By David Bressoud
Nine chemists met at Macalester College in November 2000 for the MAA Curriculum Foundations Workshop in Chemistry. Their charge was to provide advice for the planning and teaching of the mathematics curriculum as it affects chemistry majors. Two of the participants (Craig and Engstrom) were members of the American Chemical Society’s (ACS) Committee on Professional Training, the committee that sets the requirements for the ACS accredited major. Three mathematicians (myself, Tom Halverson from Macalester, and Roger Howe from Yale) were present to answer questions and probe for clarification.
With remarkable efficiency and unanimity, the chemists identified and then fleshed out six themes that they feel are essential for the mathematical preparation of chemistry majors. Quotes are taken from the report cited at the end of this article.
Multivariable relationship?The mathematics requirement for most chemistry majors is at least two and often three semesters of calculus. Some students also take linear algebra or differential equations. Almost all problems in chemistry are multivariate. The most pressing concern that the chemists voiced was that their students see multivariable functions early and often so that they are comfortable with them. They noted that while a course in linear algebra is seldom required of chemistry majors, concepts of bases, orthogonality, and eigenvectors are important in chemistry and should be among the highest priorities.
Numerical Methods?These are at the heart of the mathematics used most frequently.
’Technology makes it possible to address old questions more quantitatively and more realistically than was possible in the past. The complexities of real chemical material can be approached more fully. [...] In general, solving these problems depends on multivariate analysis and numerical methods. Use of computers is assumed.â?
Visualization?Geometric visualization is one of the highest priorities for chemists. ’Chemistry is highly visual. Synthetic chemistry ... depends on being able to visualize structures and atomic and molecular orbitals in three dimensions.â? The chemists deplored the fact that ’geometry has been largely squeezed out of the secondary school curriculum. Little background in geometry helps explain why chemistry students have growing difficulty with the spatial relationships that are at the heart of much chemical thinking.â?
Scale and estimation?Chemists work across scales that range from subatomic to cosmic. Students need a working sense of orders of magnitude and the ability to do order-of-magnitude estimation.
Mathematical reasoning ?The chemists wrote: ’Students must be able to follow and apply algebraic arguments, that is, ’listen to the equations’, if they are to understand the relationships between various mathematical expressions, adapt these expressions to particular applications, and see that most specific mathematical expressions can be recovered from a few fundamental relationships in a few steps. Logical, organized thinking and abstract reasoning are skills developed in mathematics courses that are essential for chemistry.â?
Learning how to reason mathematically requires writing mathematics. ’Today’s mathematicians and chemists agree on the value of having students write to learn mathematics and chemistry.â? The report explains that this fosters critical thinking skills and builds student confidence in using mathematics as an active language.
Data analysis?Few chemistry students take a course in statistics, but statistical inference runs throughout courses in analytical chemistry and, to a lesser extent, courses in physical chemistry. The topics that chemists feel their students need include probability, combinatorics, distributions, uncertainty, confidence intervals, and propagation of error.
Calculus is still at the core of the mathematics that chemistry majors need, but the chemists pared the essential techniques down to integration and differentiation of polynomials, logarithms, exponentials, and trigonometric functions, differentiation of inverse functions, and integration by parts. Beyond these techniques, what they considered most important are the ideas of calculus: derivative as slope or rate of change, integral as area or accumulator, knowing what is held constant in a partial derivative, understanding the interplay of graphical, symbolic, and numerical interpretations, being able to read and write calculus as a language for describing complex interactions. And they expressed a profound desire that students not come out of calculus thinking that the variable has to be x and the function labeled f.
As in most science and technical majors, it is not possible to require more math classes. Chemists teach many of these essential mathematical topics on the fly within the relevant course. But the chemists present wonderful opportunities for mathematicians. A course that combined three-dimensional visualization with linear algebra and drew on the rich set of examples within chemistry would entice many of their students.
Norman C. Craig. 2001. Chemistry Report: MAA-CUPM Curriculum Foundations Workshop in Biology and Chemistry. Journal of Chemical Education 78. 582?6.
David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College. He was the local organizer for the Curriculum Foundations Workshop on Biology and Chemistry held at Macalester College in November 2000.