Teaching Mathematics to Students of Chemistry with Symbolic Computation

We explain how the use of mathematical software improves the teaching of mathematics to, and its understanding by, students of chemistry, while greatly expanding their capabilities to solve realistic chemical problems. After an explanation of the need to improve this teaching and the opportunity with symbolic computation for this purpose, we outline the content of curriculum and its implementation, and provide examples of pertinent applications from thermodynamics and chemical kinetics. Why should we teach mathematics with computers? Students of chemistry find mathematics difficult: some students entering a postsecondary institution even select chemistry rather than physics because they think that they might thereby avoid much mathematics. Even while chemistry has become more mathematical during the past half century, largely because of an increasing prominence of statistics in analytical chemistry and chemometrics and of quantum mechanics in physical chemistry that diffuses into inorganic and organic chemistry, there has been a tendency for the number of courses in mathematics required of a student with chemistry as major subject to decrease significantly. For instance, at Simon Fraser University, in 1997 the requirements for chemistry as a major subject included five courses in mathematics – two first-year courses in differential and integral calculus, two second-year courses in multivariate calculus and linear algebra and a third-year course in differential equations; in 1998 the course on differential equations became no longer required. During the same period, the methods of undertaking calculations have likewise altered, in a progression from use of tables of logarithms and of slide rules, through pocket calculators with basic arithmetical operations, to powerful and large digital computers with software possessing ever increasing capabilities, eventually to ubiquitous graphic calculators for the pocket and computers on most desks and on many shoulders. Whereas before 1970 children in primary school learned how to extract square roots manually, since that era the topic has practically vanished from curricula: the standard method to calculate a square root now involves depressing an appropriate button on a calculator. Likewise, during the latter decades computers have evolved from being rare, huge and expensive machines devoted to mainly scientific and technological applications to become compact and inexpensive devices for which, at least in a 2 common domestic or commercial environment, technical applications are typically peripheral, even while their computational power and other properties have enormously increased. Within the same past half century there has been some evolution in the teaching of mathematics, from a formal and abstract approach based largely on theorems to a more pragmatic and less systematic development, and to service courses, with decreased numbers of courses or hours of classes notwithstanding their intent to cover material over an increased range. The use of computers in the present conditions is non-uniform: in some institutions courses are taught with greater or lesser invocation of computer algebra; in North America, the programs Maple (1) and Mathematica (2) predominate. Even within a particular university this practice might vary from one instructor to another; the result is that students progressing from one course to the next are subject to conflicting philosophies of pedagogy and disparate expected standards of competence related to manual or machine execution. In many cases, when computers have become involved, the content and delivery of standard courses have simply been developed in an isolated context, retaining a traditional sequence and scope of topics. Taking into account both the learning capabilities of students of chemistry and the heuristic applicability of computer software, we contend that a radical reorganization of the teaching of mathematics to these students is both timely and feasible (3). Our concern here is with the mathematical material typically taught by mathematicians, rather than the mathematics of chemistry, such as solutions of Schrodinger's equation for prototypical systems and ‘group theory’ or symmetry that are generally taught within particular chemistry courses. Programs for computer algebra or symbolic computation that operate readily on all current computers, even some devices small enough to fit in a pocket, not only possess embedded mathematical knowledge accumulated over thousands of years during the development of civilization but also might include material primarily directed toward the teaching of that knowledge. Moreover, new features are being continually added to some products specifically for instruction; instructors who were disappointed with software available for teaching purposes a decade or more ago should reexamine the current programs. We assert also that an holistic approach to the teaching of these mathematics at a postsecondary level is obligatory, so as to optimize the progress of a student through not only the newly encountered mathematical topics but also their implementation with the selected software: instead of merely trying to convert existing courses within a traditional pattern, we must consider the total extent of mathematical knowledge and capability reasonably expected to be acquired by chemistry students, and chart a course through that material in association with chosen software. The scope of applications is not only their immediate chemical courses but even their entire technical career to follow, for which undergraduate studies are a direct or indirect preparation; we must organize the content of mathematical courses accordingly. The teaching of mathematics that is strongly based on symbolic computation allows an instructor to explore a topic or principle according to four points of view: $ a formal statement is devised in words, just as according to tradition, but with increased emphasis on explanations of both pertinent terms and their inter-relations according to an accessible dictionary or encyclopaedia of mathematics;