Interdisciplinary Quantitative Curriculum Development: Lessons from a Project in the Life Sciences Louis J Gross Professor of Mathematics and Ecology University of Tennessee - Knoxville gross@math.utk.edu This is a talk presented at the American Mathematical Society/Mathematical Association of America Workshop on "Changing Colleagiate Education: Mathematical Sciences and their uses in other Disciplines", March 27-28, 1994 at the Washington, DC Marriott. ABSTRACT: I review the progress on a curriculum development project aimed towards increasing the quantitative skills of life science students. This includes a summary of a workshop on quantitative courses for life science students, a description of entry-level quantitative course development, the software evaluation and gopher site development component of the project, and a description of an upcoming workshop for life scientists. Some general lessons are abstracted from this with the emphasis on curriculum development across disciplines. Included are some results of a survey of mathematics faculty dealing with the issue of professional interest in applications of mathematics to the real world. Some conclusions are drawn regarding the difficulties associated with enhancing the exposure of students to significant applications of mathematics. Finally I discuss the general structure of interdisciplinary reform of quantitative training, using the CPA approach, as well as alternative paradigms for such reform. Note that a brief summary of the goals of this project was published in the February 1994 issue of BioScience 44:59. BACKGROUND: With support from the National Science Foundation (NSF Grant USE-9150354), I have been working since 1991 to develop a quantitative curriculum for life science students. The life sciences include students in all pre-health professions (pre- medical, pre-dental, pre-pharmacy, etc.), nursing, biology, biochemistry, nutrition, agriculture, forestry, wildlife, and all areas of natural resource management. The general project goal is: To produce a flexible curriculum of quantitative courses for undergraduate life science students, able to be integrated with the biological courses these students take and utilizing examples from recent biological research, thus creating a unified curriculum which enhances a students appreciation of the utility of quantitative approaches to address problems in the life sciences. This would serve a serve a dual role of both introducing new quantitative methods and reinforcing key concepts in modern biology. My procedure in carrying out this project has been: a. Conduct a survey of quantitative course requirements of life science students; b. Conduct a workshop 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; c. Develop an entry-level quantitative course sequence based upon recommendations from the workshop; d. Implement the course in an hypothesis-formulation and testing framework, coupled to appropriate software; e. Evaluate the use of a wide variety of biological software in quantitative courses through the use of individual student projects and make these evaluations and software (if they are in the public domain or Shareware) available via gopher/ftp; and f. Conduct a workshop designed specifically for life science faculty to discuss methods to enhance the quantitative component of their own courses. SURVEY OF QUANTITATIVE COURSE REQUIREMENTS FOR LIFE SCIENCE PROGRAMS: I conducted a survey to determine the current quantitative requirements for undergraduates in the life sciences at many institutions. The initial survey was done by looking at current college catalog requirements in 1991, but this sample was expanded by an open request on the ECOLOG-L listserve group for further information. This information was compiled by Aaron Ellison of Mt. Holyhoke College, and I below summarize the results. The sample includes 86 life science programs from 47 institutions, of which 4 programs had no math requirements. The majority of the institutions reported were research-level, doctoral granting major universities, which may have biased the sample. The average semesters required per life science program were: Computer Science .128 Statistics .157 Calculus 1.26 Pre-calculus .337 Total semesters of quantitative courses 1.88 11 programs required computer science (13% of sample) 14 required statistics (16%) 68 required some calculus (79%) 26 required precalculus (30%) There were 5 programs in which all requirements could be met just by precalculus. Thus there were 9/86 = 10% of the programs in which no quantitative skills above high school level were required. The above assumes, if students had an option, the order in which they took courses was precalculus, calculus, statistics, computer science. Only 16 of the 86 programs allowed some options as to which course to take. In a subset of 34 programs at 21 universities included in the above, obtained through my survey of college catalogs, approximately 1/3 of the calculus courses required (or an option) were of the social science/business/life science type. SUMMARY OF CONCLUSIONS FROM THE WORKSHOP: 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. 2. As one means to foster the inclusion of more quantitative topics within the curriculum, it is proposed that a Primer of Quantitative Biology be developed to be used in conjunction with the General Biology sequence typically included in most life science curricula. 3. Exploratory data analysis should be included in several ways as part of a life science curriculum. Methods to do this would be as (i) part of laboratory exercises within a biological course; (ii) a short-course available for credit ; and/or (iii) a formal biometry course. 