LINK
CU/LTER Scholars Program in Biology and Applied Math for Undergraduates
Introduction
A central challenge to inter-disciplinary researchers and educational programs frequently seems to be the need to find effective, connecting links among the disciplines, hence our short title for this program LINK. Ideally, such a connecting theme needs to function as a strongly motivating factor for students and faculty alike, while simultaneously affording ample opportunities for focused 'slices' of traditional disciplinary work.
The central theme of this supplementary proposal to NSF can be distilled into the word model: bright, industrious young undergraduates + skilled disciplinary researchers from Applied Math and Ecology and Evolutionary Biology + a coordinated, closely integrated, inter-disciplinary theme + a broad, authentic, hands-on research program+ NSF support for students will produce excited and enriched students who can become research leaders in mathematics and biology.
In this proposal, we describe our plan to organize an extensive, deep, broad, experiential undergraduate research project for ten undergraduate students (five from biology and five from applied mathematics and engineering). We will employ several specific pedagogical strategies enumerated in the following paragraphs. Our over-arching goal for this supplement will be to graduate students who plan to become career research scientists and who leave this institution carrying an authentic understanding, a genuine appreciation for, and an abiding commitment to the investigative model employing links between mathematics and biological science (NAS Press, 1997). A longer term goal is to forge the links and establish the infrastructure that produces twenty such educated and highly motivated graduates per year.
Many thoughtful people and organizations have stressed the significance of undergraduate education along several different routes. The National Science Foundation, for example, expresses commitment in its strategic plan: 'NSF is determined that all students at all levels …are well and broadly trained … the process of learning does not end with the classroom. … The undergraduate level plays a pivotal role' (emphases ours). Further, from NSF's Shaping the Future, 'All students {should} have access to supportive, excellent undergraduate education in science, mathematics, engineering and technology and {that} all students {should} learn these subjects by direct experience with the methods and processes of inquiry ' (emphasis ours). The National Research Council publication (1992) focused on educating mathematical scientists argues that 'Early research experience through problem solving, experimentation or computation gives students a better idea of how to apply and create mathematics and often applies additional motivation.' Many other authors, e.g. Levin et al (1997), Hastings and Palmer (2003), have elegantly articulated and described some of the main threads of areas where the intersection of applied mathematics and biology are crucially connected. In our view, the case has been made to many scholars but, in contrast, that case has yet to be successfully made to a large share of incipient professional biologists. Applied math students typically hunger for real world applications and problems; many are quite enthusiastic about doing so in a biological context.
Part of the motivation for the recent emphasis on undergraduate science and mathematical education derives from the recognition that the American K-12 educational system, all too frequently, produces students who score near the bottom on international assessments (e.g. Third International Math and Science Study, 1999). Of even greater concern, for the purposes of this proposal, we point out that biology majors tend to be math phobic to a far greater degree than most other science majors, e.g., astronomy, chemistry, physics or engineering. This tendency to avoid mathematics may often be modeled by some (hopefully, a decreasing number) of established biology faculty which significantly weakens the discipline of biological science. Students who are well-schooled in both biology and mathematics do, indeed, have an exceptionally bright future (Hastings and Palmer, 2003).
A second thread of motivation for this response to the NSF initiative is to strengthen the quality of the undergraduate science experience at CU-Boulder. In all-too-many cases, the mode of instruction can be pejoratively labeled in bumper sticker style as 'the stand and deliver model' or 'the sage on the stage model'. To the best of our knowledge, all serious investigations of science pedagogy support the following as highly desirable strategies: (1) directly involve the students in solving authentic, hands-on research problems, (2) challenge them to think creatively, (3) require them to communicate their thoughts orally and in writing, (4) mandate they assimilate and integrate seemingly disparate bits and threads of knowledge, and (5) expect them to test their knowledge for validity and robustness. Further, a strong case has even been made (Seymour and Hewitt, 1997) that a major reason why we have 'under represented groups' (women and minorities) in science and mathematics explicitly derives from the traditional weaknesses in science education methodology.
Biological and Mathematical Thinking
Biological and mathematical sciences have made enormous progress over the last several decades in developing deeper, more complete understandings of fundamental processes of many areas of biology (Levin, et al, 1997). The general intellectual framework for the vast majority of the biological work over the past century can be fairly characterized as analytical in mode of thought: isolate, separate, and characterize component parts of a complex system. Analytical, in this sense, implies a taking apart, a disassembling in order to simplify, focus and, hopefully, to understand.
