SUMMARY REPORT WORKSHOP ON A QUANTITATIVE SCIENCES CURRICULUM FOR LIFE SCIENCE STUDENTS* KNOXVILLE, TENNESSEE FEBRUARY 6-8, 1992 OBJECTIVES: The Workshop was designed to bring together a group of researchers and educators in mathematical and quantitative biology to discuss the quantitative component of the undergraduate curriculum for life science students. The key goal of the overall project of which this Workshop is a part involves the development of a curriculum that would emphasize the great utility of quantitative approaches in analyzing biological problems, use examples from recent biological research, and serve the dual role of introducing new quantitative methods and reinforcing key concepts in modern biology. The main goal of the Workshop was to develop a set of basic quantitative and biological concepts that should be included in the curriculum. Specific questions addressed at the Workshop included: 1. What are the possible goals for an entry-level quantitative course for life science students? 2. What are the appropriate biological concepts we need to include to meet these goals? 3. What are the appropriate quantitative concepts we need to include to meet these goals? 4. How do we design a modular course flexible enough so that it could reasonably be taught in three different modes: through a mathematics department, a biology department, or co- taught by both? 5. What are the most appropriate advanced topics around which to build follow-up courses for upper-division students and what should be the content of this set of courses? 6. What is the role of technology in all of the above, and are there particular software packages/programs which will be most useful in supporting various aspects of the curriculum? SUMMARY OF CONCLUSIONS: 1. It is not sufficient to isolate quantitative components of the curriculum in a few courses on quantitative topics, but rather the importance of quantitative approaches should be emphasized throughout the undergraduate curriculum of life science students. This implies that courses typically considered part of the biology curriculum should contain quantitative components appropriate for the topics addressed in the course. Thus we should encourage the introduction of quantitative skills at all levels in the life science curricula. 2. As a means to foster the inclusion of more quantitative topics in the curriculum, it is proposed that a Primer of Quantitative Biology be developed to be used in conjunction with the typical General Biology sequence included in most life science curricula, with appropriate quantitative examples developed for each section of the course. This Primer would be at the level of high school mathematics, but would focus on examples of non-intuitive results of biological importance derived from quantitative approaches. 3. Exploratory data analysis should be included in several ways as part of a life science curriculum. This can be done as (i) part of laboratory exercises within a biology course; (ii) a short-course available for credit ; and/or (iii) a formal biometry course. The last option should be constructed around key biological questions, rather than statistical methods. * Supported by the National Science Foundation's Undergraduate Course and Curriculum Program through grant #USE-9150354 to the University of Tennessee, Knoxville 4. An entry-level quantitative skills course should be developed as a specialized year-long sequence for life science students. Discrete methods should be the first topics covered in this course, followed by the calculus, but the course should have a problem-solving emphasis throughout. 5. Upper-division modeling and biological data analysis courses should be encouraged, with extensive use of computers an integral part of such courses. Modules, based on diverse biological topics, for use in illustrating key quantitative concepts should be developed for these courses as well as for the entry-level course. General Project Goal: To produce a flexible curriculum of quantitative courses for undergraduate life science students, that can be integrated with the biological sciences courses these students take, thus creating a unified curriculum which enhances a students' appreciation of the utility of quantitative approaches in addressing problems in the life sciences. A Brief History: There were a number of courses developed during the late 1960's and early 1970's designed to provide an introduction to the calculus specifically for life science students. These courses were based on a number of texts which often included topics neglected in a standard science and engineering calculus course, such as basic probability theory, difference equations, and qualitative theory of ordinary differential equations. The inclusion of these topics, along with examples of mathematical concepts using biological rather than physical examples, came at some cost however. Namely, the level of mathematical rigor as well as the number of techniques discussed were lower than in the standard sequence. Were these courses successful? Many would answer in the affirmative, particularly those individuals involved in constructing such courses. However, success was mixed given: (i) the fact that almost none of the books on "Biocalculus" or "Calculus for the Biological Sciences" are still in print, and (ii) very few institutions either adopted or maintained these courses in their curricula. There are several reasons why few students benefitted from these courses. First, the courses were often tied to particular mathematics faculty members with research interests in biological areas. If the faculty member moved on, the course disappeared because it was not intimately coupled with the biology curriculum, but rather considered essentially a mathematics course. Courses which were closely coupled to requirements for biology students could survive a change in faculty, as could courses taught by biology faculty themselves. Second, mathematics programs with strong research components in mathematical