Graduate College

Carlos Castillo-Chavez

Regents' Professor and Joaquin Bustoz Jr. Professor of Math Biology

Powerful mentoring strategies have been identified for generations and yet the fact remains that their effectiveness is primarily mentor-dependent. Individual mentorship philosophies are not built overnight, or identified from a directory of possibilities, or mass “produced” from a cadre of documented successful faculty role models. Mentoring philosophies are shaped and re-shaped with the assistance of tinkering “tools” aimed at increasing the likelihood that each advisor-advisee pairing, in a world of asymmetric student-mentor complex interactions, is successful. The evolution of my mentorship philosophy has been driven by my strong desire to increase the participation of underrepresented groups in the mathematical sciences. Tracing the evolution of this philosophy requires that the context and results of these efforts be described in this essay.

I joined the graduate faculty of Cornell University in 1988 and Arizona State University’s in 2004. I have chaired or co-chaired the PhD committees of 22 students [9 at Cornell University (1993-2008); 1 at the Universidad de Buenos Aires (1993); 1 at the Universidad del Valle Colombia (2006); 10 at Arizona State University (2007-2011), and 1 at New Mexico State University (2011)]. Students have earned PhDs (all theses including substantive mathematical/modeling components) in the fields of ecology/evolutionary biology (3), biological statistics/computational biology (7), applied mathematics (1), mathematics (4), and applied mathematics in the life and social sciences (7). This group of PhDs includes 9 women [4 international, 5 US a subgroup that includes 4 US Latinas] and 13 men [6 international, 7 US a group that includes 6 US Latinos]. Specifically, 10 US Underrepresented Minorities (URMs) have earned a PhD in 2005-2011.

My first two students, both international women, earned their PhDs in 1993, three years before the start of long-term efforts aimed at addressing issues of underrepresentation in the mathematical sciences. The success of the new program hinged on our ability to establish an environment that welcomes students not traditionally recruited by graduate programs in the mathematical sciences. The engine of this national effort was placed in the hands of an intense never-ending mentorship model that admitted students into a community of future PhDs in the mathematical sciences while they were still enrolled as undergraduates (at ASU since high school). The Mathematical and Theoretical Biology Institute or MTBI was born in 1997, a direct outcome of a successful first large-scale mentorship summer program that brought 36 undergraduates (30 US URMs) for eight weeks of intense research in 1996[1].

The evolution of my mentorship philosophy has since been shaped from my involvement with hundreds of undergraduates in a mentoring-through-research model linking high school to the postdoctoral level[2]. The wealth of experiences accumulated can be summarized as follows: over 70 summer research co-mentees have earned a PhD, a group that includes over 44 US URMs, since 2005. The timeline of my involvement as the advisor of my “PhD students” has changed as a result of this mentorship experiences. For example, I became the primary mentor of 8[3] (all US URMs, 3 Latinas) of my 22 PhD students while they were still juniors in college. The first MTBI undergraduate mentee earning a PhD under my supervision did so in 2005.

Mentorship Model[4] All my graduate students participate in MTBI’s summer institute every summer working directly for the program for about 10 hours a week, collaborating on research projects that lead to publications 50 hours per week in an environment that typically includes 20 undergraduates, 15 graduate students, 3 postdocs, 5 long-term regular visitors (all distinguished applied mathematicians), about 10 short-term distinguished visitors and a few young faculty wishing to develop their own research program in mathematical biology by directly experiencing this model. A common core of knowledge in the first three weeks of this sequential eight-week summer experience is taught—with strong participation of my PhD graduate students (part of the mentorship model). Some students have participated for seven consecutive summers—even when completing their PhDs at Princeton or Purdue or …

Common language: The first three weeks of the program are devoted to the study of dynamical systems in the context of ecology, epidemiology, immunology, and conservation biology. “Review" lectures are provided on the essentials of linear algebra, probability, writing some simple programs and areas where subgroups of students feel that they need help—guided by graduate students. The preparatory phase ends with a pre-project that forces the students go beyond the material covered in class. Students are involved in lectures, problem and modeling sessions and computational labs for an average of five hours per day supported by graduate students. They work in assignments and the pre-projects for extended periods every day including during the weekends. PhD students play a critical role mentoring students while being mentored by distinguished visitors, ASU faculty and myself.

