SIAM Life Sci/Soc. for Math Biol. Summer Biocalculus: Reflecting the needs of the students Stephen J. Merrill Dept of MSCS Marquette University Milwaukee, WI

SIAM Life Sci/Soc. for Math Biol. Summer Three Basic Approaches Calculus – the subject itself is fine. Adding artificial “bio-applications” would water down the course and make it like business calculus. Intro to Biomathematics – Calculus has shown itself to be useful in biology, let’s show them calculus through that context. Calculus for Biology Students – students have immediate needs to deal with their data – and data comes in columns.

SIAM Life Sci/Soc. for Math Biol. Summer Difficulties with each approach - 1 Calculus Students have no idea where the algebraic expressions come from. Algebra is central to understanding Calculus. Algebraic difficulties cause more failure than would be expected given the quality of the students. Students sell the books (they actually have no value to them). Instructors not able to connect the course to biology and as a result, students see it as a liberal arts course.

SIAM Life Sci/Soc. for Math Biol. Summer Difficulties with each approach - 2 Biomathematics Requires knowledgeable instructors Level of several topics (e.g. differential equations) is beyond what many students can immediately use and appreciate Importance of topics does not correspond to their biology courses (biomath has not made uniformly important contributions across biological topics) The idea of model is central to understanding calculus

SIAM Life Sci/Soc. for Math Biol. Summer Difficulties with each approach - 3 Calc for biology students Topics not as clearly defined as one does not have a canon as in The Calculus Lack of course material makes a knowledgeable instructor important Starting with columns of data, it is not clear where the “calculus” is (algebra is deemphasized) Potential lack of rigor in the course Somewhat dramatic change in point of view

SIAM Life Sci/Soc. for Math Biol. Summer Biocalculus at Marquette Before 1980, first option used (Biology students in the first semester of the general calc sequence) Math meeting with Biology (1979) Bio wanted a complete one semester 3 credit course (e.g. logs and exponentials). At least as rigorous as the general calc course. Course should have some contact with biological applications Option used ( ) was the Biomath approach, made possible by Biomath faculty and Ph.D. students. In 2000, Biology asked about other options to provide continuity with the second course, Biostatistical Models, and introduction of new topics such as bioinformatics, both were hard to add to a 3 credit course already full of content.

SIAM Life Sci/Soc. for Math Biol. Summer New Course (using 3 rd option) As rigorous as the general calc sequence Uses tables of numbers as the primary object of study (not algebraic expressions) Does not require an instructor expert in biology Provides natural transition to statistics Students see immediate applicability of the techniques and ideas to the data being collected in chemistry and biology lab – Excel is a primary tool for calculations. Course based on starting from data instead of starting from model. Discussions on what to do if the “data” is a sequence of letters are natural. But, can this be a “math” course with algebra minimized?

SIAM Life Sci/Soc. for Math Biol. Summer Things to remember In the applied sense, algebraic expressions are approximations derived from models or data. What is the “glory” of getting exact values of approximate quantities? Functions with algebraic formulas are still used as examples (often to generate a table of numbers). Function is still central, you just have sampled from it and may not have a formula Strong emphasis on equations of lines, interpolation, plotting (semilog, loglog) Data in other forms (such as sequences of letters) is easy to discuss

SIAM Life Sci/Soc. for Math Biol. Summer Examples Approximating the derivative Finding where the derivative is zero Antiderivatives

SIAM Life Sci/Soc. for Math Biol. Summer The derivative – some quotes The derivative is generally a theoretical construct, in that assumptions on the nature of a functional relationship must be made in order for one to expect the derivative to exist. This is the case because the limiting process cannot be duplicated exactly in an experimental setting. It is also the case that any (algebraic) form for a relationship between two variables is either arrived at through a model (which is approximate and simplified by its nature) or through a curve which "fits" the data, also an approximate process as the data contains "errors". If a functional form is available, as noted previously, there are rules for computing the derivative based on application of the limit theorems. It is rare to use them in practice, but some facility with them is assumed. For that reason, formulas for the derivative of a sum, product, quotient, and composition of functions will be given in the appendix.

SIAM Life Sci/Soc. for Math Biol. Summer Derivative Definition given. Using forward and backward differences, the slope of secant line is used to approximate. Using the idea of the linear approximation, the central difference approximation is derived and used in a spreadsheet (for equally spaced data) and a weighted sum of the secant slopes if not equally spaced.

SIAM Life Sci/Soc. for Math Biol. Summer Integrals and antiderivatives Easy to show relationship between Riemann sums and antiderivatives Standard methods for approximating integrals are used (trapezoidal rule) Emphasis on accumulation in the definition – derivatives are rates, integrals are accumulation, examination of the units in understanding the quantities.

SIAM Life Sci/Soc. for Math Biol. Summer Conclusions It is possible to give a calculus course minimizing the use of algebra A course built on data allows the students to immediately use what they learn (often the same day!) This approach uses tools familiar to the students (spreadsheets) and allows easy transition to statistics The ideas of calculus may be more clear in this context A cost is the students not able to continue in the calculus sequence (there are ways to deal with this)