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). Furthermore statistics is a special branch of mathematics which has its own specialized software.

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 standard Graphical User Interface (GUI) environment savvy (Windows or Mac) and exploration easily accomplishes. By using the palettes to build mathematical expressions and equations and quantities, and using the context sensitive menu on those mathematical inputs, a new user who knows no Maple syntax can do almost all the calculations needed for the calculus, differential equations and linear algebra sequence courses in which it is a required tool. However, a clear understanding of proper mathematical notation and vocabulary is required to use this tool effectively.

Any mathematical activity that is not able to be addressed in this way 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.

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.

Computer Algebra Systems

The natural tool for doing mathematics is a computer algebra system, of which there are two leading competitors: Maple and Mathematica. Villanova has an unlimited license for Maple and a limited license for Mathematica. Mathematica was created by one smart guy, Stephen Wolfram, and he built a corporate structure around it which early on was much more expensive than Maple. Maple was created by a group of educators at the University of Waterloo who morphed their software into an independent corporation, but one shaped by the university environment and pedagogical issues from the start.

Villlanova mathematics chose to use Maple in the late 1990s when it was much less user-friendly (command driven with no GUI-interface), but thanks primarily to an educator, Robert Lopez, the clickable calculus interface of palettes and context sensitive menus was developed, making Maple into a tool which reinforces standard mathematical notation and lends itself to relatively quickly for new users to become familiar with.

Other software

Software which is more specific not to understanding and learning mathematics symbolically, numerically and graphically, which a computer algebra system facilitates together with a fully typesettable text documentation mode as well, is available and in use here at Villanova, but not designed for the same goals.

MathCad is a numerical/graphical software with a graphical interface (GUI) aimed at solving concrete engineering mathematical problems. It has limited symbolics which is a bit awkward to use. For example, one can define a function f(x) :=x^2, but then to evaluate it, one assigns a value to x, namely x:=2, and then f(x) evaluates to 4, but standard math function notation f(2) does not work. All engineering students eventually learn how to use MathCad, though not immediately as entering freshmen.

MatLab is a non-GUI character mode command driven software with specific goals. Its name derives from Matrix Laboratory, namely instead of working with continuous functions, essentially (this is a simplification) it works with sampled functions for a discrete set of values of the independent variable, say x = {x1,...xn } so that f(x) = {f(x1),...,f(xn)} and one has a 2xn matrix to work with. Thus the first row of the discrete matrix representation of the function holds the x values and the second row the function values. Signals processing was one of the original motivations for this approach, and so it is important in electrical engineering. Only the electrical engineering students use it here mostly as upper classmen.

Excel is a spreadsheet that is really good at handling matrices of data in rows and columns and evaluating formulas either row-wise or column-wise, as well as fitting curves to data easily. This is useful for most STEM  students to be familiar with, often introduced in high school. It is great for instructors to manage grades, an activity is was really well designed for.

Other math software tools exist with different goals. Python and R are interesting examples of programming languages with no GUI interface, where computer code can be used to accomplish many tasks, not necessarily mathematics, although they can do certain specific math activities well. These can be used for statistical applications, along with the more traditional Minitab and SAS programs.