biology would also tend to maintain these courses. These comments, though based on limited information, point out the importance of close coupling to the biology curriculum if a quantitative component is to be successful. Given that there are few programs offering quantitative entry-level courses for life science students, what are these students now taking? A survey of college catalogs of 21 universities, including 34 life science programs, indicates a great diversity of requirements. Approximately thirty per cent of the programs require two terms of science calculus, another thirty per cent require one term of science calculus, and the rest require calculus for business students or just algebra and trigonometry. Only about ten per cent require a statistics course. There is no evidence among the sampled schools of courses designed specifically for life science students. Are the courses offered at these schools serving the needs of life science students? A major tenet of this curriculum development project is that they do not. Furthermore, we believe that we can do a much better job of providing quantitative training for these students that makes use of their interest in biology, and enhances their training in biology by giving them additional exposure to the use of theory. In recent years, considerable attention has been paid to reforms in the undergraduate mathematics curriculum, particularly the calculus. The National Science Foundation has extensively funded a variety of programs aimed at changing the calculus from being a filter into a pump for upper division science and engineering courses. It is perhaps too early to say a great deal about the successes and failures of the various reforms being attempted, many of which make use of calculators and computer to aid the learning experience. If there is one lesson that might be drawn thus far, it is the success of a variety of models for mathematical training. These appear to be at least as effective as the lecture/recitation format that has been the standard model over the past several decades. The success of a particular model depend significantly upon the availability of resources and willingness to carry out a change at a particular institution. On the biology side, there also have been a variety of attempts to carry out reform in the curricula. These are not generally oriented towards improving the quantitative reasoning skills or training of students, though that is sometimes a component. Topics addressed during workshop sessions: A: Give examples of how quantitative methods allow specific biological problems to be addressed, and how these couple with experiment and observation to drive the field. For faculty, provide reasons why their students, to appreciate the scope of modern biology, need quantitative skills. For students, point out the general role of mathematics in aiding the development of biological theory. Discussion summary: The faculty, specifically the biology faculty, must initiate the development or restructuring of introductory quantitative courses for life science students. Yet there has been very little effort by biology faculty at most institutions to carry out this development. Why has there been such a lack of interest in this? Some of the points made included: 1) The lack of quantitative training of agricultural/biological faculty. 2) The inherent difficulties when students become more literate in quantitative thinking than the faculty. 3) Too few mathematicians realize the importance of mathematical applications in biology, and many feel secure only when teaching a theorem-proof type course. One difficulty discussed was the lack of an adequate pool of biologically-literate quantitatively-oriented teachers capable of teaching quantitative courses which motivate students by including important biological issues. To develop a larger teacher pool for a more quantitative curricula several ideas were mentioned. 1) Workshops and training sessions could be provided to faculty to foster a higher degree of mathematical sophistication among life science faculty. 2) Motivating examples should be provided for both life science and mathematics faculty to illustrate situations in which mathematics is directly applicable. 3) Encourage research across disciplines. It was stated that some faculty had difficulty getting recognition for their research simply because it was not directly related to their field, and was not published in a journal of their discipline. 4) An obvious place to start incorporating more quantitative ideas was in the standard statistics course. Discussion on the topic "Should there be a specialized entry level course for life science majors?" was left unresolved, but did point out several difficulties that a mathematics department would face in instituting a course specifically for life science students. 1. There is little quality control. Some mathematics departments have difficulty finding qualified instructors for the introductory courses they offer. Many of the present introductory courses are taught by adjuncts and graduate students. Would this type of instructor be adequate for a mathematics/quantitative biology course?. 2. There is a lack of transportability. Assuming a freshman were to switch majors, would a calculus for biology course be acceptable for a management curriculum? 3. There are the bureaucratic constraints. Can a diversity of introductory courses be scheduled effectively? 