Christopher's...

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Mathematical Research Interests

My mathematical research interests involve applications of dynamical systems to epidemic modeling. More specifically, my interests include the transmission of Chagas' disease, applications to sociological phenomena driven by peer pressure (such as grassroots political movements, eating disorders, and cooperative learning), and theoretical models of bistability in population biology. Following are brief descriptions of some of these projects.

Dynamical Systems for Biological Modeling: An Introduction -- undergraduate textbook, 2015, Chapman and Hall/CRC Press, ISBN 9781420066418

Transmission of Trypanosoma cruzi

I first began studying the protozoan parasite Trypanosoma cruzi, which causes Chagas' disease in humans, during a year-long visit to the Universidad de Colima in Mexico in 2003-2004, supported by a Fulbright-Garcia Robles Scholarship. In addition to my collaborations with colleagues in Colima, I also now work with biologists in Georgia (US) and France who are studying the sylvatic (wild) transmission cycles which serve as reservoirs of the parasite. T. cruzi is native to the Americas, and Chagas' disease is endemic throughout Latin America, transmitted by a blood-sucking insect vector to humans as well as peri-domestic hosts such as dogs and sylvatic hosts such as raccoons, opossums, and woodrats. In the southern US, two different strains of T. cruzi are endemic: one, associated with opossums and the few documented human cases, is thought to be more virulent than the other, which is associated with raccoons. The two strains, which have distinct properties, are in competition with each other, and my research group studies different aspects of their spread and transmission in the US and Mexico, including the potential for geographic invasion driven by changes in climate and land use patterns. My T. cruzi research has been supported by grants from the Norman Hackerman Advanced Research Program and the National Science Foundation.

Sociological applications of dynamical systems

Through my long-time (since 1999) involvement with the Mathematical and Theoretical Biology Institute, I have developed an interest in applying modeling techniques developed for epidemiology to sociological phenomena driven by peer pressure. Both infectious diseases and peer pressure-driven behaviors are spread by contacts between individuals who are part of the process and individuals who are susceptible to becoming part of it, and the epidemiological metaphor has proven powerful in helping to understand why certain collective behaviors work the way they do. I have supervised research projects on cooperative learning, eating disorders, drug use, political behaviors, and the effects of so-called "three-strikes" laws. Many of these have been published as journal articles.

Backward bifurcations

In many instances where populations have variable susceptibility to infectious disease (that is, some individuals are at higher risk than others), the corresponding mathematical models exhibit a common phenomenon, referred to in mathematical epidemiology as a backward bifurcation, which implies that a particular phenomenon (usually an outbreak of an infectious disease) can persist in a population under certain conditions which would not normally (that is, according to its basic reproductive number R0) allow it to do so. Mathematically, this implies that there are multiple transition cycles within the population as a whole, and that although the phenomenon is not "reproducing" well within the population as a whole, it is reproducing sufficiently well within some subgroup of the population to create a "reservoir", if that subgroup begins sufficiently large. Biologically speaking, this means that these processes have a robust structure which makes them more difficult to eradicate. This complicates the control of such diseases (or other phenomena, such as eating disorders, which can be considered to be peer-pressure-driven), but also promises survivability of other phenomena with this type of structure (e.g., cooperative learning environments).

Publications


Educational Research Interests

My educational research interests focus on the development of mathematics teachers' attitudes, beliefs and teaching practices as they move toward instructional practices based on knowledge about children's learning. Many mathematics teachers, especially in the elementary grades, have had bad experiences with mathematics in their own education, and much research in recent years has gone into determining what (knowledge, experiences, skills) they need in order to be able to help children learn mathematics meaningfully and effectively. My joint appointment allows me to see both the content and the pedagogical sides of this issue. My current interests involve identifying the mathematics (content) particular to teaching, and observing how the changed view some students develop of what it means for themselves to do mathematics interacts with their later education coursework and K-12 classroom experiences to shape their approaches to teaching.

Publications


Teaching Interests

It was during the winter of 1990, while teaching electrical engineering lab courses at Georgia Tech, that I first began to realize that teaching was my calling. Since then, I've taught almost every semester, at either the high school or college level, and even gotten into curriculum design and teacher preparation.

My teaching interests focus on making students active learners in the classroom. It is a real challenge to bring one's students into a dialogue in the classroom, especially in natural sciences such as mathematics, and to get them to understand the motivation and justification for a topic. I have had some success in this area with a variety of different types of class. I am also a firm believer in anchoring mathematical knowledge to real-world applications (especially modeling, which is a big part of my own research). In calculus and linear algebra classes, it is natural to discuss engineering applications, and expose the students to design issues. In classes usually taken by future teachers, the applications to the math they will teach are typically a more integral part of the course. And with regard to both types of classes, I am also interested in bringing appropriate technology into the classroom: it not only puts more mathematical power in students' hands, but helps make it easier to discuss realistic applications and models as well.

Currently I am teaching courses to prepare future elementary school teachers to teach math: a sequence of courses in the math department designed to prepare them mathematically, and one in teacher education designed to prepare them pedagogically. Before pursuing my Ph.D., I taught high school science and math for a year, in a school for principally Native American students, so I bring that experience to both of these classes. My math students work in cooperative small groups to develop their problem-solving skills and intuitive reasoning. These activities give them a firm foundation for the math they will be teaching. I helped develop a sequence of such courses at the University of Wisconsin as a graduate student, and have done so here at UTA as well.


If you have questions about any of this, just drop me a line at kribs@uta.edu

Last modified October 13, 2010