Mathematics is more than just arithmetic. It is about symbolic relationships between numbers and more complicated mathematical quantities based on numbers. Computer algebra systems (CAS) were designed as a tool for handling symbolic mathematical quantities and relations exactly, as well as in representing their numerical and graphical aspects, with mathematical word processing (verbal) later added as a bonus to weave all three of these aspects of mathematics (algebraic, numerical, graphical) together into a powerful platform for effective mathematical calculation and communication, backed up by an enormous amount of mathematical knowledge available to the intelligent user. Without understanding the mathematical concepts underlying its rich command structure, it is a useless tool—it does not substitute for learning the mathematical foundations, although it does free us from many routine and often pointless calculations, freeing us to give our attention to charting a path towards the solution of the real mathematical problems which confront us. It also gives us the opportunity to visualize many crucial ideas that can make dry mathematical notation come alive.
A computer algebra system (as opposed to other kinds of mathematical software) is the clear choice as the primary technology tool for aiding learning in a science/engineering based calculus, differential equations and linear algebra sequence, as well as in higher level mathematics courses (although more specialized tools are appropriate for advanced statistical applications). This is not to say that other choices of mathematical software are not also important in applications of this mathematics to science and engineering. Indeed Excel, MathCAD, and MatLab (all in use here at Villanova) all have important roles to play in various technical disciplines. Our choice of CAS is Maple, perhaps the most widely used CAS in college education (together with Mathematica, also in use here at Villanova).
Goals in the Educational Process
Our goal in using Maple in this course sequence (1500, 1505, 2500, 2705) is not to teach Maple for its own sake, but to use it as a tool in aiding your learning. Since some familiarity with a CAS is an important part of your background as a student in science and engineering in the 21st century, what you do learn about using Maple can and should prove useful even after you complete these courses.
In order to be able to use Maple as a tool to aid learning, it is important to get familiar with the most useful aspects of its worksheet interface which a little Windows environment savvy and exploration easily accomplishes. Any mathematical activity that is not able to be addressed by intuitively building standard mathematical expressions and referring to a few help tips should be guided with a template from your instructor or a textbook technology supplement. By asking your instructor when necessary, and referring to Maple help and MLRC help personnel when necessary, whatever difficulty you have in using Maple can be overcome. There is no need to be frustrated. Work with a partner, and if you get stuck, get help but do not waste time.
Fortunately in Standard Maple a reliable extremely useful set of palettes and context dependent right-click menus have been introduced ("clickable calculus") which enable a new user who knows no Maple syntax to do almost all the calculations needed for the calculus, differential equations and linear algebra sequence courses in which it is a required tool. Within Standard Maple, the Tools Menu, Tutors submenu combines these features into an even more powerful popup applet window for many basic concepts that allows parameters for the activity to be set in the window, delivering a final result to the worksheet, but allowing mouse selection, Control C copying, and Control V pasting of extra window content into text regions of the worksheet as well. Feedback from Maple helps unblock a student who does not see how to choose the next step in a multi-step Tutor activity. Similarly the Tools Menu, Tasks, Browse option allows you to browse the Calculus list of typical activities that are provided with worksheet examples that may be easily inserted into your worksheet where you need them.
While outside projects using Maple can enhance your ability to use Maple and show you how it can help you solve much more interesting and realistic problems, it is important to use Maple in selected homework problems from your textbook on a regular basis so that it can help you in learning the concepts. In order to be effective, these problems should be chosen carefully keeping in mind the goal of enhancing your learning of the concepts, although it is also useful to do some routine problems to get familiar with using certain commands to substitute for the "back of the book" odd answer key as a check on your hand calculations.
For general tips on using Standard Maple and example files for the MAT1500-1505-2500-2705 sequence see:
For student/faculty download and install of Maple see