4. Finally, there is the question: will mathematics education still be as effective if business administration, engineering, life sciences, social sciences, education, pure mathematics, and honors programs all have separate introductory courses? From a student's perspective, mathematics and biology typically appear to be disjoint subjects, with few interconnections evident in the undergraduate curriculum. This difficulty is not limited to life science students, but is enhanced by the relative lack of quantitative emphasis in biological courses in the curriculum. Perhaps the strongest recommendation on which there was consensus at this Workshop was to include quantitative concepts in both the General Biology sequence and upper division courses in which there are natural connections to quantitative methods. Suggested topics for which quantitative methods could be utilized include: basic genetic, incorporating the use of simple probability theory as well as difference equations for gene frequency changes; biochemistry, emphasizing the derivation of Michaelis-Menten kinetics and the notion of a quasi-steady- state; molecular biology, incorporating a variety of discrete methods for sequence analysis; ecology, incorporating matrix methods to analyze population structure in addition to difference and differential equations for species interactions and population growth; crop science, including compartmental models utilizing linear and non-linear systems theory; and ethology, including matrix applications in developing evolutionary stable strategy ideas. Inclusion of regular connections with quantitative topics at many different points in the life science curriculum will rapidly reinforce for students the importance of mathematical concepts, and do so much more effectively than if such topics were isolated in a specialized set of courses. B: Are there key quantitative topics that we all agree should be included in an entry-level course? Can we set priorities for these and other concepts that might be included? Similarly, are there biological concepts that naturally are best introduced in a quantitative course, that enhance a students exposure to biological topics, and augment the topics covered in a general biology course? How can we assure that the biology introduced to illustrate quantitative topics doesn't add to confusion because the students have little prior biological training? Discussion summary: Mathematical Concepts: The following discussion assumes a prerequisite for an entry-level course is a standard pre-calculus course, including trigonometry. There were three general points of agreement: (1) The course should have a "problem-solving emphasis" and should include simple modeling problems that can be approached from many points of view. Thus in the beginning of the course problems stemming from empirical biological data could be analyzed from a descriptive statistics point of view. Later, the same underlying biological problem might be analyzed using difference or differential equations, using the data to either estimate parameters or evaluate model predictions. (2) Discrete and continuous methods should be integrated into the course, but discrete mathematics should take the lead in the first course. (3) The following broad course outline was developed: Semester I: 1. Descriptive Statistics: including curve fitting and non-linear, least-squares regression. 2. Matrix Algebra: up to and including eigenvalues and eigenvectors. 3. Discrete Probability. 4. Difference Equations: limits would be introduced in this context. 5. Differential Calculus: final 25% of course, including limits, continuity and derivatives. Semester II: 1. Differential Calculus: rates of change, exponential growth and decay presented from both discrete and continuous perspective. 2. Integral Calculus: closed-form solutions de- emphasized and numerical integration given equal status. 3. Differential Equations: including numerical techniques such as Euler, phase plane analysis, and stability notions for equilibrium points. Caveats: One should guard against the course becoming a hodgepodge of topics. Natural connections between topics should be exploited. Modeling problems that can be approached using different tools can serve to unify the course. Also, the course should try to preserve precalculus skills during the discrete math portion, for example using logarithms in the curve fitting. Biological concepts: A general goal is to lower the life science students' math anxiety level and help the student become comfortable with looking at biological phenomena from a quantitative perspective. Concepts which pervade all the biological sciences were discussed. A reasonable list is: 1. growth 2. feedback 3. variation 4. interaction 5. time series A key point regarding the utility of mathematics is that it aids in reducing the volume of biological information necessary to analyze a biological problem by providing unifying conceptual formulations. Quantitative methods should be included in a General Biology course, and should focus on cases in which the mathematics is necessary either to understand the biological problem or to reduce the volume of information presented, thus aiding comprehension. The following is a list of motivating examples to consider for further development. These could serve as a reserve that could be drawn upon as necessary to illustrate either quantitative points or explicate a biological problem. The list is not intended to be complete. 1-locus, 2-allele with heterozygote inferior 2-species predator-prey (effect of a perturbation such as pesticide on a system at equilibrium) the counter-intuitive effect of pesticide application when a natural enemy controls the pest gene-flow migration through a population enzyme kinetics (concept of reaction rates that depend upon concentrations) population structure (age and size) competitive and non-competitive interactions (for chemical as well as biological species) faunal and floral equilibria population growth and harvesting (from equilibrium to dynamical situations) alternative hypotheses for aging morphology (effects of parameter changes on developmental pathways) population growth and maximum sustainable yields Hardy-Weinberg and deviations from it mutation-selection balance sex-ratio and group selection arguments A key point is that the introduction of quantitative concepts in General Biology should not be done in a way which increases the content, but rather helps to decrease the amount of descriptive material covered. Quantitative content might reasonably be increased, even with the loss