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 included as an integral part. THE ENTRY-LEVEL QUANTITATIVE COURSE - BIOCALCULUS REVISITED: In response to recommendation #4 from the workshop, a pilot version of an entry-level quantitative course for life science students was constructed and has been taught over the past two years to approximately 130 students, by three different instructors (the author and two graduate students working in mathematical ecology). The prerequisites assumed were Algebra, Geometry, and Trigonometry. Goals: Develop a Student's ability to Quantitatively Analyze Problems arising in their own Biological Field. Illustrate the Great Utility of Mathematical Models to provide answers to Key Biological Problems. Develop a Student's Appreciation of the Diversity of Mathematical Approaches potentially useful in the Life Sciences. Methods: Encourage Hypothesis Formulation and Testing for both the Biological and Mathematical Topics covered. Encourage Investigation of Real World Biological Problems through the use of Data in class, for homework, and examinations. Reduce Rote Memorization of Mathematical Formulae and Rules through the use of Software such as MATLAB and MicroCalc. Encourage Investigation of Quantitative Approaches in Biological areas of Particular Interest to each Student through Projects Utilizing Software from diverse of Bio- logical areas. THE PILOT FIRST-YEAR COURSE Syllabus: Semester 1: Descriptive Statistics - Means, variances, using software, histograms, linear and non-linear regression, allometry - 3 weeks Matrix Algebra - Matrix algebra, using linear algebra software, matrix models in population biology, eigenvalues, eigenvectors, Markov Chains, compartment models - 4 weeks Discrete Probability - Experiments and sample spaces, probability laws, conditional probability and Bayes' theorem, population genetics models - 3 weeks Sequences and difference equations - limits of sequences, limit laws, geometric sequence and Malthusian growth, linear first and second order difference equations, equilibria, stability, logistic map and chaos, population models - 3 weeks. Limits of functions - numerical examples using limits of sequences, basic limit principles, continuity - 2 weeks Semester 2: Derivatives - as rate of growth, use in graphing, basic calculation rules, chain rule, using computer algebra software - 3 weeks Curve sketching - second derivatives, concavity, critical points and inflection points, basic optimization problem - 3 weeks Exponentials and logarithms - derivatives, applications to bio-physics, population growth and decay - 2 weeks Antiderivatives and integrals - basic properties, numerical computation and computer algebra systems, various applications - 3 weeks Trigonometric functions - basic calculus, applications to medical problems - 1 week Differential equations and modeling - individual and population growth models, linear compartment models, stability of equilibria, phase-plane analysis - 3 weeks From a student's perspective, mathematics and biology typically appear to be disjoint subjects, with few interconnections evident in the undergraduate curriculum. The above described course was an attempt to provide these interconnections, within the constraints imposed by a course specifically required in the curriculum to be a quantitative one. In conjunction with this however, it is important to keep in mind that perhaps the strongest recommendation on which there was concensus at the Workshop was to include quantitative concepts in both the General Biology sequence typical of most curricula, as well as making use of quantitative methods in many upper division courses in which there are natural connections. Many upper-division biology courses can include a quantitative component: Basic genetics - simple probability theory as well as difference equations for gene frequency changes Biochemistry - derivation of Michaelis-Menten kinetics and the notion of a quasi-steady-state, Molecular biology - discrete methods for sequence analysis, Ecology - matrix methods to analyze population structure , difference and differential equations for species interactions and population growth Crop science - compartmental models utilizing linear and non-linear systems theory Ethology - matrix applications in developing evolutionary stable strategy ideas. A WORKSHOP TO FOSTER QUANTITATIVE CONCEPTS DIRECTLY IN LIFE SCIENCE COURSES: In an attempt to foster the inclusion of mathematical concepts directly in life science courses, thus meeting recommendation #1 of the workshop summarized above, a second workshop will be hed in May of 1994. This Workshop is designed to bring together a group of life science, mathematics, and statistical researchers and educators to focus on the inclusion of more quantitative concepts directly in life science undergraduate courses. The majority of participants will be life science faculty, or faculty involved in the quantitative training of life science students. The basic tenet of the workshop is that it is not sufficient to isolate quantitative components of the curriculum in a few courses on such topics, rather, quantitative methods should be a component of courses throughout the undergraduate life science curriculum. The workshop will be held May 19-21 at the University of Tennessee, Knoxville. Additionally, there will be a special Symposium at the 1994 American Institute of Bological Sciences Annual Meeting in August with two components. One will be session of speakers who have developed quantitative curricula at their home institutions. The second component will be an open computer lab, displaying the wide range of software available to aid the quantitative training of biologists. APPLICATIONS WITHIN MATHEMATICS COURSES: In his book "How to Teach Mathematics: a personal perspective" (AMS, Providence, 1993), on the subject of applications of mathematics Steven Krantz states ""don't get sucked into doing trivial, artificial applications", and suggests that one should "talk to experienced faculty in your department about what resources are available to help you present meaningful applications to your classes". In regard to this, I was interested in knowing how faculty in my home math department regarded the importance of knowledge of some applications of math for themselves and their colleagues. This was motivated by the NSF Initiative which I viewed as attempting to foster major efforts by math departments to more directly couple the mathematics curriculum to applications. We continually make the argument that at a major university, despite the fact that low-level math courses can readily be taught by instructors with MS degrees, the undergraduates benefit by having faculty teach these courses who themselves do research in math. A natural extension of this argument, if one wants to do as the new NSF initiative in Mathematical Sciences and their Applications Throughout the Curriculum proposes, and include important applications of math at all levels of the undergrad math curriculum, is that the math faculty doing this should be researchers in areas of application. I thought myself that this is too much for NSF to ask, but that it might be reasonable to expect that the