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0,R&___PPT968 = O PPQ2R?%~85 Overview:
Summary of BIO2010 quantitative education recommendations
Different viewpoints from the Math/CS Panel summary
Overview of the University of Tennessee programs
Future directions: Suggestions to implement BIO2010 ideas and impediments
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7Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.hZ"H"747Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.Z"")&%96Major BIO2010 Recommendations`Schools should reexamine current approaches to see if they meet the needs of today s undergraduate biology students. Those selecting the new approaches should consider the importance of building a strong foundation in mathematics, and the physical and information sciences to prepare students for research that is increasingly interdisciplinary in character. This implementation should be accompanied by a process of assessment.
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2. Concepts, examples, and techniques from mathematics, and the physical and information sciences should be included in biology courses, and biological concepts and examples should be included in other science courses. Faculty must work collaboratively to integrate mathematics and physical sciences into life science courses as well as providing avenues for incorporating life science examples that reflect the emerging nature of the discipline into courses taught in mathematics and physical sciences.,&";8
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4. Laboratory courses should be as interdisciplinary as possible, since laboratory experiments confront students with real-world observations do not separate well into conventional disciplines * =:
5. All students should be encouraged to pursue independent research as early as is practical in their education. They should be able to receive academic credit for independent research done in collaboration with faculty or with off-campus researchers * >;
6. Seminar-type courses that highlight cutting-edge developments in biology should be provided on a continual and regular basis throughout the four-year undergraduate education of students. Communicating the excitement of biological research is crucial to attracting, retaining, and sustaining a greater diversity of students to the field. These courses would combine presentations by faculty with student projects on research topics. Z ?<
7. Medical school admissions requirements and the Medical College Admissions Test (MCAT) are hindering change in the undergraduate biology curriculum and should be reexamined in light of the recommendations in this report. , @=
8. Faculty development is a crucial component to improving undergraduate biology education. Efforts must be made on individual campuses and nationally to provide faculty the time necessary to refine their own understanding of how the integrative relationships of biology, mathematics, and the physical sciences can be best melded into either existing courses or new courses in the particular areas of science in which they teach.
<" A>'Summary of Quantitative Recommendations((' 1.5 Biology majors headed for research careers need to be educated in a more quantitative manner, which may require the development of new types of courses. We recommend that all biology majors master the concepts listed below and that life science majors become sufficiently familiar with the elements of programming to carry out simulations of physiological, ecological, and evolutionary processes. They should be adept at using computers to acquire and process data, carry out statistical characterization of the data and perform statistical tests, and graphically display data in a variety of representations. Students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases. (Z"B?.Concepts of Mathematics and Computer Science //"(.Calculus
Complex numbers
Functions
Limits
Continuity
The integral
The derivative and linearization
Elementary functions
Fourier series
Multi-dimensional calculus: linear approximations, integration over multiple variables
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qLinear Algebra
Scalars, vectors, matrices
Linear transformations
Eigenvalues and eigenvectors
Invariant subspaces.c d A&DA
Dynamical Systems
Continuous time dynamics equations of motion and their trajectories
Test points, limit cycles, and stability around them
Phase plane analysis
Cooperativity, positive feedback, and negative feedback
Multistability
Discrete time dynamics mappings, stable points, and stable cycles
Sensitivity to initial conditions and chaosDZGZ""H"6*pEB
Probability and Statistics
Probability distributions
Random numbers and stochastic processes
Covariation, correlation, and independence
Error likelihood@Z~Z" $]0FC
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Information and Computation
Algorithms (with examples)
Computability
Optimization in mathematics and computation
' Bits : information and mutual information@ZZ" GE
Data Structures
Metrics: generalized 'distance' and sequence comparisons
Clustering
Tree-relationships
Graphics: visualizing and displaying data and models for conceptual understanding < "QNHD?Additional Quantitative Principles Useful to Biology Students @@"(?Rate of change
This can be a specific (e.g., per capita) rate of change or a total rate of change of some system component.
Discrete rates of change arise in difference equations, which have associated with them an inherent time-scale.
Continuous rates of change arise as derivatives or partial derivatives, representing instantaneous (relative to the units in which time is scaled) rates.
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The process of abstracting certain aspects of reality to include in the simplifications we call models.
Scale (spatial and temporal) different questions arise on different scales.
What is included (system variables) depends on the questions addressed, as does the hierarchical level in which the problem is framed (e.g., molecular, cellular, organismal).
There are trade-offs in modeling no one model can address all questions. These trade-offs are between generality, precision, and realism.
Evaluating models depends in part upon the purpose for which the model was constructed. t ZhZNZZ""hN"$`JG
Equilibria and stability
Equilibria arise when a process (or several processes) rate of change is zero.
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Equilibria can be dynamic, so that a periodic pattern of system response may arise.
There are numerous notions of stability, including not just whether a system that is perturbed from an equilibrium returns to it, but also how the system returns (e.g., how rapidly it does so).
Modifying some system components can lead to destabilization of a previously stable equilibrium, possibly generating entirely new equilibria with differing stability characteristics. BZZ""R
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Grouping components of a system affects the kinds of questions addressed and the data required to parameterize the system.
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Geometry of the aggregation can affect the resulting formulation.
Symmetry can be useful in many biological contexts to reduce the complexity of the problem, and situations in which symmetry is lost (symmetry-breaking) can aid in understanding system response.B
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There are relatively few ways for system components to interact. Negative feedbacks arise through competitive and predator-prey type interactions, positive feedback through mutualistic or commensal ones.
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Though ultimately everything is hitched to everything else, significant effects are not automatically transferred through a connected system of interacting components locality can matter.
Sequences of interactions can determine outcomes program order matters.B
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Data and measurement
Only a few basic data types arise (numeric, ordinal, categorical), but these will often be interconnected and expanded (e.g., as vectors or arrays).
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A variety of statistical methods exist to characterize single data sets and to make comparisons between data sets. Using such methods with discernment takes practice \ZZ"""NK
WStochasticity
In a stochastic process, individual outcomes cannot be predicted with certainty. Rather, these outcomes are determined randomly according to a probability distribution that arises from the underlying mechanisms of the process. Probabilities for measurements that are continuous (height, weight, etc.), and those that are discrete (sex, cell type) arise in many biological contexts.
Risk can be identified and estimated.
There are ways to determine if an experimental result is significant.
There are instances when stochasticity is significant and averages are not sufficient.ZZ-ZZ
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There are diverse methods to display data.
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These are rules that determine the types of interactions in a system, how decisions are made, and the time course of system response.
These can be thought of as a sequence of actions similar to a computer program, with all the associated options such as assignments, if-then loops, and while-loops.