Salt and pepper: Relevance seems to be the key to motivation and success. A modeling seminar is conducted twice a week by program alumni describing the process that they followed as participants in identifying and selecting their own project as well as in convincing a group of colleagues (three to four) to join in their efforts. Alumni put an emphasis on identifying/isolating a key research question - a process that precedes the selection of the appropriate modeling framework. Students are assisted by graduate students and resident faculty and highly encouraged to brainstorm together. Following the general Oberwolfach or BIRS model, the lectures, seminars and talks are followed by a community dinner where students interact with faculty, graduate students and visitors. Paper tablecloths serve a double function - they are also used as writing or drawing pads. Napkins are not sufficient in these learning communities. Graduate students over multiple summers develop their own research programs using these experiences as a model. This is novel, as most graduate students in math never develop their own research programs.

Absence of hierarchies: By design, the general research agenda of this summer institute is set by the undergraduate participants (graduate students work also independently on their own agendas with the support of resident faculty). Students try to sell their projects to two to three additional participants during the first three weeks. There are no rules regarding the formation of such groups except that they must include three to four individuals. Once the groups are formed students present orally their ideas and preliminary results to a group of faculty, graduate students and visitors—often twice a day. We all give input and suggestions. The initial role of these sessions is to help students narrow the scope of their initial, typically quite ambitious project. Project defining questions have included: What is the impact of alcohol on brain activity?  What are the dynamics of eating disorders?  What conditions will guarantee the survival of the monarch butterfly?  What are the effects of different social structures on the spread disease of HIV in Nigeria?  Once a sub-question that captures the essence of the students’ project is selected, efforts to build an appropriate model are carried out. We often have two or three meetings per day. Modeling challenges may move the students into the world of networks or stochastic processes or agent-based simulations. During this process, students are assigned faculty advisors and graduate student participants/mentors. The incorporation of the faculty/grad students is based on their desire to get involved in the enterprise. Most often, MTBI participants work on problems for which the faculty has no answer (often he/she does not know where to start). Students often see the MTBI faculty struggle. To sum it up, faculty, graduate students, and undergraduate participants become true scientific collaborators. The building of a research community around research challenges where being a graduate student or a professor do not put you on an intellectually advantageous position, is at the heart of my mentorship philosophy—it assumes that most students are smarter than me. My PhD students have identified their own PhD questions (not common in math) and taught themselves or with the help of the community the tools needed to address them.

Meeting expectations: Regular open meetings are conducted where each group presents and defends their effort; a system of rotation in the presentations that report on the group progress is established at the outset to guarantee that the project is a group effort. After three weeks a series of results (numerical, analytical and statistical) that shed some light on the question of interest are completed. Students then work hard on writing a technical report (25-45 pages) that captures the problem, the model, the methods, their results and their conclusions. Graduate students play a pivotal role on all aspects and since they often participated as undergraduates, their role grows year after year. Further, graduate students often take turns presenting to the audience of visitors and regular faculty the results of their own research (most often PhD dissertation work).

The product: The participants write a technical report (146 in fifteen years), which can be found at ( The students prepare a thirty-minute presentation and a professional poster. For example, the eleventh summer program began on June 6, 2006 and concluded on July 29, 2006. Seven groups of participants made oral presentations of their results at the joint meeting of the Society for Industrial and Applied Mathematics (Life Sciences Group) and the Society for Mathematical Biology--held in Raleigh, North Carolina from July 30 to August 4, 2006. Seven posters were presented. These posters were also presented at the annual SACNAS meeting in Tampa, Florida (October 27, 2006) and at the annual AMS meeting in January of 2007. Students regularly have presented their research at their universities and at local conferences during the academic year, following the completion of the project. An average of 2-3 awards per year have been given to MTBI projects. For example, in 2010, 10 technical reports were produced and a fraction was presented at the annual SACNAS meeting. The Los Fires poster won a SACNAS award on October 1, 2010. Graduate students do the same but with their own projects. As a result, we have students completing their PhDs with 2-9 publications in refereed journals, quite uncommon in the mathematical sciences. This model of mentorship has in many instances accelerated the time-to-completion of the PhD. Further, this model of mentorship has the inherent ability to generate resilient communities of learning and in fact, we have published a mathematical model of the building community process and proved that communities so constructed are highly resilient[5].

The most important component of the MTBI’s mentorship model is tied in to its intimate relationship with its community model. The MTBI community is a deliberately created social structure (see footnote five), which is critical to the success of students in the program and later on while they find themselves in graduate school. This community model is the cornerstone of my mentorship model. So how do I explain it using some theories of education?

Piaget: Our perspective on Piaget’s theory of learning is colored by our background in mathematical biology (Piaget’s himself had a strong biological background[7]) and seems largely concerned with the internal aspects of learning, specifically the ways in which children’s models of their experiences change as they develop. We see Piaget as modeling learning as a complex adaptive system. As the human body experiences stimuli, it begins to organize and anticipate stimuli, creating complex systems of mental actions and anticipated results in an effort to predict and control stimuli to generate more favorable results. Piaget described the process of intellectual development with the words “intelligence organizes the world by organizing itself.”