of biological content, because the earlier that students see that quantitative methods are an integral part of the life sciences, the more willing they will be to pursue courses with a greater mathematical content. Also, a typical curriculum contains many opportunities for biological training to be enhanced in courses succeeding General Biology, but very few opportunities to enhance quantitative training. Some general comments are that we need to find methods to counter the widely held notion that biology is "math free". Students are well aware that historically biology has been taught in a non-quantitative framework. It is viewed as the last math-free refuge for the science major. We need to prepare students for the increasing impact of quantitative methods in the life sciences. One means to overcome mathematics illiteracy in the biology community is to focus some efforts on increasing the mathematical competency of current life science faculty. A series of workshops or courses could do this, with appropriate support. C: How might the structure of an entry-level course vary for general biology students versus those with more specific life science interests? Are there particular topics/formats which lend themselves well to an audience of general biology students, but should be substituted in some way for pre- health or agriculture students? Disciplines including agriculture, pre-health, and resource management are differentiated from general biology by their inclusion of people, decision making, and social and socioeconomic systems. Students in these applied disciplines, therefore, require a quantitative curriculum with an emphasis on decision making, data analysis, probability, and risk analysis. Despite their interest in applied biology, however, students in agriculture, pre- health, and resource management (some felt life science students in general) enter college poorly prepared in mathematics, often have an aversion to mathematics, and often have chosen to enroll in a life science or related program specifically to avoid mathematics. The curricula in these disciplines do not now incorporate very much mathematics despite the fact that these disciplines are highly involved with describing and interpreting data, modeling, and dealing with uncertainty and complexity. These disciplines have a tradition of using quantitative science, especially statistics. Further, the increasing complexity of these disciplines requires students to become more adept at using quantitative analytical techniques. The group felt that the specific curricular needs in mathematics include: decision analysis, probability and statistics, risk analysis, linear algebra, difference equations, differential and integral calculus, and differential equations. The focus in these areas should be on data analysis and decision making, motivated through the use of realistic examples, avoiding formalism and stressing fundamentals, building conceptual understanding and basic foundations for upper level courses, and avoiding unnecessary computational complexity. Such a mathematics course could be common for all life science students and substantial integration with other courses and years of the students' curriculum can easily be obtained. Thus there is no need at the entry-level for separate courses designed specifically for pre-health or agricultural students. Rather, sections of a common introductory course could provide the emphasis on uncertainty and decision making required for these applied students. D: Provide some alternative methods to structure upper division courses to build upon the biological training of students as well as their quantitative skills developed in the entry-level course. How can we best increase the biological content of these courses to ensure that biostatistics isn't merely a cook-book course, but rather is closely tied to real biological problems? Can a modeling course be run in a true problem-solving, exploratory mode for the students? An overall goal for upper division quantitative courses is to prepare students to read, with comprehension, the basic literature in their field. Goals and Structures for a Modeling Course 1) WHAT IS MEANT BY MODELING? The group examined a number of variants of the term "model", including the statistical parameter fit-to-data approach, conceptual models in which physical processes are described from first principles, and models that capture the essence of a some behavior in a more metaphorical way. It was agreed that a modeling course should expose the student to a variety of approaches. The continuum from "strategic " to "tactical " models was discussed. This refers mainly to the use to which a model is put, but "stategic" models often emphasize possible mechanisms and "tactical" models tend to focus on the detailed understanding of a particular system, such as a pond. The appropriate model depends on the specific aims or goals of the modeler. A good mixture of general conceptual as well as particular system models should be included. Any course on modeling should include some problem-solving, some exposure to "real data" and data analysis as well as modeling efforts by the students on topics that they develop and work on individually. 