ZZ,ZZZ"
"""ROPotential curricula@Four examples were given:
First-year math, perhaps a mixture of discrete and calculus
Quantitative emphasis with first year of math, plus probability/biostatistics, DE plus advanced course
Math I plus CS first year, then Math II plus advanced math/CS course later
Calc and DE in year 1, biostatistics and 1 CS course6Z'" Z'6
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SPMath/CS Panel Comments" There is a distinction between the quantitative biologist, who works at the interface of math/computer science and biology, and the research biologist, who needs familiarity with a range of mathematical and computational ideas without necessarily being expert. Thus, the panel felt that flexibility in offerings is more advisable than a fixed curriculum.
Consider offering quantitatively intensive versions of standard biology courses, with extra credit2Z"g"hTQ
Consider new math courses which condense much of undergrad math into 3-4 semesters
Encourage interdisciplinary modeling courses at both introductory and advanced level, possibly as a first-year and senior seminar
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Success in quantitative life science education requires an integrated approach: formal quantitative courses should be supplemented with explicit quantitative components within life science courses. Z"(
Life science students should be exposed to diverse quantitative concepts: calculus and statistics do not suffice to provide the conceptual quantitative foundations for modern biology."(
H We can t determine a priori who will be the researchers of the future educational initiatives need to be inclusive and not focused just on the elite. Assume all biology students can enhance their quantitative training and proceed to motivate them to realize its importance in real biology.D%Z"$"$"$$URJThe CPA Approach to Quantitative Curriculum Development across DisciplinesKK
As a summary of the approach I have taken in this life sciences project, and in hope that this will be applicable to other interdisciplinary efforts, I offer the CPA Approach:
Constraints, Prioritize, Aid
Understand the Constraints under which your colleagues in other disciplines operate - the limitations on time available in their curriculum for quantitative training.ZZ" Z$$$$$$
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z Work with these colleagues to Prioritize the quantitative concepts their students really need, and ensure that your courses include these.
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.
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M 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.
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Collaborators$
Drs. Beth Mullin and Otto Schwarz (Botany), Susan Riechert (EEB)
Monica Beals, Susan Harrell - Primer of Quantitative Biology
Drs. Sergey Gavrilets (EEB) and Suzanne Lenhart (Math) NIH Short Courses
Drs. Thomas Hallam (EEB) and Simon Levin (Princeton) International Courses
Society for Mathematical Biology Education Committee www.smb.org4`Z"^"~3> (xProject activities:$Conduct a survey of quantitative course requirements of life science students;
Conduct a workshop with researchers and educators in mathematical and quantitative biology to discuss the quantitative component of the undergraduate life science curriculum;
Develop an entry-level quantitative course sequence based upon recommendations from the workshop;
Implement the course in an hypothesis-formulation and testing framework, coupled to appropriate software;|ZNbbbbbbbkb"
Conduct a workshop for life science faculty to discuss methods to enhance the quantitative component of their own courses;
Develop a set of modules to incorporate within a General Biology course sequence, illustrating the utility of simple mathematical methods in numerous areas of biology;
Develop and evaluate quantitative competency exams in General Biology as a method to encourage quantitative skill development;
Survey quantitative topics within short research communications at life science professional society meetings.
hZZbbqbbB$# < The Entry-level Quantitative Course: Biocalculus Revisited4= :b$b$' In response to workshop recommendations, a new entry-level quantitative course for life science students was constructed and has now become the standard math sequence taken by biology students. The prerequisites assumed are Algebra, Geometry, and Trigonometry.&!`$!Goals: 2 b(!!C 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 DZbe````b```u``C%"Methods: b(a 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 Maple. bZbbbebbbb|bbbbtbbbb$K&#RCourse 1 Content Discrete Math Topics:**b ) Descriptive Statistics - Means, variances, using software, histograms, linear and non-linear regression, allometry
Matrix Algebra - using linear algebra software, matrix models in population biology, eigenvalues, eigenvectors, Markov Chains, compartment models
Discrete Probability - Experiments and sample spaces, probability laws, conditional probability and Bayes' theorem, population genetics models Sequences and difference equations - limits of sequences, limit laws, geometric sequence and Malthusian growth
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T'$RCourse 2 Content Calculus and Modeling:**b )] Linear first and second order difference equations - equilibria, stability, logistic map and chaos, population models
Limits of functions - numerical examples using limits of sequences, basic limit principles, continuity
Derivatives - as rate of growth, use in graphing, basic calculation rules, chain rule, using computer algebra software
Curve sketching - second derivatives, concavity, critical points and inflection points, basic optimization problem Exponentials and logarithms - derivatives, applications to population growth and decay Antiderivatives and integrals - basic properties, numerical computation and computer algebra systems
Trigonometric functions - basic calculus, applications to medical problems
Differential equations and modeling - individual and population growth models, linear compartment models, stability of equilibria
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In many ways the course is more challenging than the standard science calculus sequence, but students are able to assimilate the diversity of concepts.
It is still necessary to review background concepts (exponentials and logs), but this is eased through the use of numerous biological examples.
Despite much experience with word-processing and game software, students have difficulty utilizing mathematical software and developing simple programs.
YZYbX)&RAlternative Routes to Quantitative Literacy for the Life Sciences: General Biology*SAb b R! Determine the utility of alternative methods to enhance the quantitative components of a large-lecture format GB sequence using:
Quantitative competency exams developed specifically to evaluate the quantitative skills of students taking the GB sequence for science majors;
Modules comprising a Primer of Quantitative Biology designed to accompany a GB sequence, providing for each standard section of the course a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. "ZbSbb.bfrbbffbb!*'Quantitative Competency Exams:b(W Multiple choice exams based upon the skills and concepts appropriate for the Organization and Function of the Cell and the Biodiversity (whole organism, ecology and evolutionary) components of GB. Given at beginning and end of the course to track changes in skills. Require only high-school math skills, with questions placed in a GB context.VXZb`x```+(Goals of Competency Exams:b( (i) inform students at the beginning of a course exactly what types of math they are expected to already be able to do;
(ii) help students be informed about exactly what concepts they don't have a grasp of, so they can go back and refresh their memory; and
(iii) ensure that the class is not held back through having to review material that the students should know upon entering. ,bb$},)Pre- and post-testing were done in GB sections taught by collaborators on this project, emphasizing quantitative skills, and other sections taught by faculty in a standard manner, as a control.`
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Conclusion:( Inclusion of a quantitative emphasis within biology courses can aid students in improving their quantitative skills, if these are made an inherent part of the course and not simply an add-on.