Over these last few decades, an exciting, powerful new theme in science has begun to emerge, capturing the imaginations and energy of an ever-expanding number of research scientists: this theme may be labeled as a synthetic 'systems' approach to {biological} science. Among the principal intellectual threads of this movement are: (1) an emphasis on complexity in both the intuitive and the mathematical senses, (2) an explicit framework of investigation which studies the integration among component parts, (3) a recognition that such interaction among parts of a system is almost invariably non-linear, (4) an understanding that those interactions constitute core properties which cannot be justifiably ignored, and (5) that the intellectual tools that successful research 'synthesists' will need for the foreseeable future include a strong background in and explicitly training in mathematics and computer experimentation to complement and enhance more traditional areas of biological research training. In a complementary fashion, mathematicians are looking increasingly toward biological systems for ideas, processes, and systems properties to inspire their mathematical thinking and modeling (e.g., Arbib, 2003).
This theme of interaction, of non-linear dynamics, of mathematical rigor, of 'Life at the Edge of Chaos' (Kauffman, 1992) can best be instilled early in the developmental trajectory of all students to include the potential scientist. Undergraduates come to the academy, and to research opportunities, without the well-developed walls of specialization or focus and without the all-to-human vested interests of a particular research theme, method or program sometimes found in more experienced faculty. While certainly not mentally tabula rasa, they tend to be quite open, even merrily enthusiastic, about crossing traditional disciplinary boundaries.
We propose to capitalize on this open-minded undergraduate enthusiasm, our experience as faculty, NSF support, and our commitment to experiential research models of biological science and applied mathematics education.
We propose to organize, co-ordinate, direct, and lead a vigorous, rigorous program wherein ten students recruited from the following traditional academic majors of our campus—from Ecology and Evolutionary Biology and Applied Mathematics in particular, but also potentially from Molecular, Cellular and Developmental Biology, Computer Science, and Biomedical Engineering—will form a cohesive, integrated, interacting cohort of undergraduates who engage in sustained research scholarship throughout their undergraduate years and shape their formal coursework according to long-term plans of becoming research scholars in mathematical biology.
We anticipate that each undergraduate will become at least a junior author on one or more scholarly publications for first-line journals by the time they graduate and to have participated in at least one scientific conference to present some of their work. We would also expect such students, upon graduation, to be in high demand by the best graduate programs in the country.
An Integrating Theme
We propose to organize this undergraduate research cadre around the broad general themes of of the NSF supported Long Term Ecological Research program, especially at Niwot Ridge. At the outset, however, we fully recognize that the goals of this proposed project do not include that of producing finished, professional biomathematical researchers --such a goal would be clearly unrealistic. For any of our students to reach that level of training, completion of a Ph.D. program supplemented with a post-doctoral experience will be needed. Rather, this project aims to utilize the framework of LTER to organize thought, to direct student projects and to offer insights into general problems common to many major biological and mathematical research themes. The LTER themes should provide ample opportunity for appropriate student research for all of the recruited LINK students and permit them to identify one or more pathways of approach which would stimulate their personal enthusiasm for knowledge while encouraging them to employ their own individual talents and skills in an authentic, research-oriented context. Further, the advantages of a multi-disciplinary strategy should be prominent. Faculty counsel and guidance will be crucial here, especially in the early stages of the project, hence the sustained weekly meetings with the entire cohort to be described shortly.
As this is a request for supplementary funding under a current NSF LTER grant, we include a brief section from the proposal itself, quoted below.
“DEB 9810218: The Niwot Ridge Long Term Ecological Research
Program 1998-2004: Control on the Structure, Function and Interactions of
Alpine and Subalpine Ecosystems of the
INTRODUCTION AND CONCEPTUAL FRAMEWORK
The importance of alpine tundra to regional biogeochemical and atmospheric
processes is far greater than that indicated by its small area. The temperate
alpine and subalpine region of
The research proposed here is a continuation of our current theme: to understand the influence of increased snowpack and atmospheric N deposition on ecosystem processes and landscape patterns, with increased emphasis on the biotic feedbacks and ecosystem responses to these changes. As detailed below, we now propose to exploit more fully the long-term data sets available from the pre-LTER, LTER and non-LTER data collected in the greater Niwot Ridge area, and to include not only the alpine but the adjacent subalpine ecosystems as well in our analyses. Our goals are to 1) provide a hierarchical perspective to the ecology of high elevational ecosystems, 2) explore the coupling
between the alpine and the subalpine, and 3) document the biotic responses to anthropogenic influences occurring within each ecosystem and compare these responses between ecosystem types. “
Example Possibilities for Student Projects
We have identified a short list of examples for illustrative purposes only. The actual scope greatly exceeds what we articulate below and those individual research threads will need to be developed jointly with the students during the fall semester, 2003. Biologists tend to lean toward a mental framework of conditional, context-specific conclusions, often with many caveats; mathematicians more commonly seek conclusions with broad, exception-free generalities. Each outlook has strengths and weaknesses with which, our students (and we, too!) will need to grapple regularly.