2) HOW SHOULD MODELING BE TAUGHT? We considered three separate issues related to teaching students how to model: (A) HOW TO CONVERT VERBAL INFORMATION TO A MODEL Students have difficulty with word-problems and the steps through which a verbal description of a process is translated into mathematics. The point was made that models need not always consist of equations or mathematical symbols, but could include graphical or diagrammatic (e.g. flow chart) models. These structural modeling approaches can provide insight, even for students who have weaker backgrounds in mathematics. It was postulated that most problems are solved by recognizing a connection with previously encountered problems, and that the first part of such a modeling course should introduce numerous examples. The ideas of proceeding from the simple to the complex in gradual increments, and the process of trial and error, were emphasized. (B) HOW TO POSE THE RIGHT QUESTIONS Even more difficult than the above is teaching the approaches one might take when it is not obvious or predetermined what the right questions are. We discussed the fact that defining the right variables, selecting the appropriate modeling strategy, and arriving at a sufficiently tractable verbal caricature of a system are fundamental problems - indeed, they are the same problems that we face as researchers. There was conflicting opinion about whether the four-step approach in Polya's "How to Solve It" truly depicts the way we, as scientists, solve problems. In our limited discussion time, these issues were not resolved. We did, however, note the importance of good case studies and "realistic" or captivating examples to motivate modeling. Examples from the physical sciences include: (a) A Disney movie of ping-pong balls and traps; (b) the drinking duck oscillator; (c) a three-magnet pendulum; (d) the height of foam on a glass of beer; and (e) a black-box containing an electronic device (a nine-volt battery) to be analyzed by external measurements of voltage. A similar list of examples from the biological sciences could include cases in many of the topic areas mentioned above under biological concepts. (C) HOW TO ANALYZE THE MODEL Our discussion did not emphasize this issue since, presumably, one goal of the course would be to teach the appropriate analytical techniques. Developing models that are powerful, without unnecessary complexity is a talent that evolves gradually, with increasing experience. (D) HOW TO INTERPRET THE PREDICTIONS A good model is one which contributes to understanding or insight about the behavior of the system. Students need to be able to convert their symbolism back to clear verbal statements about the system being modelled. We strive for examples in which the mathematics leads to conclusions that might have escaped us had we been restricted to simple verbal arguments, not those in which the predictions are trivially related to the assumptions we made at the outset. 3) WHAT SHOULD BE INCLUDED IN A MODELING COURSE? A list of possible mathematical topics, with the associated biological case studies and examples appears below: ____________________________________________________________ ____________ MATHEMATICAL TOPIC BIOLOGICAL EXAMPLES ____________________________________________________________ ____________ Optimization chemical equilibrium (free energy) Linear programming allocating resources (root/shoot) Decision analysis harvesting, fisheries, bioeconomics optimal foraging medical decisions ------------------------------------------------------------- ----------------------------------- Matrix Methods age distributions (Leslie Matrices) life cycle models (see book by Hal Caswell) compartment models (see book on linear models by Michael Cullen) ESS allocation of reproductive effort (see book by Eric Charnov) ------------------------------------------------------------- ----------------------------------- Dimensional analysis Michaelis Menten kinetics chemostat allometry (see book on Scaling by Knut Schmidt-Nielsen) Spruce Budworm models (see various papers by Donald Ludwig and collaborators e.g. J. Anim. Ecol. 47:315-332) ------------------------------------------------------------- ---------------------------------- Phase Plane Analysis above examples in dimensional analysis grazing models (see I. Noy-Meir J. Appl. Ecol. 15:809-835) classical Lotka-Volterra models spread of epidemics temperature control in animals ------------------------------------------------------------- --------------------------------- Nonlinear Behavior discrete logistic Steady states population dynamics Stability log logistic discrete vs. continuous dynamical diseases 1 and 2D maps Leisch-Nyham defect parameter sensitivity sickle cell anemia chaos patch-dynamics models ------------------------------------------------------------- --------------------------------- Finite Markov chains epidemic models coding of point mutations population genetics migration behavioral sequence analysis density dependent succession island biogeography rainfall events ------------------------------------------------------------- --------------------------------- Birth and Death Processes molecular evolution coalescence theory ------------------------------------------------------------- --------------------------------- Multivariate Models migration Partial differential equations molecular diffusion Diffusion pheromones genetic drift random walks (see book by Howard Berg) pattern formation (Turing models) ------------------------------------------------------------- -------------------------------- Projective Geometry multivariate statistics community descriptions numerical systematics & taxonomy ------------------------------------------------------------- -------------------------------- Data Analysis hare-lynx data Meakin vs. Gompertz growth dissociation curves (DNA) drug clearance rates ------------------------------------------------------------- -------------------------------- Model Evaluation successful vs. less successful case studies Goals and Alternative Structures for Biostatistics Premise: Many biostatistics texts and courses are carried out in a "cookbook" manner which teaches students a number of tools but leaves them with little idea of overall statistical principles and with few data analysis skills. To remedy this we propose the addition of a laboratory module or short course to either a concept-based statistics course or a data analysis rich biology course. In some form such a course should be a required component of a life-science curriculum. Laboratory module contents: The module would be constructed around an easy-to-use, flexible, statistical computing package with superior graphics capability. The focus would be on data analysis and re-analysis, both at interim points and at the conclusion of a