R`b$``@5Do students retain the quantitative skills developed?N We surveyed a sophomore level Genetics class a year after the students had been in the General Biology course, and determined student performance on another quantitative competency exam. We compared exam scores of students who had been in a GB course which emphasized quantitative ideas to those who had been in a standard GB course.OZO !Thus the available evidence suggests that students retain quantitative skills obtained within biology courses through later courses. 6$`!_
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Modules in GB(
The objective is to provide, for each standard section of GB, a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. Most examples are at the level of high-school math, though there are some calculus-level and above examples. A standard format for each module was established and a collection of 57 modules have been developed. Zb``?``8``g`.+Use of Modules within GB(TThese modules have been implemented in a variety of ways in GB.
(i) in lectures as a supplement to lecture material.
(ii) assigned to students as outside reading assignments.
(iii) students have been asked to turn in formal reports as homework assignments based around the additional questions to be answered at the end of each module. |UZS``1``@```B$?63"What quantitative topics are used?##@" Surveys were done at annual meetings of the Ecological Society of America and the Society for the Study of Evolution. The most important quantitative topic for each poster was assessed as well as a listing of all quantitative concepts used for each poster.@4152
Some lessons:
!\1. It is entirely feasible to include diverse mathematical and computational approaches in an entry-level quantitative course for life science students. This can be successful, even though it is in many respects more difficult than a standard science and engineering calculus course, if students see the biological context throughout the course.
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2. Inclusion of a quantitative emphasis within biology courses can aid students to improve their quantitative skills, if these are made an inherent part of the course and not simply an add-on. Evidence suggests that students retain these quantitative skills through later courses.
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3. Instructors can utilize quantitative competency exams to encourage students early in a course to focus on skills they should have mastered and see the connection between these skills and the biological topics in the course.`1.
4. The key quantitative concepts that are used in short scientific communications are basic graphical and statistical ones that are typically covered very little in a formal manner in most undergraduate biology curricula. Visualization/interpretation of data and results are critical to the conceptual foundations of biology training and we should give them higher priority in the curriculum. This might include a formal course on Biological Data Analysis, but needs to be emphasized throughout the science courses students take.
.Zbb2/Future Directions:b$ The BIO2010 Report gives numerous recommendations on quantitative skill development. Accomplishing these above can be aided through:
a. Agreed upon quantitative competency testing across courses.
b. Setting up teaching circles involving the key faculty involved in appropriate groups of courses.
c. Encouraging projects either formally within courses or as part of labs that require quantitative analysis involving the concepts deemed critical for comprehension.
d. Including key quantitative ideas from the beginning in basic entry-level courses - expecting students to utilize skills developed in high school and providing mechanisms to aid those who need remediation. ZbUb/bb@bbfbbbbbbXUImpediments to progressb( Few math faculty at research universities have any appreciation (or interest) in real applications of math
Few biology faculty (not including many recently hired) have strong quantitative skills except in statistics
Multiculturalism of math departments creates problems for faculty/student interactions
Cultures are different few undergrads in math are expected to work on research with faculty, while it is expected that the better biology undergrads will have some exposure to research in filed/lab situations with faculty
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Louis J. Gross
Departments of Ecology and Evolutionary Biology and Mathematics, The Institute for Environmental Modeling, University of Tennessee Knoxville
Financial Support: National Science Foundation (DUE 9150354, DUE 9752339)
National Institutes of Health (GM59924-01)
www.tiem.utk.edu/bioed
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Overview:8Main components of quantitative life science education:8Main components of quantitative life science education:Major BIO2010 RecommendationsPowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPowerPoint Presentation(Summary of Quantitative Recommendations/Concepts of Mathematics and Computer Science PowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPowerPoint Presentation?Additional Quantitative Principles Useful to Biology Students @ Additional Quantitative Principles Useful to Biology Students PowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPowerPoint PresentationPotential curriculaMath/CS Panel CommentsPowerPoint Presentation
Key Points:PowerPoint PresentationPowerPoint PresentationKThe CPA Approach to Quantitative Curriculum Development across DisciplinesPowerPoint PresentationPowerPoint PresentationCollaboratorsProject activities:PowerPoint Presentation= The Entry-level Quantitative Course: Biocalculus Revisited Goals: Methods:*Course 1 Content Discrete Math Topics:*Course 2 Content Calculus and Modeling: Results:SAlternative Routes to Quantitative Literacy for the Life Sciences: General BiologyQuantitative Competency Exams:Goals of Competency Exams:Pre- and post-testing were done in GB sections taught by collaborators on this project, emphasizing quantitative skills, and other sections taught by faculty in a standard manner, as a control.PowerPoint PresentationPowerPoint PresentationConclusion:6Do students retain the quantitative skills developed?PowerPoint PresentationThus the available evidence suggests that students retain quantitative skills obtained within biology courses through later courses. Modules in GBUse of Modules within GB#What quantitative topics are used?PowerPoint PresentationPowerPoint PresentationSome lessons:PowerPoint PresentationPowerPoint PresentationPowerPoint PresentationFuture Directions:Impediments to progressFonts UsedDesign Template
Slide TitlesD 8@_PID_HLINKSAthttp://www.tiem.utk.edu/bioed
&_Louis J. GrossLouis J. Gross.@2
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Summary of BIO2010 quantitative education recommendations
Different viewpoints from the Math/CS Panel summary
Overview of the University of Tennessee programs
Future directions: Suggestions to implement BIO2010 ideas and impediments
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7Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.hZ"H"747Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.Z"")&%96Major BIO2010 Recommendations`Schools should reexamine current approaches to see if they meet the needs of today s undergraduate biology students. Those selecting the new approaches should consider the importance of building a strong foundation in mathematics, and the physical and information sciences to prepare students for research that is increasingly interdisciplinary in character. This implementation should be accompanied by a process of assessment.