Piaget is sometimes considered to have focused too much on the individual, to the point of neglecting to detail the influence of environment and society on learning (Chapman, 1988). A strength of the MTBI community model is the diversity of its students and mentors who come from different cultural and educational backgrounds and bring varying degrees of mathematical and biological understanding. This inherent/innate diversity and the emphasis on collaborative work places students in an environment that is ideal for the building of robust understandings.

If we take as given that each student arrives to MTBI with their own unique packages of experience, expertise, and understandings, then the difficult homework assignments and research questions provide ample opportunity for those initial understandings to fail. However, without appropriate stimuli, there is no guarantee that student’s new understandings will be any more than incrementally different from their initial understandings. The community model provides opportunities for conversations that advance student understandings well beyond what a student would do individually.

We see the role of collaboration in this process as two-fold: that the collaboration of students with different perspectives and experience provides conversational stimuli for developing new understandings; and that the collaboration of students provides opportunities for students to take their own actions as objects of discussion and study, which facilitates reflecting abstraction – a process Piaget considered essential to learning[8]. Vygotsky, citing Piaget, describes a similar process: “Such observations [of child argumentation] prompted Piaget to conclude that communication produces the need for checking and confirming thoughts, a process that is characteristic of adult thought.”[9]

Unlike our perspective of Piaget as a biologist, we see Vygotsky as being primarily a social psychologist. Both Piaget and Vygostky studied the internal conceptual development of students and the environment influencing that development, but their focus was different. Piaget primarily examined the details of the mechanisms within the student by which learning occurs. Vygotsky’s focus was more outward, focusing less on the details of the student’s learning mechanism, and more on the details of the environmental effects, specifically social effects. In particular, we see Vygotsky as being focused primarily on how students grow into and become members of a pre-exiting society (Vygotsky, 1978; 1986[10]), or in Vygotsky’s (translated) words: “Human learning presupposes a special social nature by which children grow into the intellectual life of those around them,” (Vygotsky, 1978).

An influential component of Vygotsky’s theory of learning is the zone of proximal development (ZPD), based in part on the work of Piaget (Vygotsky, 1978). Vygotsky describes the zone of proximal development as the difference in the development level of a student working on their own and the developmental level of the same student working with assistance. Vygotsky illustrates this with the hypothetical example of two children, who on their own both work at the level of an eight year old, but with assistance one works at the level of a solitary nine year old and the other works at the level of a solitary twelve year old (Vygotsky, 1978). Vygotsky considered collaborative work in the zone of proximal development to be a critical component of the learning process. Vygotsky included both mentors and peers as collaborators for accessing a student’s zone of proximal development, and the MTBI collaborative mentorship model bears a striking similarity to this. Students work in collaborative groups with both the assistance of their peers as well as a broad spectrum of mentors in the form of returning students, graduate students, and faculty. Furthermore, the MTBI practice of giving students extremely difficult assignments targets these zones of proximal development precisely. The fact that students eagerly collaborate on these projects indicates that the problems are beyond their capacity to complete individually, while the fact that the students do complete their assignments satisfactorily indicates that the problems are within their ability to do collaboratively. This places MTBI assignments and research projects squarely in Vygotsky’s hypothesized learning zone and this set up hence provides an excellent model for training young researchers, particularly my PhD students.

We think that Vygotsky would have agreed that collaboration with a mathematician and collaboration with a biologist would enable a student to solve different sets of problems, as these two collaborators come from different cultures and areas of expertise. In MTBI’s collaborative groups the participants all work together to solve problems, with each participant contributing their own unique experiences and expertise to the collaboration. We believe that this makes the zones of proximal development for each participant exceedingly large, as each participant makes contributions to the zone of every other participant by the means of differing expertise. This enables the group to solve problems that are far more difficult that the group would have been able to solve as a student mentor binary pair. In fact, anecdotal observation shows that these zones are large enough that the students can and do frequently make contributions to the understandings of the faculty. Since most of my PhD students are engaged in multi-disciplinary research and plan to build interdisciplinary or transdiciplinary research programs, we believe that this model of mentorship is ideal for the school philosophy model championed by ASU’s President Michael Crow.

Although Bandura refers to his theory as “social cognitive theory” (A. Bandura, 1996) we find it useful to classify Bandura’s theory as a theory of behavior rather than as a theory of cognition. Although cognition certainly plays a role in Bandura’s theory, we see Bandura as more concerned with social factors that affect behavioral change rather than the details of mental activity and its development. Unlike Piaget and Vygotsky, who based their theories primarily on case studies, much of Bandura’s work is based on statistical analysis (Albert Bandura, 1977; Albert Bandura, et al. 1996[11]; Capara, et al 2000[12]).