(1) The importance of scale in biology has become
a topic of great interest over the last few decades, partly because of
mathematical insights such as fractal dimensions, (West et al, 1997, van
Gardigen, 1997). The LTER work covers
several orders of spatial-scale magnitude ranging from isolation of DNA
fragments from soil Archeae (10-6
m) to complete river systems (106 m) and beyond. Studies which span several orders of
magnitude in time scales also take place.
Traditional biological training does not well equip our undergraduates
to deal with the essential mathematical nature of these kinds of data sets
either in a statistical analysis framework or in a dynamic modeling
framework. We anticipate that both of
these themes will be chosen by some of our undergraduates and they will have
abundant opportunity to shape their formal course work for necessary academic
training and for tackling genuine research questions at the same time.
(2) Computer simulation and graphical analyses provide one of the most promising of all themes of mathematical biology in that approaches such as agent based modeling, parameter manipulation and experimentation of predictive models and the ability to run ‘experiments’ over thousands of generations provides opportunities patently impossible to accomplish in an empirical, real-time manner. Even non-linear dynamical systems can be modeled, explored, validated, tossed-out where analytical approaches might prove completely intractable (Kauffman, 1992, Epstein, 1997).
(3) An important element of the multi-disciplinary approach will entail an emphasis on broad awareness of different tools of study. Given that the LTER approach contains very strong geographic components, we will likely encourage some of the cadre to employ the rapidly developing computer technologies generally falling under the rubric 'Geographic Information Systems'. Basically these methodologies employ spatially relevant data to do topological and spatial analysis in a multi-dimensional context. Overlay analyses, for example, of temperatures, rainfall, El Nino effects, land use, and topography onto other biological traits of interest can provide valuable insights. It is also true that biologists tend not to be exposed to a wealth of spatial analyses techniques commonly employed by geologists and geographers but which would be potentially valuable and powerful in these contexts.
For example, we have very
active links with the McMurdo Dry Valleys LTER to provide opportunities for one
of biology’s most powerful techniques: comparative analysis. The McMurdo Dry
Valleys LTER project has been conducting integrated research and monitoring in
the
(4) The applied math and
computer science students are likely to be naturally drawn to approaches
employing non-linear computer models, Bayesian inference, and semi-Bayesian
techniques in spatial or temporal analysis contexts where signal to noise
ratios tend to be low. Applied
mathematicians have become increasingly aware of the 'realities' of biological analyses which commonly have
incomplete, censored or poorly resolved data points and are developing and
employing procedures to effectively deal with these problems. A simple, but illustrative case study was
recently provided by the so-called 'Seahenge' research project. After discovery of a group of submerged timbers
in the
(5) Specially designed projects. In addition to the regular LTER program and myriad opportunities therein, we will provide the students with several specially designed projects to develop the LINK concept. Here is one such example.
In-stream conservative tracer experiment
One way to promote cohesiveness
within the LINK student cohort is to have them conduct an experiment together
in a stream ecosystem. Streams integrate the inputs from the watershed and are
a clear, distinct system that can be studied as a “snapshot” in time in a way
that emphasizes complex spatial connections. The multiple goals of cohort
cohesiveness and integrated math and biology experience can be enhanced by
conducting an experiment in which a conservative chemical tracer is injected
into a stream system for a period of several hours. The stream will be a small
headwater stream in the sub-alpine zone, possibly Como Creek which runs through
the
Stream discharge: The students will
(1) use the conductivity data to determine stream discharge at the downstream
points and (2) compare that calculation with discharge measurements made at the
same time using the area-velocity approach and (3) the velocity in a culvert on
Como Creek. These three methods involve different equations and underlying
assumptions which will help the students learn about quantifying physical
processes in natural systems.
Q is the discharge, Ct is the initial
tracer concentration, Cb is the background concentration of the tracer
in the stream, Vt is the volume of tracer added to the stream, and Cd(t)
is the tracer concentration measured at the downstream site over time. By
examining the physical meaning of the integral of conductivity over time, we
know that value of the denominator in the above equation can be derived from
area that is bounded by the conductivity curve on a conductivity-time graph,
where the downstream concentration, Cd, is plotted on the y-axis, verses
time on the x-axis.