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2. Concepts, examples, and techniques from mathematics, and the physical and information sciences should be included in biology courses, and biological concepts and examples should be included in other science courses. Faculty must work collaboratively to integrate mathematics and physical sciences into life science courses as well as providing avenues for incorporating life science examples that reflect the emerging nature of the discipline into courses taught in mathematics and physical sciences.,&";8
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4. Laboratory courses should be as interdisciplinary as possible, since laboratory experiments confront students with real-world observations do not separate well into conventional disciplines * =:
5. All students should be encouraged to pursue independent research as early as is practical in their education. They should be able to receive academic credit for independent research done in collaboration with faculty or with off-campus researchers * >;
6. Seminar-type courses that highlight cutting-edge developments in biology should be provided on a continual and regular basis throughout the four-year undergraduate education of students. Communicating the excitement of biological research is crucial to attracting, retaining, and sustaining a greater diversity of students to the field. These courses would combine presentations by faculty with student projects on research topics. Z ?<
7. Medical school admissions requirements and the Medical College Admissions Test (MCAT) are hindering change in the undergraduate biology curriculum and should be reexamined in light of the recommendations in this report. , @=
8. Faculty development is a crucial component to improving undergraduate biology education. Efforts must be made on individual campuses and nationally to provide faculty the time necessary to refine their o
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0,R&___PPT968 = O PPQ2R?%~85 Overview:
Summary of BIO2010 quantitative education recommendations
Different viewpoints from the Math/CS Panel summary
Overview of the University of Tennessee programs
Future directions: Suggestions to implement BIO2010 ideas and impediments
0
7Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.hZ"H"747Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.Z"")&%96Major BIO2010 Recommendations`Schools should reexamine current approaches to see if they meet the needs of today s undergraduate biology students. Those selecting the new approaches should consider the importance of building a strong foundation in mathematics, and the physical and information sciences to prepare students for research that is increasingly interdisciplinary in character. This implementation should be accompanied by a process of assessment.
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2. Concepts, examples, and techniques from mathematics, and the physical and information sciences should be included in biology courses, and biological concepts and examples should be included in other science courses. Faculty must work collaboratively to integrate mathematics and physical sciences into life science courses as well as providing avenues for incorporating life science examples that reflect the emerging nature of the discipline into courses taught in mathematics and physical sciences.,&";8
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4. Laboratory courses should be as interdisciplinary as possible, since laboratory experiments confront students with real-world observations do not separate well into conventional disciplines * =:
5. All students should be encouraged to pursue independent research as early as is practical in their education. They should be able to receive academic credit for independent research done in collaboration with faculty or with off-campus researchers * >;
6. Seminar-type courses that highlight cutting-edge developments in biology should be provided on a continual and regular basis throughout the four-year undergraduate education of students. Communicating the excitement of biological research is crucial to attracting, retaining, and sustaining a greater diversity of students to the field. These courses would combine presentations by faculty with student projects on research topics. Z ?<
7. Medical school admissions requirements and the Medical College Admissions Test (MCAT) are hindering change in the undergraduate biology curriculum and should be reexamined in light of the recommendations in this report. , @=
8. Faculty development is a crucial component to improving undergraduate biology education. Efforts must be made on individual campuses and nationally to provide faculty the time necessary to refine their own understanding of how the integrative relationships of biology, mathematics, and the physical sciences can be best melded into either existing courses or new courses in the particular areas of science in which they teach.
<" A>'Summary of Quantitative Recommendations((' 1.5 Biology majors headed for research careers need to be educated in a more quantitative manner, which may require the development of new types of courses. We recommend that all biology majors master the concepts listed below and that life science majors become sufficiently familiar with the elements of programming to carry out simulations of physiological, ecological, and evolutionary processes. They should be adept at using computers to acquire and process data, carry out statistical characterization of the data and perform statistical tests, and graphically display data in a variety of representations. Students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases. (Z"B?.Concepts of Mathematics and Computer Science //"(.Calculus
Complex numbers
Functions
Limits
Continuity
The integral
The derivative and linearization
Elementary functions
Fourier series
Multi-dimensional calculus: linear approximations, integration over multiple variables
D ZZZ""$T}C@
qLinear Algebra
Scalars, vectors, matrices
Linear transformations
Eigenvalues and eigenvectors
Invariant subspaces.c d A&DA
Dynamical Systems
Continuous time dynamics equations of motion and their trajectories
Test points, limit cycles, and stability around them
Phase plane analysis
Cooperativity, positive feedback, and negative feedback
Multistability
Discrete time dynamics mappings, stable points, and stable cycles
Sensitivity to initial conditions and chaosDZGZ""H"6*pEB
Probability and Statistics
Probability distributions
Random numbers and stochast
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Covariation, correlation, and independence
Error likelihood@Z~Z" $]0FC
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Information and Computation
Algorithms (with examples)
Computability
Optimization in mathematics and computation
' Bits : information and mutual information@ZZ" GE
Data Structures
Metrics: generalized 'distance' and sequence comparisons
Clustering
Tree-relationships
Graphics: visualizing and displaying data and models for conceptual understanding < "QNHD?Additional Quantitative Principles Useful to Biology Students @@"(?Rate of change
This can be a specific (e.g., per capita) rate of change or a total rate of change of some system component.
Discrete rates of change arise in difference equations, which have associated with them an inherent time-scale.
Continuous rates of change arise as derivatives or partial derivatives, representing instantaneous (relative to the units in which time is scaled) rates.
Jn
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The process of abstracting certain aspects of reality to include in the simplifications we call models.
Scale (spatial and temporal) different questions arise on different scales.
What is included (system variables) depends on the questions addressed, as does the hierarchical level in which the problem is framed (e.g., molecular, cellular, organismal).
There are trade-offs in modeling no one model can address all questions. These trade-offs are between generality, precision, and realism.
Evaluating models depends in part upon the purpose for which the model was constructed. t ZhZNZZ""hN"$`JG
Equilibria and stability
Equilibria arise when a process (or several processes) rate of change is zero.
There can be more than one equilibrium. Multiple stable states are typical of biological systems.
Equilibria can be dynamic, so that a periodic pattern of system response may arise.
There are numerous notions of stability, including not just whether a system that is perturbed from an equilibrium returns to it, but also how the system returns (e.g., how rapidly it does so).
Modifying some system components can lead to destabilization of a previously stable equilibrium, possibly generating entirely new equilibria with differing stability characteristics. BZZ""R
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wStructure
Grouping components of a system affects the kinds of questions addressed and the data required to parameterize the system.
Choosing different aggregated formulations (by sex, age, size, physiological state, activity state) can expand or limit the questions that can be addressed, and data availability can limit the ability to investigate effects of structure.
Geometry of the aggregation can affect the resulting formulation.
Symmetry can be useful in many biological contexts to reduce the complexity of the problem, and situations in which symmetry is lost (symmetry-breaking) can aid in understanding system response.B
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(Interactions
There are relatively few ways for system components to interact. Negative feedbacks arise through competitive and predator-prey type interactions, positive feedback through mutualistic or commensal ones.
Some general properties can be derived based upon these (e.g., 2-species competitive interactions), but even relatively few interacting system components can lead to complex dynamics.
Though ultimately everything is hitched to everything else, significant effects are not automatically transferred through a connected system of interacting components locality can matter.