study. Ties to a strongly conceptual statistics course or a biology course would be necessary to provide a framework to avoid over- evaluation of data. The links between biological principles, experimental design and statistical analysis would be emphasized. Topics which are not usually included in a course at this level, but should be included in this course are: multivariate statistics, presented from conceptual and graphical viewpoints; nonlinear least squares estimation for mathematical models. Some instructors may choose to incorporate simulation or resampling based, computer intensive methods for nonstandard problems (e.g. bootstrapping). The corresponding statistics or biology course would need to emphasize the following ideas: sampling, experimental design, observational versus experimental studies, bias and confounding. A Sample Curriculum for Agricultural and Biological Engineers Premise: Essentially all curricula for these students follow the same basic mathematics courses as generally required for physical science and engineering students, consisting of approximately two years of calculus, linear algebra and differential equations. Thus the objective of the curriculum briefly described below is to couple these students' prior training in mathematics to realistic and practical applications in biology. The curriculum would be modified appropriately for students in particular programs, such as biomedical engineering. The below list of topics might represent distinct courses in the curriculum, or could be combined in a shorter sequence of courses. 1. Properties of materials in biological systems 2. Transport processes in biological systems 3. Instrumentation and controls for biological systems 4. Biological science for engineers 5. Modeling and simulation of biological and agricultural systems. E: Is a computer-based course necessary? What are its advantages, if any, over a course using graphing calculators? How can instructors find appropriate software and incorporate it in the courses? Entry-level Courses: The majority of students will use computers after they graduate, therefore a computer element should be included in any quantitative course. Furthermore, the use of calculators and computers will enable an instructor to spend less time on details such as techniques of integration. This allows for relatively more time to be spent on conceptual ideas as well as understanding and exploring more complicated non- linear models of biological processes and their critical evaluation. In particular, the use of graphing facilities such as sketching and curve fitting, and the use of computer algebra systems (Derive, Mathematica, Theorist, Maple, etc.) was stressed. Although these packages are not perfect, and may require a student to reformulate a problem to enable a solution to be found by the algebra system, this provides new incentives for a student to work on basic ideas such as substitution methods for integration. It was felt that both calculators and computers have a role in an entry-level course. Relatively inexpensive graphing calculators are readily available, and provide advantages for in-class work when computer laboratory facilities are in short supply. Often the abilities of students in a class vary widely, from "Nintendo kids" to those who suffer from technology anxiety. One solution is to split the students into small work groups early in the course, each group containing students with varying computer experience. Though there is an abundance of potentially useful software for biological problem solving , the quality of these is very mixed. A resource where faculty could obtain reviews of software useful in entry-level courses would be quite beneficial. Faculty as well as students would require the time and training in such technology, and faculty would perhaps need help in learning how to cope with more open- ended, project oriented courses that the use of this technology can foster. Upper Division Courses: Computers have become a valuable tool for understanding and solving problems associated with modeling biological phenomena. Therefore computing should be an important part of a modeling course so that students have an opportunity to observe the interplay between computation and analysis that is typical of modern research in quantitative biology. Computers will affect both the way modeling courses are taught, and the order of included topics. This is especially evident in the computer classroom in which the traditional lecture style gives way to individual exploration of problems, although most faculty will find it appropriate to mingle the two teaching approaches, either by holding computer laboratory sessions or by assigning homework which requires computer use. The use of calculators alone was thought to be inappropriate in courses at this level. There are several potential misuses of technology in these courses: (a) failure to think carefully enough about a problem to ensure that the computer is a necessary tool to investigate it; (b) having too much confidence in the computer so that one uncritically accepts misleading and incorrect output; (c) having too little knowledge of algorithms used or programming, so that they are used inappropriately; students may well lack the ability or confidence to modify or develop their own small programs. (d) spending excessive time on programming details - it is unrealistic to expect that life science students will have extensive programming experience at the undergraduate level, and the variety of languages and systems available would make teaching a class that relied on the students' abilities in this area very difficult. Despite this, some knowledge of programming is extremely useful, if only to reinforce the use of logical approaches to formulate the solution to a problem. There is a diversity of software available for use in courses in the life sciences. Biologists and mathematicians teaching upper division courses need access to a clearinghouse or listing service for such