I do math modeling opportunistically. That is I attack problems where I think I can make significant progress. I enjoy working on problems that have a substantial empirical base and where conceptual understanding has been minimal. I try to construct simple models yet models which "make sense" conceptually, and which have interesting and hopefully important empirical as well as analytical implications. There are many formal languages which are used for modeling. Virtually all of my work uses probability theory as the formal language. Sometimes such models are called "statistical models." Most of my work has been within the area of cognitive growth and development, although I have done work in psychophysics, meta-analysis and other areas. Lately I have become increasingly interested in artificial neural networks.
Students often seem baffled as to how it might be possible to construct mathematical models of psychological phenomena. The problem is with the perception: Anyone who has taken elementary statistics with a slant toward psychology has studied mathematical psychology. But students typically do not think of it that way. They view statistical procedures as "tools" for data analysis without a substantive interpretation. But any data analysis procedure should be viewed as a potential theory of behavior. For example, suppose we are interested in assessing how well a method of teaching reading improves students' achievements. Of course children vary in their classroom abilities. But would we expect all children to improve the same amount? That is, would it be expected that the amount by which the better students improve would be the same as the amount by which poorer student improve? Most people would probably say no, expecting the more able students to learn more. But commonly used methods for assessing change (e.g., the two sample t test) make the assumption that the incremental improvement is the same for all children, except for random variation. That is a theory of behavior!
Students sometimes think they could not become involved in such research programs. Not true! Students are be able to contribute in an active way to such research. If you think you might be interested, I would like to hear from you.
Most of the mathematics needed is in the undergraduate curricula. A beginning would include one year of introductory calculus, a course in linear algebra, and a mathematical statistics sequence (Statistics 414, 415, and perhaps 416 at Penn State) . Psychology 449 at Penn State, Introduction to Mathematical Psychology, introduces students to a number of different formal theories and attempts to help students view psychology from a more formal perspective.
Coombs, C. H. (1983). Psychology and Mathematics. Ann Arbor: University of Michigan Press.
Thomas, H. & Lohaus, A. (1993). Model ing Growth and Individual Differences in Spatial Tasks. Chicago: University of Chicago Press. (Monographs of the Society for Research in Child Development, Vol. 58, No. 9). Pages 1-8.
Thomas, H. (1995). Modeling class inclusion strategies. Developmental Psychology, 31, 170-179.
Thomas, H. (1996). Between sex differences are often averaging artifacts. Behavior and Brain Science, 19:2, 265.
Thomas, H. and Horton, J. J. (1997). Competency criteria and the class inclusion task: Modeling judgments and justifications. Developmental Psychology, in to appear later this year.