Sequences of interactions can determine outcomes program order matters.B
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Data and measurement
Only a few basic data types arise (numeric, ordinal, categorical), but these will often be interconnected and expanded (e.g., as vectors or arrays).
Consistency of the units with which one measures a system is important.
A variety of statistical methods exist to characterize single data sets and to make comparisons between data sets. Using such methods with discernment takes practice \ZZ"""NK
WStochasticity
In a stochastic process, individual outcomes cannot be predicted with certainty. Rather, these outcomes are determined randomly according to a probability distribution that arises from the underlying mechanisms of the process. Probabilities for measurements that are continuous (height, weight, etc.), and those that are discrete (sex, cell type) arise in many biological contexts.
Risk can be identified and estimated.
There are ways to determine if an experimental result is significant.
There are instances when stochasticity is significant and averages are not sufficient.ZZ-ZZ
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There are diverse methods to display data.
Simple line and bar graphs are often not sufficient.
Non-linear transformations can yield new insights.>""PM
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Algorithms
These are rules that determine the types of interactions in a system, how decisions are made, and the time course of system response.
These can be thought of as a sequence of actions similar to a computer program, with all the associated options such as assignments, if-then loops, and while-loops.
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"""ROPotential curricula@Four examples were given:
First-year math, perhaps a mixture of discrete and calculus
Quantitative emphasis with first year of math, plus probability/biostatistics, DE plus advanced course
Math I plus CS first year, then Math II plus advanced math/CS course later
Calc and DE in year 1, biostatistics and 1 CS course6Z'" Z'6
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SPMath/CS Panel Comments" There is a distinction between the quantitative biologist, who works at the interface of math/computer science and biology, and the research biologist, who needs familiarity with a range of mathematical and computational ideas without necessarily being expert. Thus, the panel felt that flexibility in offerings is more advisable than a fixed curriculum.
Consider offering quantitatively intensive versions of standard biology courses, with extra credit2Z"g"hTQ
Consider new math courses which condense much of undergrad math into 3-4 semesters
Encourage interdisciplinary modeling courses at both introductory and advanced level, possibly as a first-year and senior seminar
R" $ Key Points:
Success in quantitative life science education requires an integrated approach: formal quantitative courses should be supplemented with explicit quantitative components within life science courses. Z"(
Life science students should be exposed to diverse quantitative concepts: calculus and statistics do not suffice to provide the conceptual quantitative foundations for modern biology."(
H We can t determine a priori who will be the researchers of the future educational initiatives need to be inclusive and not focused just on the elite. Assume all biology students can enhance their quantitative training and proceed to motivate them to realize its importance in real biology.D%Z"$"$"$$URJThe CPA Approach to Quantitative Curriculum Development across DisciplinesKK
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
Understand the Constraints under which your colleagues in other disciplines operate - the limitations on time available in their curriculum for quantitative training.ZZ" Z$$$$$$
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z Work with these colleagues to Prioritize the quantitative concepts their students really need, and ensure that your courses include these.
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.
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M 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.
.NZI$T
Collaborators$
Drs. Beth Mullin and Otto Schwarz (Botany), Susan Riechert (EEB)
Monica Beals, Susan Harrell - Primer of Quantitative Biology
Drs. Sergey Gavrilets (EEB) and Suzanne Lenhart (Math) NIH Short Courses
Drs. Thomas Hallam (EEB) and Simon Levin (Princeton) International Courses
Society for Mathematical Biology Education Committee www.smb.org4`Z"^"~3> (xProject activities:$Conduct a survey of quantitative course requirements of life science students;
Conduct a workshop with researchers and educators in mathematical and quantitative biology to discuss the quantitative component of the undergraduate life science curriculum;
Develop an entry-level quantitative course sequence based upon recommendations from the workshop;
Implement the course in an hypothesis-formulation and testing framework, coupled to appropriate software;|ZNbbbbbbbkb"
Conduct a workshop for life science faculty to discuss methods to enhance the quantitative component of their own courses;
Develop a set of modules to incorporate within a General Biology course sequence, illustrating the utility of simple mathematical methods in numerous areas of biology;
Develop and evaluate quantitative competency exams in General Biology as a method to encourage quantitative skill development;
Survey quantitative topics within short research communications at life science professional society meetings.
hZZbbqbbB$# < The Entry-level Quantitative Course: Biocalculus Revisited4= :b$b$' In response to workshop recommendations, a new entry-level quantitative course for life science students was constructed and has now become the standard math sequence taken by biology students. The prerequisites assumed are Algebra, Geometry, and Trigonometry.&!`$!Goals: 2 b(!!C 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 DZbe````b```u``C%"Methods: b(a 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 Maple. bZbbbebbbb|bbbbtbbbb$K&#RCourse 1 Content Discrete Math Topics:**b ) Descriptive Statistics - Means, variances, using software, histograms, linear and non-linear regression, allometry
Matrix Algebra - using linear algebra software, matrix models in population biology, eigenvalues, eigenvectors, Markov Chains, compartment models
Discrete Probability - Experiments and sample spaces, probability laws, conditional probability and Bayes' theorem, population genetics models Sequences and difference equations - limits of sequences, limit laws, geometric sequence and Malthusian growth
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T'$RCourse 2 Content Calculus and Modeling:**b )] Linear first and second order difference equations - equilibria, stability, logistic map and chaos, population models
Limits of functions - numerical examples using limits of sequences, basic limit principles, continuity
Derivatives - as rate of growth, use in graphing, basic calculation rules, chain rule, using computer algebra software
Curve sketching - second derivatives, concavity, critical points and inflection points, basic optimization problem Exponentials and logarithms - derivatives, applications to population growth and decay Antiderivatives and integrals - basic properties, numerical computation and computer algebra systems
Trigonometric functions - basic calculus, applications to medical problems
Differential equations and modeling - individual and population growth models, linear compartment models, stability of equilibria
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(%Results: b$XThis sequence is now taken by approximately 150 students per semester, and is taught mostly by math instructors and graduate students in math biology.
In many ways the course is more challenging than the standard science calculus sequence, but students are able to assimilate the diversity of concepts.
It is still necessary to review background concepts (exponentials and logs), but this is eased through the use of numerous biological examples.
Despite much experience with word-processing and game software, students have difficulty utilizing mathematical software and developing simple programs.