math faculty should at least be knowledgable in some area of application, even if not actively pursuing research in it. Thus the below question which I to the faculty in my department in an anonymous survey: Do you consider the below to be an appropriate departmental expectation of all professorial-level faculty? All faculty members are expected to be knowledgable of the application of an area of their mathematical expertise to real-world situations. Knowledgable here does not mean that the faculty member is necessarily expected to carry out research in such an area of application. Rather, the faculty member shall be sufficiently knowledgable so as to be able to read with understanding the primary literature (e.g. journals) in the area of application outside of mathematics, and be able to explain the significance of the mathematical work in this application area to other faculty and to students. Please mark one response given below (or add your own if you want!): I consider this a reasonable request and support it 14 (45%) I consider this a reasonable request, but would not follow it myself 2 (6%) I do not consider this a reasonable request, and would ignore it 7 (23%) I consider this an infringement of a faculty members right to choose their own research program, and would actively oppose it 6 (6%) No classification checked 2 (6%) The numbers and percentages above indicate the responses from my colleagues. There were a number of comments expressed on this, including: it should be an expectation that all faculty be able to teach doctoral level courses in the areas of analysis, algebra and topology; I thought all faculty engage in interpreting mathematical applications when asked; this should be an expectation for all new faculty, not the older ones; it is unreasonable to expect me to expend the time and effort necessary to do this in my field of mathematical interest; totally unreasonable, this is a mathematics department, not a trade school; teaching applications to undergraduates is remotely connected to knowing applications of our own research field; I support this absolutely, emphatically, unconditionally; I consider this a reasonable expectation, not necessarily a requirement for hiring, tenure, promotion. LESSONS FROM THE ABOVE: It is unrealistic to expect most math faculty to have any strong desire to really learn significant applications of math that students will readily connect to their other course work in other fields. The majority of math faculty do not feel particularly comfortable with realistic applications, and cannot be expected to expend the effort necessary to learn about them. Despite this, there is a core group of faculty who support such efforts (who may or may not be individuals involved in the subfields typically associated with applied mathematics). So what do we do to enhance quantitative understanding across disciplines? Below is what I say to life science faculty: Who can foster change in the quantitative skills of life science students? Only you, the biologists can do this! You have two routes: 1. Convince the math faculty that they're letting you down 2. Teach the courses yourself Note: Math faculty will not take you seriously unless you show them how the quantitative topics you insist that they cover will be used in your own courses! This means biology courses must become less of a "litany of conclusions", and more an exploration of how and why natural systems came to be as they are. Unfortunately, this battle must be fought over and over at each institution. As a non-biology example in regards to the above, the Business College faculty at my University recently decided to require all their students to take the standard science and engineering calculus sequence. This change was made despite the fact that the math department had been asked by the Business college a few years earlier to design a special sequence for their students, including certain topics in multivariate calculus and optimization not covered in the previous course. The change was also made despite the fact that many incoming business students do not have quantitative skills equivalent to the entering science and engineering students. In fact, the Business College has had to allow a sizable fraction of their students to continue to take this prior course, due to their lack of preparation. My colleagues and i viewed this change as an overt attempt to apply a filter to their students, though the Business College swears that this is not the objective. My colleagues and i would be much more willing to accept this if there were any indication that the Business College had made any effort to utilize the different skills which students get in the standard science and engineering calculus course by revising the business courses - no such effort has been made by any department inthe College. THE CPA APPROACH TO QUANTITATIVE CURRICULUM DEVELOPMENT ACROSS DISCIPLINES: As a summary of the approach I have taken in this life sciences project, and in hope that this will be applicable to other interdisciplinary efforts, I offer the CPA Approach: CONSTRAINTS, PRIORITIZE, AID 1. Understand the Constraints under which your colleagues in other disciplines operate - the limitations on time available in their curriculum for quantitative training. 2. Work with these colleagues to Prioritize the quantitative concepts their students really need, and ensure that your courses include these. 3. Aid these colleagues in developing quantitative concepts in their own courses that enhance a students realization of the importance of mathematics in their own discipline. This could include team teaching of appropriate courses. Note: The above operates under the paradigm typical of most U.S. institutions of higher learning - that of disciplinary compartmentalization. An entirely different approach involves real interdisciplinary courses. This would mean complete revision of course requirements to allow students to automatically see connections between various subfields, rather than inherently different subjects with little connection. Such courses could involve a team approach to subjects, which is common in many lower division biological sciences courses, but almost unheard of in mathematics courses. Comments about the above should be sent to me at: Louis J. Gross Department of Mathematics University of Tennessee Knoxville, TN 37996-1300 gross@math.utk.edu (865) 974-4295 (865) 974-3067 (FAX) (865) 974-3065 (Secretary)