A specific aim of the computer module portion of the Mathematics Learning Center project is to provide the student enrolled in a given Calculus course with a review of the subset of material from a previous course which is needed to succeed in the current course. For example, at our University, a student in Calculus 2 needs to understand what a derivative is and to be able to take derivatives (from Calculus 1), but does not need an understanding of the concept of linear approximation. Newton's Method is occasionally needed but concavity is not. Curve sketching is needed but the Mean Value Theorem is not. The project's aim is to provide a quick review of those topics and only those topics, which form the essential background for success in the student's current course. The materials therefore in no way provide a review of the entire prerequisite course.
The reason for the limitation described above regarding the scope of the module at this level is strictly psychological. The student in need of the Mathematics Learning Center (MLC) is very likely feeling discouraged and overwhelmed. We want that student to be able to visit the MLC, either in person or via the web, and find a manageable menu of background skills, which he or she must master in order to succeed in the current course. If this list appears too long, the student will likely be discouraged instead of encouraged, and abandon the task as hopeless. A short list of topics to be mastered, each of manageable size, is an important goal of the project, especially at the freshman / sophomore level.
Consider the Mathematics Learning Center from the viewpoint of a student enrolled in Calculus 2, who suspects that some of the trouble he or she is having in that course is due to a lack of mastery of material from Calculus 1 (or perhaps a teacher has instructed him or her to review a specific topic from Calculus 1.) The student enters the Mathematics Learning Center, signs in, and boots up the machine (or if at home, accesses the Center via the web) and selects CALCULUS 2 from the menu. After viewing a short introductory page stating the goals and limitations of these review modules, the student sees the MAIN MENU of topics from Calculus 1 which are essential for Calculus 2. From the choices presented there, the student makes his or her selection. The choices are:
Let us suppose that the student is having trouble taking derivatives, and let us further suppose that it is the concept of the derivative which is confusing the student. The student would click on the first item in the MAIN MENU above.
NOTE: The remainder of this Level 2 section of the paper deals with how a student interacts with the materials in our web site. For maximum benefit, you should go to our web site and once there arrange the windows on your screen in such a way that you can view both this paper and the web site. In that way you can read along the paper while at the same time viewing what the student would view. Alternatively, you can get a printout of this paper and refer to that as a guide while viewing the web site. Most of the web site is still under construction, but access is permitted to the various parts referred to in this paper. Note that in order to view certain graphics you will need to use Netscape Navigator ver 4.x with the Quicktime plug-in available at www.apple.com/quicktime .
Please access our web site now, at www.umd.umich.edu/casl/math/sample/mlc2/index.htm and select the module entitled "The Concept of a Derivative" by clicking on it in the Main Menu. The written presentation of this concept that you see there is standard, subject however to the following two guidelines for computer-based learning at this level: 1) information is presented in as concise manner as possible to minimize the "many words on the screen make the mind wander" effect, and 2) the presentation is interactive wherever possible to engage the student in active participation in the learning process.
The total length of this module on The Concept of a Derivative is about eleven screens (each screen amounts to half a page in a textbook), and of these eleven screens, approximately half are devoted to interactive windows in which the student can manipulate what is happening on various graphs. One such interactive window shows a sequence of secant lines converging to the tangent line, while at the same time, the numeric value of the slopes of the secant lines, [ f(x+h) - f(x) ] / h , are displayed, together with the corresponding value of h. The student controls how many secant lines are shown, and when. After some initial play, the student is asked to estimate the value to which the slopes of the secant lines seem to be converging. A pop up window, again at the studentÕs command, tells the student whether or not his or her answer is correct.
Interactive windows illustrating the concepts via moving pictures which are under the student's control, are incorporated throughout the module to engage the student's interest. This potential learning and teaching technique is what can be used to advantage in computer learning (as opposed to a static textbook presentation.) Seeing things move and change is particularly important and appropriate for Calculus which is, after all, the mathematics of change and motion. Another example of interactive student learning comes later in this module when the graph of the function f(x) = x 2 is displayed and the student can then control a moving tangent line which rides back and forth along the curve. Not only is the moving tangent line displayed, but also the numeric value of the slope of this tangent line is displayed. In this way, students begin to realize that slope can be considered as a function of position. Examples showing student-controlled riding tangent lines are then repeated for two more functions, f(x) = sin(x) and f(x) = abs(x) / (abs(x) + 1).
The next step in developing the concept of the slope as a function of position, is taken through another interactive window closely related to the one described in the previous paragraph. This new window again shows the riding tangent lines, but this time it appends a second coordinate system directly below the graph of the function, and on that lower coordinate system, for each value of x, a dot is inserted at a height equal to the numerical value of the slope of the original curve at x. This is an important concept, since it is exactly the slope function, that is, the derivative function. It is conveyed much more convincingly via the moving graphs and interactive windows than can possibly be done using words and static pictures. When the student (and you) is finished with this module, click on the "Return to MAIN MENU" button at the end of the module. From the MAIN MENU, click on the other module available, "Taking Derivatives: The Four Basic Rules ( +, -, x, / )." This module reviews the Sum Rule, the Difference Rule, the Product Rule, and the Quotient Rule for students in Calculus 2 who may have lost some of these Calculus 1 skills over time. This would be especially useful for students who have had a long break between Calculus 1 and Calculus 2. At the beginning of this module, there is a 3 by 3 grid which lists the functions that the student should already be able to differentiate. The student can test his or her memory of these derivatives by clicking on the function, at which time the derivative is temporarily displayed. The rest of the presentation in this module is standard. There are five graded problem sets at the end of the module. These problem sets are slightly interactive in that the student is first queried about whether the basic form of the problem is a sum, difference, product, or quotient, and based the student's answer, a dialogue takes place to make clear what the proper approach is. Once the proper approach is chosen, the student is then asked to work the problem with pencil and paper. Upon request, for comparison purposes, the correct answer is revealed.