software. Although reviews appear regularly in a variety of journals, there seems to be no compendium of these that biologists and mathematicians might subscribe to. F: Is a modular approach appropriate for the entry-level course? If so, how can we aid an instructor in choosing appropriate modules that cover key concepts and can be unified, as opposed to a piecemeal assortment of biological examples coupled to mathematical concepts? For upper division courses, should these modules suggest avenues for independent research by students, or quickly present various biological topics that would often be glossed over in non-quantitative biology courses? I. In selecting examples to use in an entry level course, it is important to keep the following points in mind: a. remind the class that the mathematical structure provides unity for the course; b. select examples that can be be approached by more than one mathematical technique; c. use some examples as problems for in-class workshop sessions, or class projects that can be feasibly completed by groups of students outside the formal class sessions. II. Examples should come from at least the following four major areas of biology: a. photosynthesis; b. population dynamics and ecology; c. genetics; d. cell and organismal physiology. III. The relationship between continuous and discrete processes can be emphasized at many points; examples include population dynamics, and photosynthesis (at the level of the leaf vs. the chloroplast). In addition, for some applications, discrete models are exact (seasonality, population dynamics of organisms with synchronous patterns of birth/death), whereas continuous models provide an approximation. IV. The following suggested examples are of biological topics meant to accompany mathematical topics. Letters in parentheses denote the biological areas described above. Topics in square brackets are other examples that do not fit into the area classification but could be included at the instructor's discretion. descriptive statistics, least squares regression (a) rate of photosynthesis vs. light (c) quantitative genetics [allometry, scaling, log-transformations] [classification of organisms, numerical taxonomy] matrix algebra and eigenvalues (b) life tables and stage structured populations (b) plant succession (b) ecosystem models (c) population genetics [pharmacokinetics] discrete probability (a) arrangement of leaves/chloroplasts in photosynthesis i.e., the probability of a photon being utilized in the photosystems (c) mutation at the molecular level (c) Mendelian genetics (d) immunological response [animal behavior models] [Shannon-Simpson index] difference equations and stability of difference equations (b) population growth (c) operons and genetics limits there are many applications on this list that illustrate the process of going from a discrete to a continuous formulation, e.g. genetic variance, light hitting leaves vs. chloroplasts, population dynamics rates of change (discrete and continuous) (b) population dynamics (b) equilibrium biogeography (c) divergence of DNA sequences and the molecular clock (c) mutation-selection balance (d) enzyme kinetics (d) physiological homeostasis (d) diffusion from the cell membrane (d) uptake of substances from fish gills, chemical exposure exponential growth and decay (b) population dynamics (d) dye-dilution techniques, drug physiology, alcohol clearance from the blood integrals (a) photosynthesis measured as a function of degree days (other processes also) (b) community productivity (b) estimating depletion of natural resources [volumes of objects such as bird eggs, tomatoes, clams] numerical integration (d) areas under peaks of a curve in chromatography [probability density functions] differential equations and stability (including phase plane analysis) (b) population dynamics (b) ecological interactions (b) chemostat [diffusion, as in diffusion vs. convection, or movement through xylem or phloem] G: Are there natural ways that the quantitative courses might best be coupled to entry-level general biology, and more advanced biology courses? How do we foster integration? How do we increase the availability of appropriate material? A theorem postulated at the Workshop was that if mathematicians and biologists are brought together, great wonders will occur. This general conclusion was based on the experiences of many of the participants who found that collaborative ventures lead to entirely new approaches to old problems, in addition to pointing the way to entirely new problems. A clear conclusion was that fostering an increase in quantitative skills of life science students must come from the life science faculty themselves - in general, this increase will not occur if the impetus is left solely to colleagues in mathematics. A variety of programs were suggested to encourage life science faculty to increase their own quantitative competence, and that of the graduate students they are training. These include: setting up a modeling center in which biologists might work jointly on problems with mathematicians (in a similar manner to the statistical consulting centers at many institutions), short-courses on mathematical modeling in biology designed specifically for life science faculty, post-doctoral and sabbatical programs in quantitative biology tenable, and encouragement of team-taught courses in which a biologist and a mathematically trained colleague collaborate. Throughout, agreement was reached that biology needs to develop its own cadre of mathematically- competent researchers and teachers, and that this should be viewed as a natural progression within biology. Incorporating more quantitative concepts in biology courses at all levels has the important effect of improving students' capabilities to read with comprehension the literature in their own field of interest. This in turn might entice motivated students (i.e. those going on to graduate work in life science) to invest the necessary effort in quantitative topics, but it may not be