How do you know where to begin?

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How Do You Know Where to Begin? C. N. Yang, in a n attempt t o describe the relationship between mathematicians and physical scientists tells the following story: A man carrying a bundle of clothing and searching for a place to do his laundry walked down a city street until he came to a store with a sign in the window. The sign read "Clothing Washed Here, 500 a bundle". The man entered the stare and asked the clerk hehind the desk how long i t would take to launder the clothing. The clerk replied "We don't do laundry here". The man protested and pointed t o the sign in the window. The clerk answered "We don't do laundry, we paint signs". So the dispute over the DurDoses of mathematics continues. In the undergr&iuete educa;ronofphy*ical scientists, this dispute contributes to an unfortunate effect: Cullrge students in [he sciences often grasp the operations of mathemat~cobut frequently miss the cunnectim hetwecn mathematical op. eratrons and the physical systems they can describe. Thus we we students who cannot tran4ete problems in chemistry or physics into the mathematical forms that will yield new knowledge {the answer', For example, i t is disrouraglng to find so many students who understand the mechanics of solving simultaneous equations hut who are unable to decide when and how to search for such equations in the solution of a problem. On other occasions, in approaching the solution to a problem, students may select the appropriate mathematical operation, hut then exhibit little awareness of its limitations. Thus many students use integration t o calculate enthalpy or entropy changes and integrate temperature-dependent functions through limits that include the temperature of a phase change for the substance of interest. The notion of limiting mathematical behavior and its connection to a conclusion obtained by extrapolation of experimental data eludes a surprising number of upper-level science students. For example, students frequently have difficulty in making this connection when they are asked to calculate activity coefficients from emf data describing the limiting-concentration behavior of electrochemical cells. The reasons for these developments are, I am sure, many and complicated. Discussion with my colleagues in mathematics and science suggests that the translation from words to mathematical symbols involves a leap of the imagination that manv people find hard t o make. While this mav he a n inherent diffieultv for manv students. one thine" seems clear: students LpeLience in high school and in the first itages of college too fen eonnert& hetween mathematics and the behavior of the real world. At Westminster College, we are heginning to recognize this and are moving caut~ourlyto ease the problem. Teachers from mathematics, physics, and chemistry are meeting and searching for common concerns in their teaching. From our discussions, we are learning bow differing emphases in the classroom of mathematicians and physical seientists often preserve the integrity of the disciplines but ignore students as they try to navigate the troubled waters hetween disciplines. We are hednning to realize that teachers of mathematics and ~hvsieal . . science.. hv" sharine the strateev and content of their classes, can heip science students grasp relationships between mathematics and science. As a result of these discussions, the sciences and mathematics faculty are beginning t o cooperate in the evaluation of the entering students' mathematical skills. Science and mathematics facult" are reviewing the mathematical diffieulties that science students encounter most frequently and exchanging this information with each other. Finally, science teachers are selecting specific groups of problems, requiring a range of mathematical approaches for their solutions. The mathematics faculty will review these problems and incorporate them into appropriate locations in their courses.We helieve these activities should help in two important ways: They will enrich the experiences of the science student in the mathematics classroom and will help insure that the science teachers use the t w l s of mathematics as the mathematician intended. Nowhere in this activity do we wish todimrnish the rtudents'appreeiatron for pure mathematics ar a fl~ghtof imagination guided by logical constraints Yet we know our science students will never develop their abdity rosolw pn,hlems until they find that ineffable answer to their own question: "How do you know where to begin"" ~~~

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Westminster College New Wilmington, Pennsylvania 16142

Robert P. De Siena

Volume 52, Number 12, December 1975 / 783