Hyporheic exchange: The students
will use the data and a one-dimensional solute transport model with storage
computer code (OTIS) to quantify the hydrologic interaction of the stream with the
area under and adjacent to the stream, called the hyporheic zone.
This process can be modeled for a
conservative solute as:
where C is the tracer concentration in the stream, U is the velocity in the stream, D’ is the dispersion coefficient, a is the exchange coefficient, A is the cross-sectional area of the stream channel and As is the cross-sectional area of the hyporheic zone. The exchange of water between these zones strongly influences the biogeochemical processes in the stream, especially for nutrients. The existence of a hyporheic zone can be seen in the water quality data (conductivity and Na and Cl concentrations) from the experiment as the attenuation of the pulse from the tracer injection. For example, after the injection has stopped, the Na and Cl stored in the hyporheic zone will bleed back out into the stream. The students will then be able to use values for input into the OTIS computer model and evaluate the dependence of the output on the time step and spatial step selected as well as be able to manipulate other physical parameters such as dispersion and stream bed roughness (Manning’s n).
Program Mechanics
This proposal will be
exclusively directed toward undergraduate majors who profess a strong interest
in biology and mathematics. We will recruit,
initially, sophomores and juniors who have demonstrated strong academic success
with careful attention to student populations which are under-represented in
science. The campus has active programs
for Engineering as well as Arts and Sciences students through which such
students can be recruited. Similarly, we
have an academic honorary society, the
Initial faculty supporters
include Dr. Timothy Seastedt, Principle
Investigator for the Niwot LTER. Co-PIs for this supplementary request include
Dr.
We will expect LINK students to work full time during eight weeks of the summer for which they will receive both an education and a stipend (from which they will have to pay their own expenses). We envision the students working largely on the main campus in our excellent computational laboratories exploring, examining, analyzing and modeling of results being provided to them by various parts of the LTER program. We also envision them actually participating in some field work directly although empirical data collection will not be a major theme for these students—analysis, interpretation and modeling will be the main theme. We recognize that they will need substantial faculty mentoring, guidance, and supervision, especially during their first summer of research work.
We expect each student to continue their research project work during the academic year and will have weekly meetings led by one of the Co-PIs (mostly Grant) in which students will report their progress, interact with each other as research colleagues, identify gaps in their training which we, as faculty, will try to address and generally stimulate their interest in mathematics and biology.
In addition, we anticipate that
we will have more formalized Chautauqua-like workshops on specific topics
taught by
Assessment of the student's progress in LINK will be continual, mostly in the context of the mandatory weekly meetings of the whole group but also by the supervising faculty from the various disciplines. The student’s academic scholarship, research productivity, scientific maturation, motivation levels and future plans will be continuously examined using a regularly updated 'dossier' approach for each student. Students and faculty will regularly contribute. Individual future career plans will constitute an important part of the dossier. We will expect (perhaps over optimistically) every student to graduate and pursue graduate training in a mathematically oriented field of biology or applied mathematics with special interest in biology.
We expect that some of our LINK students may want to take advantage of CU’s dual BA/MA program in which undergraduates can simultaneously earn a baccalaureate and a Master’s degree in five calendar years (they remain formally undergraduates until completion of the entire program). This opportunity would mesh very well with the need for students to take a substantial undergraduate mathematics sequence and a substantial undergraduate biology sequence. Further, the Dept. of Applied Mathematics and the Dept. of Ecology and Evolutionary Biology have previously held faculty discussions about constructing special courses and special tracks for students interested in joining these two majors. A similar coordinated track in applied ecology that was implemented through the Dept. of Ecology and Evolutionary Biology and the Environmental Engineering program has already had about ten seniors graduate and was approved through ABET. This LINK program could obviously serve as a very valuable stimulus both to the students and to the faculty.
We faculty are excited about this potential opportunity and expect to be able to generate a good deal of excitement in our students as well, if funded.
Literature Cited
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Hastings,
A. and M. Palmer. 2003. A bright future
for biologists and mathematicians? Science 299: 2003-2004
National Research Council. 1992. Educating Mathematical
Scientists: Doctoral Study and the Postdoctoral Experience in the
Seymour, E. and N. Hewitt. 1997. Talking about leaving: Why undergraduates leave the sciences. Westview Press, Boulder, Co.
Schmidt, William. 1999. http://nces.ed.gov/timss/
Van Gardigen, P.R., Foody and P.J. Curran. 1997. Scaling Up:
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