We have only recently discovered Bandura’s work, but we find that taking Bandura’s theory, as a theory of behavior is particularly useful when examining the MTBI community mentorship model. The goals of MTBI, the recruitment and retention of graduate students in the mathematical sciences, are primarily behavioral goals. Although better mathematical and scientific understanding certainly play a role in the success of students in graduate school, Bandura’s work directly addresses issues of behavioral change and academic success relevant to these goals.

The two of Bandura’s theoretical constructs that we find most useful in studying the MTBI community model are the ideas of self-efficacy, and prosocialness (not discussed here), both of which Bandura correlates with academic achievement. On the topic of self-efficacy[13], distinguishes between outcome expectations and efficacy expectations, where outcome expectations are the person’s estimate that a certain behavior will lead to a certain outcome, while an efficacy expectation is a person’s belief that they can successfully engage in that behavior.

Self-efficacy, the belief that one can carry out certain behaviors, touches directly on the central mission of MTBI. In order for students to attempt graduate school, they must believe that they can successfully regulate their life as a researcher. The MTBI community model addresses this idea in two ways. Firstly, by requiring the students to work on challenging, self-selected and self-directed research projects, while mentors work primarily to insure that the students’ research goals are attainable. We recognize this method of building self-efficacy in the following quote by Bandura:

People motivate and guide their actions by setting themselves challenging goals and then mobilizing their skills and effort to reach them. After people attain the goal they have been pursing, those with a strong sense of efficacy set higher goals for themselves.”[14]

In above quote, Bandura specifically describes challenging goals that are set by the students themselves, which is precisely what an MTBI research topic is. The mentor works to make sure the student’s choose attainable goals, with the hope that upon succeeding, the students will then take on related, more challenging goals such as publishing a paper or writing a dissertation, which is precisely the process that Bandura describes. This is the way we move students from an “undergraduate mind frame” into graduate school mode rather rapidly and effectively.

The second way in which the MTBI community model builds self-efficacy for graduate school is in underrepresented minority recruitment. Although there is a great deal of diversity in the MTBI social population, there are also plenty of similarities. MTBI students need only look across the room to find an advanced student, a graduate student, or a faculty member who comes from a similar cultural, educational, or socioeconomic background. The student finds it easy to identify with these mentors, all of whom are engaging in research, attending graduate school, and succeeding. We feel that this “she’s just like me” feeling is critical to building the student’s belief that “I can do it too.”

We see Piaget as largely concerning himself primarily (but not exclusively) with the internal mechanisms of learning, from the sensory neurons inward; Vygotsky as largely concerning himself with the bridge between external and internal mechanisms of learning, by studying the effect of human interaction on conceptual development; and Bandura concerns primarily (but not exclusively) with the external, looking for the inner causes of external behavior.

Conflicts between these theories certainly occur, and the authors of these theories have occasionally critiqued each other. We find that on the whole, these theories are highly compatible if each is taken to be addressing a different domain of the problem of education and mentorship and this is what has helped me understand why the mentorship model that I use with my graduate students has been successful and effective. It is a community effort and I am only one of its members. My contribution has been effective in specific ways including those that facilitate the access to highly diverse human resources in ways that multiple aspects of the educational/mentorship process are simultaneously addressed.

What about MTBI’s national impact?

MTBI US Latinos/as alumni have earned 7 PhDs per year since 2005, with 3 in mathematical biology (MTBI alumni earned about 13 PhDs each year since 2005). The nation has awarded roughly between 1100 and 1200 PhDs in the mathematical sciences each year. The percentage of PhDs in the applications of mathematics starting in 1998 [US mathematical associations data] has been (10.69%; 8.83%, 9.47%, 10.02%, 11.92%, 10.52%, 10.95%, 10.39%, 12.77%, 10.54%, 15.30%, 11.19%). The average prior to 2003 was 10.2% while the average afterwards has been 11.7%. MTBI has contributed between 1.6 and 4.3 percent towards the national production of (US permanent residents or citizens) PhD’s in applied mathematics since 2005. There are no published data on the number of PhDs awarded in mathematical biology but a glance at the proportion of positions advertised in mathematical biology versus those directed to other areas of applied mathematics, suggest (if used as a proxy) that since 2005, no more than 10% of the PhDs in applied mathematics are in mathematical biology. Hence, it is likely that about 5% of the PhDs in mathematical biology awarded nationally to US students (citizens and permanent residents) are being awarded to MTBI alumni. The nation roughly awards about 20 PhDs to US Latinos/as per year, on the average, since 2005. If it is assumed that the percentage of applied mathematics PhDs awarded to US Latinos/as exceeds the national average of 11.7% by more than 100% (over the past five years), let’s say it is 25% then, one concludes that roughly 5 out of the 20 would be in mathematical biology. MTBI Latinos/as alumni have been collecting about 3 PhDs in mathematical biology every year since 2005, that is, over 50% of the PhDs in mathematical biology, awarded to US Latinos/as each year, are going to MTBI alumni according to the above rough calculation.

