Holding true to my promise to add more math, and reflecting the fact that class registration for the Spring semester begins next week at Wellesley, here are some thoughts on the math courses I have had that I find the most valuable for my work as a paleoanthropologist.
Probability Theory: Earlier today I tweeted that one thing elections (and election coverage) show that is relevant to paleoanthropology is that we could all use a little better understanding of probability. As a discipline that continually finds itself facing small sample sizes and non-normal distributions, correctly understanding probability and the mathematical properties of several commonly encountered probability distributions is essential. Hypothesis testing is all about understanding probability, and when the more traditional statistical tests associated with large, normally distributed datasets are unavailable, a good knowledge of probability helps greatly.
Calculus (differential equations): Evolutionary questions are, essentially by definition, interested in change over time. And if you want to mathematically address questions about change over time, calculus, or at least the ability to understand the concepts associated with differential equations, is unavoidable. A good working knowledge of calculus can be an important tool for turning a qualitative idea into a quantitative and testable hypothesis.
A lot of people are scared off by the idea of calculus and reflexively respond to the notion by suggesting that she/he is “not a math person.” That is crazy. I took an evolutionary theory class as a graduate student from Doug Futuyma, which was wonderful, in part, because it was math-heavy. This, despite the fact that Dr. Futuyma made it clear that he was “not a math person.” Math, particularly the concepts and language of calculus, are critical for systematically representing the actions and dynamics of evolutionary processes. We did not “do” a lot of math in the class, in the sense of solving equations or finding mathematical answers, but nearly every evolutionary concept we encountered we illustrated conceptually through mathematics.
Matrix Algebra: There are a lot of specialized mathematical approaches that can be useful depending on the exact kind of data or questions your research is focused on. In my experiences, matrix algebra, being able again to conceptually understand and operationalize the relationship between interconnected sets of data, has been highly useful. Again, there are simply a lot of evolutionary questions, particularly if you are interested in patterns of morphological variation, that can be broken down into matrix-algebra derived hypothesis tests.
That is it. I am sure a lot of people would, when asked what math class is most essential, would immediately suggest some kind of stats course. And indeed, the math class probably most required of anth/bioanth grad students probably is some kind of stats class. But given the varying kinds of data and questions encountered in paleoanthropology, most of what you would encounter in a standard stats class is not directly applicable. Instead, I would suggest that knowing the concepts that form the basis of most statistical tests is far more useful, because it allows you to move between different techniques and evaluate different kinds of methods based on the underlying principles and not just trust the results of a particular test.
UPDATE: And one quick update…the ability to program in some kind of usable/useful language is also enormously helpful. Plug and chug statistics packages simply are not much help for asking and testing evolutionary questions. You really need to be able to construct programs to work with data sets that have regular and varying “problems” (e.g. missing data) and write code for hypothesis tests specific to the question/scenario you are interested in exploring.
