Notes to the Teacher
Introduction [ Top ]
Calculus Concepts is not only a new calculus textbook, it is a new approach to the teaching of calculus. Instead of focusing on the algebraic manipulation of contrived functions, we focus on the interpretation of differentiation and integration as they relate to real situations. A derivative is a mathematical tool used to analyze a rate of change and an integral is a tool used to analyze an accumulation of change. Therefore, we do not view either as an end unto itself but as a means to interpreting the change that is occurring in the world.
Because of our focus, we spend more time than traditional texts developing the concepts of calculus graphically and numerically before we begin to use algebraic manipulation. Once we start determining derivatives and integrals algebraically, we use technology to help minimize the algebra necessary. It has been our experience, that once students understand the concepts of calculus graphically and numerically, they have little difficulty interpreting what is done algebraically.
We have attempted in this text to bring mathematics to life for the student. Too often students get the impression that mathematics and the real world are two disjoint entities. The majority of the activities in this book are based on real situations. You should point out to your students that anywhere data or an equation is footnoted in this text, it came from a real situation. You can further help your students view mathematics as a tool for analyzing the real world by having them look for data and equations that pertain to their own interests. You as well as your students will be surprised at what they find.
Since our approach to teaching calculus relies heavily on real situations, we must have a way of modeling those situations as mathematical functions. This modeling (although not calculus itself) is an important part of bringing calculus to life. Some of you who are using Calculus Concepts for the first time may think that we spend too much time doing non-calculus mathematics before we first mention the word "derivative," and others of you may think that if we discuss modeling at all we should go into more detail about the finer points of modeling. We do not pretend to teach data analysis. If a student wishes to learn more about the details of modeling, that student should take a statistics course that covers regression. On the other hand, we have included the minimum amount of modeling necessary to bring mathematics to life for the student. Without this time spent up front learning the basics of finding a model, we would be trapped into using contrived functions with which the students cannot identify.
Besides the use of technology and modeling and the focus on interpretation and conceptual understanding, another aspect of Calculus Concepts that you may find different is the emphasis on communication. Students should be encouraged to communicate their ideas and knowledge with their peers as well as their instructors. Therefore, we suggest the use of in-class group work, individual written assignments, and group papers and presentations as well as the standard homework, quiz, and test cycle. Further discussion concerning group assignments as well as testing techniques is included later in this guide.
Since Calculus Concepts presents calculus differently from traditional texts you may find the list of suggested tools as well as the suggested syllabi helpful in your lesson plans. Also included in this part of the Instructorís Guide are sections on introducing technology to your students and locating data sets.
There have also been many changes and additions to this edition of Calculus
Concepts. A lot of the changes and additions have been made in response to comments
from those of you who have been using the preliminary edition of this text. New material
and substantial changes to the text will be highlighted in the Chapter Notes section
of this guide.
Each student will need...
You will need...
There are two syllabi; each of the syllabi is based on forty-two 50-minute class meetings. You may wish to change them to fit into the schedule of your institution or to cover material at an accelerated or decelerated rate. Either syllabi can be adapted so that only the first semester of the two-semester sequence is taught.
Syllabus A is designed for the course that must cover through single variable differentiation in the first semester and single variable integration and multivariable calculus in the second semester. If desired, sine modeling (Chapter 8) can be interlaced throughout the first and second semesters as indicated.
Syllabus B is designed for the course that must cover single variable differentiation and integration in the first semester. Sine modeling, multivariable calculus, and differential equations, as well as some applications of integration are reserved for the second semester.
Finding Data [ Top ]
One question we hear over and over is "Where can I find data to put on tests or give as extra work?" Data may crop up in the most unusual places so keep your eyes open. However, some of the sources that we have found to produce a lot of data sets of different varieties are
Of course the list could go on.
You may be surprised at the last entry in the list. As a matter of fact, students
can be the best source of data. We have found that assigning students to find nice
data sets in their area of interest not only helps us increase our bank of data sets,
but it helps them to get a feel for how calculus may be applied in their own areas
of interest. Make sure that you tell your students to properly cite the source for
the data they gather.
This time spent at the beginning of the course will benefit you in two ways. First, it will help your students to become more familiar and comfortable with the calculator and consequently help them build confidence in their ability to use it effectively. Second, it will cut down on the time you would have to spend during the remainder of the course addressing calculator-specific questions and giving calculator-specific instructions.
A good way to introduce students to their calculators is to lead them through a worksheet similar to the one at the end of this discussion. The worksheet included here is specifically written to be used with the TI-83; however, it can be easily adapted to whatever technology you may be using. Some of the issues you should cover are:
Some instructors also recommend administering a technology quiz that must be retaken until passed in the first several days of the semester. This quiz is administered during non-class hours and serves to motivate the students to become familiar with the technology quickly. A sample of such a quiz is given after the Introduction to the TI-83 worksheet.
In addition to the time you spend at the beginning of the semester acquainting
your students with their calculators, you should also encourage your students to
refer to the Graphing Calculator Instruction Guide as they are reading the
text and working the activities. The Graphing Calculator Instruction Guide
gives examples and hints as to ways the calculator can be used to help analyze a
problem. It is designed to parallel the examples and activities in the text. An open
book icon in the text indicates that a parallel example exists in the Graphing
Calculator Instruction Guide.
The theme of Calculus Concepts is using and interpreting calculus in real situations. This theme should also be stressed in your teaching and testing. Carrying this theme over to your tests may require you to change your testing techniques. Instead of having "do it" questions on your tests, where the students are asked to find the derivative or antiderivative of such and such a function, try to use real situations and ask multi-part, interpretation questions such as the activities in the text.
Writing tests that will assess the students' ability to reason through and interpret calculus in the context of a real situation takes a little thought and effort. However, with some well thought out questions and a pre-determined partial-credit grading scheme, they will not require much more time or effort to grade than the traditional "do it" test.
Students should not be awarded points only on the basis of whether or not the final numerical answer is correct or incorrect. They should be granted only partial credit for a correct number. Credit should also be awarded for the correct interpretation of the numerical answer in the context of the problem. Does the student really know what that number represents? Can the student properly label the number?
Sample tests have been included in the Chapter Notes section of this guide to help you prepare tests that have multi-part, interpretation problems. Each sample test also lists an appropriate grading scheme for each problem.
You may have the idea that we do not like "do it" questions. Not so! "Do it" questions have their appropriate place -- quizzes. Students learn the mechanics of any mathematics best when they are forced to prepare for daily (or weekly) quizzes. Our experience has been that daily quizzes of "do it" problems during Chapter 4 and Sections 6.4 through 7.2 are a wonderful teaching tool to encourage the students to learn the algebraic rules of differentiation and integration. Of course, quizzes may be appropriate at other times as well.
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