The list of MTBI instigated efforts, where elements of the MTBI mentorship model have been incorporated, includes:

  • Applied Mathematics Sciences Summer Institute ( established in 2005 by MTBI alumni Erika Camacho (2003 Cornell Ph.D.) and Steve Wirkus (Ph.D. Cornell 1999). They mentored 32 underrepresented minorities in two summers.
  • Ivelisse Rubio (Cornell Ph.D. 1998) and Herbert Medina (involved in first MTBI summer program) closed their successful The Summer Institute in Mathematics for Undergraduates, SIMU ( in 2002. It mentored 120 US underrepresented minorities over five years (1998-2002)
  • Establishment of the Cornell Summer Math Institute ( It was instigated by MTBI alumni (enrolled as graduate students at Cornell University) who wanted to maintain the MTBI mentorship model after it moved to ASU.
  • Summer Undergraduate Mathematical Sciences Research Institute (SUMSRI) established by Dennis Davenport at the Miami of Ohio University ( Its model was influenced by MTBI’s model.


[1] The American Mathematical Society recognized MTBI as a “Mathematics Program that Makes a Difference” ten years after its birth, that is, in 2007

[2] I was recognized with 12th the American Mathematical Society Distinguished Public Service Award for these efforts.

[3] I was recognized with the 2007 AAAS Mentor award for the high number of US URMs PhD students.

[4] Modified from Castillo-Chavez C. and C. W. Castillo-Garsow. "Increasing Minority Representation in the Mathematical Sciences: Good models but no will to scale up their impact," In: Doctoral Educations and the Faculty of the Future, Edited by, Ronald G. Ehrenberg and Charlotte V Kuh, pp 135-145, Cornell University Press, pp 135-145, (2009)

[5] Crisosto, M. N., C., Kribs-Zaleta, C Castillo-Chavez and S Wirkus, “Community Resilience in Collaborative Learning,” Discrete and Continuous Dynamical Systems B, Volume 14, No 1, pages 17-40, July 2010

[6] As I have tried to figure out why “my mentorship philosophy” works I have engaged my son, Carlos W Castillo-Garsow (an expert in mathematics education) on discussions that may help me understand why the community mentorship model has worked so well for 16 years. This section is the result of these discussions, if anything of substance is detected, must be my son speaking and if something is vague or possibly mistaken well it is due to my poor understanding of these theories. My son and I hope to write and submit an article on these issues once I can pass a minimal test in educational theories.

[7] Chapman, M. (1988). The seeker and the search. Constructive evolution: origins and development of Piagetʼs thought (pp. 11-30). Cambridge University Press

[8] Piaget, J. (2001). Studies in reflecting abstraction. Sussex: Psychology Press

[9] Vygotsky, L. S. (1978). Mind in Society. (M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Eds.). Cambridge, MA: Harvard University Press

[10] Vygotsky, L. S. (1986). Thought and Language. Cambridge, MA: The MIT Press

[11] Bandura, Albert, Barbaranelli, C., Caprara, G. V., & Pastorelli, C. (1996). Multifaceted Impact of Self-Efficacy Beliefs on Academic Functioning. Child Development, 67(3), 1206 - 1222. Blackwell Publishing on behalf of the Society for Research in Child Development. Retrieved March 1, 2011, from

[12] Capara, G. V., Barbaranelli, C., Pastorelli, C., Bandura, Albert, & Zimbardo, P. G. (2000). Prosocial foundations of childrenʼs academic achievement. Psychological Science, 11(4), 302-306.

[13] Bandura, Albert. (1977). Self-efficacy: toward a unifying theory of behavioral change. Psychological Review, 84(2), 191-215

[14] Bandura, Albert, Barbaranelli, C., Caprara, G. V., & Pastorelli, C. (1996). Multifaceted Impact of Self-Efficacy Beliefs on Academic Functioning. Child Development, 67(3), 1206 - 1222. Blackwell Publishing on behalf of the Society for Research in Child Development. Retrieved March 1, 2011, from

Current and Past Outstanding Doctoral Mentors