We believe our choice of Mathcad as the primary computing environment, has greatly contributed to what we perceive as a successful project. Our students overwhelmingly agree that Mathcad is easy to learn and use. It also is sufficiently powerful to handle most types of calculus problems, and flexible enough to allow for creating laboratory reports.
Having taught numerous sections of computer-based Calculus I (through techniques of integration) and Calculus II (through multiple integrals), we now have sufficient data to compare students in the computer-based sections with students in the traditional sections of calculus. While the common final examinations, administered for both computer- and non-computer-based sections, revealed no apparent difference in regard to the students' analytical skills, students enrolled in the computer-based sections do gain skills that those in the non-computer sections do not, such as:
Among the people with whom we shared news about our project, several reacted with disbelief to our claims about the virtues of Mathcad. It would typically go like this: "Mathcad? You must be kidding! I tried to play with it myself once, and couldn't get it to [[fill in the blank]], so I decided to just forget about it".
Admittedly, at first glance Mathcad can appear far from being friendly. The freedom given to the Mathcad user by the graphical interface employed by Mathcad, can make it difficult to know where to begin. However, this turns out to be one of Mathcad's strengths. The ability to intersperse text, graphics and mathematical computations in a 2-D layout is unique to Mathcad. Furthermore, the WYSIWYG output instantly allows students to see if they have entered a problem correctly (No time wasted on looking for mismatched parentheses!). From a pedagogical point of view, this allows even the weakest of students to focus on the real problem at hand: learning calculus.
Of course, Mathcad is far from being perfect. Its 3D graphics leave a lot to be desired, and it cannot do animations (these are the main two reasons why we also use Maple V in the second half of our Calculus II.) Also, its poor error handling and lack of any programmability do not help. Overall, however, we have yet to see an environment that would be better-suited for the purpose of teaching calculus.
We consider our laboratory "Mathcad Basics" to be a key ingredient of the project. This is the only lab devoted mostly (but not exclusively) to the software itself. Having completed this one week long assignment, students are familiar with about 80% of the Mathcad syntax they will need in calculus. The rest of it will be picked up during some of the other labs.
Students in our classes are neither provided with, nor use Mathcad manuals. If they forget something, they can:
We also believe, Mathcad makes an excellent tool for creating such activities and demos. Whether they are to be based on replaying or modifying an existing document, or to be created "live" by the instructor during the lecture (maybe in response to students' questions), Mathcad's use of the almost-standard-mathematical-notation, combined with its flexibility (mixing text, math and graphs, like one would on the blackboard) really make it shine. Of course, the instructor has to possess a certain amount of Mathcad skill to successfully conceive and carry out such activities. On the other hand, even if some typos creep in during a demo, it will only benefit students to see how the instructor corrects them. In the class notes included with this package, we have mentioned several opportunities for such activities and demonstrations. Clearly, many other demos are possible. While the time devoted to such activities can vary greatly, the complete absence of any such activities creates an atmosphere that can lead to the computer labs being perceived as irrelevant. An average lab assignment takes a team of two students from 2 to 3.5 hours to complete. (occasionally a lab will take a particular team as little as an hour and a half, or as much as 5 hours). At O.D.U., the initial 1-2 hours of a team's work on an assignment are done during a closed lab time, where the instructor and an assistant are present. Each team is then given a few more days to complete the assignment during the open lab hours (two assistants are present in the lab during all such hours.)
When the team finishes their work on the assignment, they submit it in electronic form to their instructor for grading (it could be done on floppy disks, but having a networked lab really makes most sense).
The instructor then uses Mathcad to grade the documents, and send them, still as Mathcad files, back to the student teams.
Students can then retrieve their file, are encouraged to correct their mistakes (Mathcad's instant recalculation comes in handy), and print the document for their record if desired.
While grading styles vary among professors perhaps as greatly as their teaching styles, there are certain universal considerations. Lab assignments are designed in such a way that students' active participation in an assignment should enhance their understanding of the subject matter. Acquiring such understanding is a process, which neither begins nor ends with any given lab activity. To reflect this, the grading scheme should impose significant penalty when a student ignores (or gives up on) a question, while not penalizing (or at least not severely penalizing) students whose answers are less-than-perfect, but who have made a genuine effort to solve the problem. Grading of students' assignments could in general be a very time-consuming activity, but if the instructor prioritizes his/her objectives, then a report on average can be graded in less than five minutes (for a section of 30 students teamed up into pairs this translates into 1 - 1.5 hour). As our project grows, we foresee the time when the grading of labs will be turned over to the lab assistants.
Please also note that 486DX (or 486SX with coprocessor) computers with 8 MB RAM are a recommended hardware minimum. These computers should be networked. A computer-driven projection system (LCD panel, 3-beam, etc.) is necessary for classroom demonstrations.
It is our intention to continue improving upon the project modules, therefore, we would appreciate comments which might help us in reaching this goal. It is for such helpful comments, that we would like to express our gratitude to several faculty members of the Department of Mathematics and Statistics of Old Dominion University, as well as our lab assistants. Please forward your comments and/or questions regarding our project to
Przemyslaw Bogacki (757-683-3262) or Gordon Melrose (757-683-3888)
Department of Mathematics and Statistics
Old Dominion University
Norfolk, VA 23529
Fax: 804-683-3885 (Attn.: P. Bogacki)