than others? Each of these questions could generate many hypotheses, and students can then go on to use Matlab to analyze the data sets they collect in order to evaluate the hypotheses. Some hypotheses do not relate to a biological area and are based on mathematics alone. For example, after linear regression is introduced, students are asked whether this regression can be reasonably used to determine the y-value for an x-value for which there are no data. This leads naturally to a discussion of interpolation and extrapolation.

As each topic is introduced, the instructor includes a brief description of how it relates to biology. This is often done by having a background biological example used for each main mathematical topic being covered, which can be referred to regularly as the math is developed. For example, in covering matrices, the material can be introduced with this example: “Suppose you are a land manager in the U.S. West, and you have satellite images of the land you manage taken every year for several years. The images clearly show whether a point on the image (actually a 500 m x 500 m plot of land) is bare soil, grassland, or shrubland. How can you use these to help you manage the system?” From this, the students develop the key notion of a transition matrix; the professor can then go on to matrix multiplication, and eigenvalues and eigenvectors for describing dynamics of the landscape and the long-term fraction in bare soil, grass, and shrubs.

Attempts are made to include real, rather than fabricated, data in class demonstrations, project assignments, and exams. For example, data of monthly CO2 concentrations in the Northern Hemisphere can be used to introduce semi-log regression, and allometry data can be used for studying log-log regressions. Students are encouraged to collect their own data for appropriate portions of the course, particularly the descriptive statistics section. Scientific journal articles that use the math under study are also provided.

Syllabus Math 151:

Descriptive statistics—analysis of tabular data, means, variances, histograms, linear regression

Exponentials and logarithms, non-linear scalings, allometry

Matrix algebra—addition, subtraction, multiplication, inverses, matrix models in population biology, eigenvalues, eigenvectors, Markov chains, ecological succession

Discrete probability—population genetics, behavioral sequence analysis

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