YZYbX)&RAlternative Routes to Quantitative Literacy for the Life Sciences: General Biology*SAb b R! Determine the utility of alternative methods to enhance the quantitative components of a large-lecture format GB sequence using:
Quantitative competency exams developed specifically to evaluate the quantitative skills of students taking the GB sequence for science majors;
Modules comprising a Primer of Quantitative Biology designed to accompany a GB sequence, providing for each standard section of the course a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. "ZbSbb.bfrbbffbb!*'Quantitative Competency Exams:b(W Multiple choice exams based upon the skills and concepts appropriate for the Organization and Function of the Cell and the Biodiversity (whole organism, ecology and evolutionary) components of GB. Given at beginning and end of the course to track changes in skills. Require only high-school math skills, with questions placed in a GB context.VXZb`x```+(Goals of Competency Exams:b( (i) inform students at the beginning of a course exactly what types of math they are expected to already be able to do;
(ii) help students be informed about exactly what concepts they don't have a grasp of, so they can go back and refresh their memory; and
(iii) ensure that the class is not held back through having to review material that the students should know upon entering. ,bb$},)Pre- and post-testing were done in GB sections taught by collaborators on this project, emphasizing quantitative skills, and other sections taught by faculty in a standard manner, as a control.`
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Conclusion:( Inclusion of a quantitative emphasis within biology courses can aid students in improving their quantitative skills, if these are made an inherent part of the course and not simply an add-on.
R`b$``@5Do students retain the quantitative skills developed?N We surveyed a sophomore level Genetics class a year after the students had been in the General Biology course, and determined student performance on another quantitative competency exam. We compared exam scores of students who had been in a GB course which emphasized quantitative ideas to those who had been in a standard GB course.OZO !Thus the available evidence suggests that students retain quantitative skills obtained within biology courses through later courses. 6$`!_
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Modules in GB(
The objective is to provide, for each standard section of GB, a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. Most examples are at the level of high-school math, though there are some calculus-level and above examples. A standard format for each module was established and a collection of 57 modules have been developed. Zb``?``8``g`.+Use of Modules within GB(TThese modules have been implemented in a variety of ways in GB.
(i) in lectures as a supplement to lecture material.
(ii) assigned to students as outside reading assignments.
(iii) students have been asked to turn in formal reports as homework assignments based around the additional questions to be answered at the end of each module. |UZS``1``@```B$?63"What quantitative topics are used?##@" Surveys were done at annual meetings of the Ecological Society of America and the Society for the Study of Evolution. The most important quantitative topic for each poster was assessed as well as a listing of all quantitative concepts used for each poster.@4152
Some lessons:
!\1. It is entirely feasible to include diverse mathematical and computational approaches in an entry-level quantitative course for life science students. This can be successful, even though it is in many respects more difficult than a standard science and engineering calculus course, if students see the biological context throughout the course.
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2. Inclusion of a quantitative emphasis within biology courses can aid students to improve their quantitative skills, if these are made an inherent part of the course and not simply an add-on. Evidence suggests that students retain these quantitative skills through later courses.
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3. Instructors can utilize quantitative competency exams to encourage students early in a course to focus on skills they should have mastered and see the connection between these skills and the biological topics in the course.`1.
4. The key quantitative concepts that are used in short scientific communications are basic graphical and statistical ones that are typically covered very little in a formal manner in most undergraduate biology curricula. Visualization/interpretation of data and results are critical to the conceptual foundations of biology training and we should give them higher priority in the curriculum. This might include a formal course on Biological Data Analysis, but needs to be emphasized throughout the science courses students take.
.Zbb2/Future Directions:b$ The BIO2010 Report gives numerous recommendations on quantitative skill development. Accomplishing these above can be aided through:
a. Agreed upon quantitative competency testing across courses.
b. Setting up teaching circles involving the key faculty involved in appropriate groups of courses.
c. Encouraging projects either formally within courses or as part of labs that require quantitative analysis involving the concepts deemed critical for comprehension.
d. Including key quantitative ideas from the beginning in basic entry-level courses - expecting students to utilize skills developed in high school and providing mechanisms to aid those who need remediation. ZbUb/bb@bbfbbbbbbXUImpediments to progressb( Few math faculty at research universities have any appreciation (or interest) in real applications of math
Few biology faculty (not including many recently hired) have strong quantitative skills except in statistics
Multiculturalism of math departments creates problems for faculty/student interactions
Cultures are different few undergrads in math are expected to work on research with faculty, while it is expected that the better biology undergrads will have some exposure to research in field/lab situations with faculty
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$,1Integrating Education and Biocomplexity ResearchPoiLouis J. GrosscLouis J. Grossc117Microsoft PowerPoint an@'Summary of Quantitative Recommendations((' 1.5 Biology majors headed for research careers need to be educated in a more quantitative manner, which may require the development of new types of courses. We recommend that all biology majors master the concepts listed below and that life science majors become sufficiently familiar with the elements of programming to carry out simulations of physiological, ecological, and evolutionary processes. They should be adept at using computers to acquire and process data, carry out statistical characterization of the data and perform statistical tests, and graphically display data in a variety of representations. Students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases. (Z"B?.Concepts of Mathematics and Computer Science //"(.Calculus
Complex numbers
Functions
Limits
Continuity
The integral
The derivative and linearization
Elementary functions
Fourier series
Multi-dimensional calculus: linear approximations, integration over multiple variables
D ZZZ""$T}C@
qLinear Algebra
Scalars, vectors, matrices
Linear transformations
Eigenvalues and eigenvectors
Invariant subspaces.c d A&DA
Dynamical Systems
Continuous time dynamics equations of motion and their trajectories
Test points, limit cycles, and stability around them
Phase plane analysis
Cooperativity, positive feedback, and negative feedback
Multistability
Discrete time dynamics mappings, stable points, and stable cycles
Sensitivity to initial conditions and chaosDZGZ""H"6*pEB
Probability and Statistics
Probability distributions
Random numbers and stochastic processes
Covariation, correlation, and independence
Error likelihood@Z~Z" $]0FC
<
Information and Computation
Algorithms (with examples)
Computability
Optimization in mathematics and computation
' Bits : information and mutual information@ZZ" GE
Data Structures
Metrics: generalized 'distance' and sequence comparisons
Clustering
Tree-relationships
Graphics: visualizing and displaying data and models for conceptual understanding < "QNHD?Additional Quantitative Principles Useful to Biology Students @@"(?Rate of change
This can be a specific (e.g., per capita) rate of change or a total rate of change of some system component.
Discrete rates of change arise in difference equations, which have associated with them an inherent time-scale.
Continuous rates of change arise as derivatives or partial derivatives, representing instantaneous (relative to the units in which time is scaled) rates.
Jn
"z"IF"(Modeling
The process of abstracting certain aspects of reality to include in the simplifications we call models.
Scale (spatial and temporal) different questions arise on different scales.
What is included (system variables) depends on the questions addressed, as does the hierarchical level in which the problem is framed (e.g., molecular, cellular, organismal).