sufficient for the general life science student. For these students, regular examples are needed in General Biology and upper division courses which illustrate, using quantitative methods, that an individual's naive biological intuition may be very wrong. Examples include situations in which pesticide application actually increases the density of pests, when projections of population growth based on age structure are wrong because size or stage is the key factor related to individual survival and fertility, and when increases in harvesting effort expended on a particular fish stock actually lowers the yield. Such counter-intuitive results should be supplemented with examples of situations in which mathematical approaches led to major new biologically significant results, for example epidemiology theory providing the basis for successful immunization programs and biocontrol models leading to predictions as to when techniques such as sterile male release will be successful. Examples related to societal and human issues have a special power to convince students of the efficacy of quantitative approaches, and should be emphasized. Listed above in Section A are several examples of quantitative methods which can be included in upper division biology courses, with the general consensus being that courses as diverse as Vertebrate Physiology, Plant Physiology, Population Genetics, Population Biology, Ethology, Evolutionary Biology, and Molecular Evolution could all be taught in a manner which includes substantive quantitative concepts. This is of course in addition to whatever statistical methods might be used in these courses. An additional suggestion was that the life science curricula should include at least one upper division course with a strong quantitative component, offering students the option to choose from a list of courses such as: Biological Data Analysis, Models in Biology, or Experimental Design. The inclusion of such a requirement in the curricula would further emphasize to students the importance of quantitative training. In addition to the above, there are ways by which professional societies can help to enhance the quantitative training of life scientists. One suggestion which would serve to enhance the prestige of quantitative research among life science colleagues would be for a society, such as the Society for Mathematical Biology, to offer occasional awards to recognize significant recent accomplishments in a variety of biological disciplines which relied upon quantitative methods. There is no need for these awards to involve any amount of funds, since the prestige alone may help to convince those colleagues in biology who have some antipathy towards mathematics that quantitative skills are essential for modern biologists. Additional suggestions included efforts to revise the quantitative standards required on such tests as the CAT, since these were viewed as outdated by many participants. PARTICIPANTS Ha. Rest Akcakaya - Applied Biomathematics, Inc. Dennis Baldocchi - Atmospheric Turbulence and Diffusion Division, NOAA Jaroslav Benedik - Department of Genetics, Masaryk University, Czechoslovakia Leslie Bishop - Department of Biology, Earlham College Mahadev Bhat - Department of Agricultural Economics, University of Tennessee William Bossert - Division of Applied Science, Harvard University Russell Butler - Department of Biology, Vanderbilt University Charles Clark - Department of Mathematics, University of Tennessee Michael Cullen - Department of Mathematics, Loyola Marymount University Jim Cushing - Department of Mathematics, University of Arizona Lothar Dohse - Department of Mathematics, University of North Carolina at Asheville Jim Drake - Department of Zoology, University of Tennessee Leah Edelstein-Keshet - Department of Mathematics, University of British Columbia Bard Ermentrout - Department of Mathematics, University of Pittsburgh Henry Foehl - Department of Mathematics, Philadelphia College of Pharmacy and Science Henry Frandsen - Department of Mathematics, University of Tennessee Lev Ginzburg - Department of Ecology and Evolution, SUNY - Stony Brook Anil Gore - Department of Statistics, University of Poona, India Louis Gross - Department of Mathematics, University of Tennessee Brian Hahn - Department of Applied Mathematics, University of Capetown Thomas Hallam - Department of Mathematics, University of Tennessee Art Heinricker - Department of Mathematics, University of Kentucky Carole Hom - Department of Mathematics, University of California - Davis Henry Horn - Department of Biology, Princeton University Dan Hornbach - Department of Biology, Macalester College Fern Hunt - National Institute for Standards and Technology James Jones - Department of Agricultural Engineering, University of Florida John Jungck - Department of Biology, Beloit College Denise Kirschner - Department of Mathematics, Vanderbilt University Mark Kot - Department of Applied Mathematics, University of Washington Suzanne Lenhart - Department of Mathematics, University of Tennessee Philip Maini - Centre for Mathematical Biology, Oxford University Chuck McCulloch - Biometrics Unit, Cornell University Michael Mesterton-Gibbons - Department of Mathematics, Florida State University Dung Ba Nguyen - Department of Therapeutic Radiology, Yale University School of Medicine John Norman - Department of Soil Science, University of Wisconsin-Madison Stuart Pimm - Department of Zoology, University of Tennessee Muriel Poston - Department of Botany, Howard University Jon Seger - Department of Biology, University of Utah Nicholas Stone - Department of Entomology, Virginia Polytechnic Institute and State University Marcy Uyenoyama - Department of Zoology, Duke University For further information about the Workshop, or any aspect of this Curriculum Development Project, contact: Dr. Louis Gross Mathematics Department University of Tennessee Knoxville, TN 37996- 1300 (615)974-4295 (615)974-2461 (Secretary) (615)974-6576 (FAX) gross@math.utk.edu (INTERNET) gross@utkvx (BITNET)