addison Wesley, 1993.
Gordon, Sheldon P., Calculus and the Personal Computer, Prentice Hall, 1986.
Gresser, John, A Maple Approach to Calculus, Prentice Hall, 1999.
Harris, Kent and Robert J. Lopez, Discovering Calculus with Maple, 2nd edition, Wiley, 1995.
Hastings, Nancy Baxter, Workshop Calculus, volume 1, Springer, 1997.
Kaput, James J. and Ed Dubinsky, eds, Research Issues in Undergraduate Mathematics Learning, MAA Notes Number 33, MAA, 1994.
Leinbach, L. Carl, et.al., The Laboratory Approach to Calculus, MAA Notes Volume 20, Mathematical Association of America, 1991.
Marsden, Jerrold and Alan Weinstein, Calculus, Springer-Verlag, 1985.
Mett, Coren L., "Writing as a Learning Device in Calculus," Mathematics Teacher, October, 1987, pp. 534–537.
Pimm, David, Speaking Mathematically. Communication in Mathematics Classrooms, Routledge, 1987.
Reynolds, Barbara E., et al, A Practical Guide to Cooperative Learning in Collegiate Mathematics, Mathematical Association of America, MAA Notes 37, 1995.
Schoenfeld, Alan H., Problem Solving in the Mathematics Curriculum, MAA Notes 1, Mathematical Association of America, 1988.
_______________, editor, Student Assessment in Calculus, MAA Notes 43, Mathematical Association of America, 1997.
Steen, Lynn Arthur, editor, Calculus for a New Century. A Pump not a Filter, MAA Notes 8, Mathematical Association of America, 1988.
Stein, Sherman K., Calculus and Analytic Geometry, McGraw Hill, 1982.
Strang, Gilbert, Calculus, Wellesley–Cambridge Press, 1991.
Treisman, Philip Uri, "Teaching Mathematics to a Changing Population: The Professional Development Program as the University of California, Berkeley," in Mathematics and Education Reform, Naomi Fisher, et.al., (eds), American Mathematical Society, 1990.
Tucker, Thomas W. (editor), Priming the Calculus Pump: Innovations and Resources, MAA Notes 17, Mathematical Association of America, 1990.
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