There are trade-offs in modeling no one model can address all questions. These trade-offs are between generality, precision, and realism.
Evaluating models depends in part upon the purpose for which the model was constructed. t ZhZNZZ""hN"$`JG
Equilibria and stability
Equilibria arise when a process (or several processes) rate of change is zero.
There can be more than one equilibrium. Multiple stable states are typical of biological systems.
Equilibria can be dynamic, so that a periodic pattern of system response may arise.
There are numerous notions of stability, including not just whether a system that is perturbed from an equilibrium returns to it, but also how the system returns (e.g., how rapidly it does so).
Modifying some system components can lead to destabilization of a previously stable equilibrium, possibly generating entirely new equilibria with differing stability characteristics. BZZ""R
+KH
wStructure
Grouping components of a system affects the kinds of questions addressed and the data required to parameterize the system.
Choosing different aggregated formulations (by sex, age, size, physiological state, activity state) can expand or limit the questions that can be addressed, and data availability can limit the ability to investigate effects of structure.
Geometry of the aggregation can affect the resulting formulation.
Symmetry can be useful in many biological contexts to reduce the complexity of the problem, and situations in which symmetry is lost (symmetry-breaking) can aid in understanding system response.B
ZnZ ""nwLI
(Interactions
There are relatively few ways for system components to interact. Negative feedbacks arise through competitive and predator-prey type interactions, positive feedback through mutualistic or commensal ones.
Some general properties can be derived based upon these (e.g., 2-species competitive interactions), but even relatively few interacting system components can lead to complex dynamics.
Though ultimately everything is hitched to everything else, significant effects are not automatically transferred through a connected system of interacting components locality can matter.
Sequences of interactions can determine outcomes program order matters.B
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MJ
Data and measurement
Only a few basic data types arise (numeric, ordinal, categorical), but these will often be interconnected and expanded (e.g., as vectors or arrays).
Consistency of the units with which one measures a system is important.
A variety of statistical methods exist to characterize single data sets and to make comparisons between data sets. Using such methods with discernment takes practice \ZZ"""NK
WStochasticity
In a stochastic process, individual outcomes cannot be predicted with certainty. Rather, these outcomes are determined randomly according to a probability distribution that arises from the underlying mechanisms of the process. Probabilities for measurements that are continuous (height, weight, etc.), and those that are discrete (sex, cell type) arise in many biological contexts.
Risk can be identified and estimated.
There are ways to determine if an experimental result is significant.
There are instances when stochasticity is significant and averages are not sufficient.ZZ-ZZ
""#,".
0OL
Visualizing
There are diverse methods to display data.
Simple line and bar graphs are often not sufficient.
Non-linear transformations can yield new insights.>""PM
;
Algorithms
These are rules that determine the types of interactions in a system, how decisions are made, and the time course of system response.
These can be thought of as a sequence of actions similar to a computer program, with all the associated options such as assignments, if-then loops, and while-loops.
ZZ,ZZZ"
"""ROPotential curricula@Four examples were given:
First-year math, perhaps a mixture of discrete and calculus
Quantitative emphasis with first year of math, plus probability/biostatistics, DE plus advanced course
Math I plus CS first year, then Math II plus advanced math/CS course later
Calc and DE in year 1, biostatistics and 1 CS course6Z'" Z'6
~
SPMath/CS Panel Comments" There is a distinction between the quantitative biologist, who works at the interface of math/computer science and biology, and the research biologist, who needs familiarity with a range of mathematical and computational ideas without necessarily being expert. Thus, the panel felt that flexibility in offerings is more advisable than a fixed curriculum.
Consider offering quantitatively intensive versions of standard biology courses, with extra credit2Z"g"hTQ
Consider new math courses which condense much of undergrad math into 3-4 semesters
Encourage interdisciplinary modeling courses at both introductory and advanced level, possibly as a first-year and senior seminar
R" $ Key Points:
Success in quantitative life science education requires an integrated approach: formal quantitative courses should be supplemented with explicit quantitative components within life science courses. Z"(
Life science students should be exposed to diverse quantitative concepts: calculus and statistics do not suffice to provide the conceptual quantitative foundations for modern biology."(
H We can t determine a priori who will be the researchers of the future educational initiatives need to be inclusive and not focused just on the elite. Assume all biology students can enhance their quantitative training and proceed to motivate them to realize its importance in real biology.D%Z"$"$"$$URJThe CPA Approach to Quantitative Curriculum Development across DisciplinesKK
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
Understand the Constraints under which your colleagues in other disciplines operate - the limitations on time available in their curriculum for quantitative training.ZZ" Z$$$$$$
WT
z Work with these colleagues to Prioritize the quantitative concepts their students really need, and ensure that your courses include these.
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.
~{Z i$VS
M 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.
.NZI$T
Collaborators$
Drs. Beth Mullin and Otto Schwarz (Botany), Susan Riechert (EEB)
Monica Beals, Susan Harrell - Primer of Quantitative Biology
Drs. Sergey Gavrilets (EEB) and Suzanne Lenhart (Math) NIH Short Courses
Drs. Thomas Hallam (EEB) and Simon Levin (Princeton) International Courses
Society for Mathematical Biology Education Committee www.smb.org4`Z"^"~3> (xProject activities:$Conduct a survey of quantitative course requirements of life science students;
Conduct a workshop with researchers and educators in mathematical and quantitative biology to discuss the quantitative component of the undergraduate life science curriculum;
Develop an entry-level quantitative course sequence based upon recommendations from the workshop;
Implement the course in an hypothesis-formulation and testing framework, coupled to appropriate software;|ZNbbbbbbbkb"
Conduct a workshop for life science faculty to discuss methods to enhance the quantitative component of their own courses;
Develop a set of modules to incorporate within a General Biology course sequence, illustrating the utility of simple mathematical methods in numerous areas of biology;
Develop and evaluate quantitative competency exams in General Biology as a method to encourage quantitative skill development;
Survey quantitative topics within short research communications at life science professional society meetings.
