It is a truism that increasingly, most jobs will involve interaction with computers. As ubiquitous computing and embedded systems technologies break additional barriers, even the everyday and commonplace activity will involve interacting with a computing device. It is disturbing then, that so few people understand, and so little of our curricula address, fundamental principles such as those of data, information, process, algorithm, logic, relation, and in general, the discrete mathematics of finite objects. Except perhaps for those students planning on higher education in one of the sciences, such topics are arguably more important than the standard (high-school) mathematics curriculum. As concrete examples, consider the problem of making a successful search query on the world-wide-web using a standard search engine, or of properly "programming" the McDonald's cash-register keypad. The inability of the layman to set their VCR clock has been the butt of many jokes. The problem is indeed that the instructions are written in a different language. However, the language is not Japanese, but "computerese": the instructions are written by people who are familiar with the simple notions of finite state and of transition, but they are written for people who do not have such familiarity. Without these easily taught notions the user finds the machine's responses unpredictable as s/he cannot build an accurate mental model of its behavior. The uneducated user must hope that the instructions are extremely well-written, accurate, and cover all possible contingencies. For all but the simplest devices, such instructions are unlikely.
Unfortunately, current mathematical educational practices have not kept up with the conceptual demands made by our increasingly technological culture and workplace: Traditional mathematics education involves either rote activities, or "cookbook" applications of an algorithm or method to solve a problem (for example, how to multiply two numbers, how to find a volume, or how to take a derivative). There is little in the way of instruction on the understanding or design of algorithms, or on the process of deciding which algorithm(s) are appropriate in a given setting. The few CS programs in the schools tend to be aimed at the elite (AP courses for the best students), and typically address only computer programming. There is virtually no understanding of what the science of computing is, of its relationship to mathematics, and what it has to offer in the way of understanding the problem solving process.
It is no surprise then that among the first several standards articulated by the National Council of Teachers of Mathematics (Curriculum and Evaluation Standards) are those of mathematics as problem-solving, mathematics as reasoning, and mathematical connections. The NCTM standards appear to be written with the ideas from discrete mathematics, the design and analysis of algorithms, and the theory of computation in mind. Computer science and algorithm design is exactly about problem solving and about abstraction and modeling of problems. Connections are made to many other disciplines because problem-solving methods are required in all disciplines. The mathematics of computation and of algorithm incorporate a rich theory that need not rely on difficult arithmetic (for example, fundamental topics such as elementary set theory, graph theory, propositional logic, and recursion and induction are best presented (even at the college level) without equations and formulas that might obscure the key ideas), nor on past exposure to computer technology.
An underlying premise upon which this proposal is based is that in our everyday experiences in the world, in society, in Nature, problems arise from discrete mathematics, mathematical problem solving, and algorithm design and analysis. Moreover, each such problem comes equipped with a "knob", which can be turned to the extreme left (pointing at "K" for kindergarten), or to the extreme right (pointing at "R", for research). When the knob is set correctly for an individual, what emerges is a set of activities that are not qualitatively different than those enjoyed by research mathematicians and computer scientists. Finally, many of these activities, when presented properly, are great fun at any level. There is a growing collection of instructional materials indicating that this premise is more fact than fantasy (e.g., "Computer Science Unplugged", by Bell, Witten, and Fellows). The materials have been used successfully in classrooms from kindergarten through the undergraduate level, in different countries (successful experiences have been reported with these learning activities in areas of Peru with no electricity, let alone computer technology), by the materials' authors, by college professors and elementary school teachers. Some of the materials can be expanded vertically (increased difficulty on same topic for higher level grades -- up through still-open research problems that are easily stated) as well as horizontally (an incredibly rich literature from discrete mathematics and theory of computation, design and analysis of algorithms, from which spring many interesting and fun low-level activities).
Much of the importance behind this approach to education is articulated quite well by Casey and Fellows [Implementing the Standards: Let's Focus on the First Four. Manuscript, 1996] In short, addressing the NCTM standards, they argue that the standards and goals can be met by involving children in the same types of mathematical experiences which have captured the imaginations and interest of mathematicians and computer scientists. In particular, it is argued that:
The metaphor mentioned in the third item is compelling: Few would argue the importance of children learning to read and write, including the learning of proper grammar and spelling. To function in a literate world, these are necessary communication skills. Similarly, few would argue that the "two R's" (reading and 'riting) are the only skills our children need, and that the only reason such skills are needed is to communicate in day-to-day life. Instead, language is universally acknowledged as something uniquely human, as a means of artistic expression, as a precursor to civilization, and as a means of communication of complex and subtle experience. It would be unthinkable for a school to teach reading and writing without ever exposing children to literature, poetry, or creative or expository writing. Yet desirable mathematics education is often viewed as the simple acquisition of the last ``R'' ('rithmetic): skills to survive everyday life. The NCTM Standards break with past models of mathematics education by adopting the view of mathematics as the language of problem solving. Casey and Fellows argue what is well-known to working mathematicians and computer scientists: doing mathematics and computer science is a means of expression, a way of communicating complex and subtle ideas, and is often an artistic endeavor. Surprisingly, they also argue convincingly, and with experience in (and out) of the classroom to back them up, that these same rich modes of mathematical experience can be made available to the very young. Finally, because the problems addressed do not rely on typical rote arithmetic mastery nor past mathematics "success", differences in gender and social class are minimized. The theory comes alive because it has immediate motivation from interesting real-world problems, and because it is rife with puzzle-solving and other game-play to which children are naturally attracted.
The Main Goals of MATHmaniaCS:
To harness the excitement of some of the instructional materials mentioned above and develop new materials for other topics in discrete mathematics and computer science.
To facilitate the incorporation into the classroom of activities and materials that