hZZbbqbbB$# < The Entry-level Quantitative Course: Biocalculus Revisited4= :b$b$' In response to workshop recommendations, a new entry-level quantitative course for life science students was constructed and has now become the standard math sequence taken by biology students. The prerequisites assumed are Algebra, Geometry, and Trigonometry.&!`$!Goals: 2 b(!!C 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 DZbe````b```u``C%"Methods: b(a 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 Maple. bZbbbebbbb|bbbbtbbbb$K&#RCourse 1 Content Discrete Math Topics:**b ) Descriptive Statistics - Means, variances, using software, histograms, linear and non-linear regression, allometry
Matrix Algebra - using linear algebra software, matrix models in population biology, eigenvalues, eigenvectors, Markov Chains, compartment models
Discrete Probability - Experiments and sample spaces, probability laws, conditional probability and Bayes' theorem, population genetics models Sequences and difference equations - limits of sequences, limit laws, geometric sequence and Malthusian growth
Zcg_cgcg{cc"gLccHi
T'$RCourse 2 Content Calculus and Modeling:**b )] Linear first and second order difference equations - equilibria, stability, logistic map and chaos, population models
Limits of functions - numerical examples using limits of sequences, basic limit principles, continuity
Derivatives - as rate of growth, use in graphing, basic calculation rules, chain rule, using computer algebra software
Curve sketching - second derivatives, concavity, critical points and inflection points, basic optimization problem Exponentials and logarithms - derivatives, applications to population growth and decay Antiderivatives and integrals - basic properties, numerical computation and computer algebra systems
Trigonometric functions - basic calculus, applications to medical problems
Differential equations and modeling - individual and population growth models, linear compartment models, stability of equilibria
^Zc2gGcgVcgncgdccg<ccgIcg6c#g`cv6
8hD
(%Results: b$XThis sequence is now taken by approximately 150 students per semester, and is taught mostly by math instructors and graduate students in math biology.
In many ways the course is more challenging than the standard science calculus sequence, but students are able to assimilate the diversity of concepts.
It is still necessary to review background concepts (exponentials and logs), but this is eased through the use of numerous biological examples.
Despite much experience with word-processing and game software, students have difficulty utilizing mathematical software and developing simple programs.
YZYbX)&RAlternative Routes to Quantitative Literacy for the Life Sciences: General Biology*SAb b R! Determine the utility of alternative methods to enhance the quantitative components of a large-lecture format GB sequence using:
Quantitative competency exams developed specifically to evaluate the quantitative skills of students taking the GB sequence for science majors;
Modules comprising a Primer of Quantitative Biology designed to accompany a GB sequence, providing for each standard section of the course a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. "ZbSbb.bfrbbffbb!*'Quantitative Competency Exams:b(W Multiple choice exams based upon the skills and concepts appropriate for the Organization and Function of the Cell and the Biodiversity (whole organism, ecology and evolutionary) components of GB. Given at beginning and end of the course to track changes in skills. Require only high-school math skills, with questions placed in a GB context.VXZb`x```+(Goals of Competency Exams:b( (i) inform students at the beginning of a course exactly what types of math they are expected to already be able to do;
(ii) help students be informed about exactly what concepts they don't have a grasp of, so they can go back and refresh their memory; and
(iii) ensure that the class is not held back through having to review material that the students should know upon entering. ,bb$},)Pre- and post-testing were done in GB sections taught by collaborators on this project, emphasizing quantitative skills, and other sections taught by faculty in a standard manner, as a control.`
4`bB
Conclusion:( Inclusion of a quantitative emphasis within biology courses can aid students in improving their quantitative skills, if these are made an inherent part of the course and not simply an add-on.
R`b$``@5Do students retain the quantitative skills developed?N We surveyed a sophomore level Genetics class a year after the students had been in the General Biology course, and determined student performance on another quantitative competency exam. We compared exam scores of students who had been in a GB course which emphasized quantitative ideas to those who had been in a standard GB course.OZO !Thus the available evidence suggests that students retain quantitative skills obtained within biology courses through later courses. 6$`!_
-*
Modules in GB(
The objective is to provide, for each standard section of GB, a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. Most examples are at the level of high-school math, though there are some calculus-level and above examples. A standard format for each module was established and a collection of 57 modules have been developed. Zb``?``8``g`.+Use of Modules within GB(TThese modules have been implemented in a variety of ways in GB.
(i) in lectures as a supplement to lecture material.
(ii) assigned to students as outside reading assignments.
(iii) students have been asked to turn in formal reports as homework assignments based around the additional questions to be answered at the end of each module. |UZS``1``@```B$?63"What quantitative topics are used?##@" Surveys were done at annual meetings of the Ecological Society of America and the Society for the Study of Evolution. The most important quantitative topic for each poster was assessed as well as a listing of all quantitative concepts used for each poster.@4152
Some lessons:
!\1. It is entirely feasible to include diverse mathematical and computational approaches in an entry-level quantitative course for life science students. This can be successful, even though it is in many respects more difficult than a standard science and engineering calculus course, if students see the biological context throughout the course.
>]ZY```@[/,
2. Inclusion of a quantitative emphasis within biology courses can aid students to improve their quantitative skills, if these are made an inherent part of the course and not simply an add-on. Evidence suggests that students retain these quantitative skills through later courses.
&`@0-
3. Instructors can utilize quantitative competency exams to encourage students early in a course to focus on skills they should have mastered and see the connection between these skills and the biological topics in the course.`1.
4. The key quantitative concepts that are used in short scientific communications are basic graphical and statistical ones that are typically covered very little in a formal manner in most undergraduate biology curricula. Visualization/interpretation of data and results are critical to the conceptual foundations of biology training and we should give them higher priority in the curriculum. This might include a formal course on Biological Data Analysis, but needs to be emphasized throughout the science courses students take.
.Zbb2/Future Directions:b$ The BIO2010 Report gives numerous recommendations on quantitative skill development. Accomplishing these above can be aided through:
a. Agreed upon quantitative competency testing across courses.
b. Setting up teaching circles involving the key faculty involved in appropriate groups of courses.
c. Encouraging projects either formally within courses or as part of labs that require quantitative analysis involving the concepts deemed critical for comprehension.
d. Including key quantitative ideas from the beginning in basic entry-level courses - expecting students to utilize skills developed in high school and providing mechanisms to aid those who need remediation. ZbUb/bb@bbfbbbbbbXUImpediments to progressb( Few math faculty at research universities have any appreciation (or interest) in real applications of math
Few biology faculty (not including many recently hired) have strong quantitative skills except in statistics
Multiculturalism of math departments creates problems for faculty/student interactions
Cultures are different few undergrads in math are expected to work on research with faculty, while it is expected that the better biology undergrads will have some exposure to research in field/lab situations with faculty
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0,R&___PPT968 = O PPQ2R?%ܥ85 Overview:
Summary of BIO2010 quantitative education recommendations
Different viewpoints from the Math/CS Panel summary
Overview of the University of Tennessee programs
Future directions: Suggestions to implement BIO2010 ideas and impediments
0
7Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.hZ"H"747Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.Z"")&%96Major BIO2010 Recommendations`Schools should reexamine current approaches to see if they meet the needs of today s undergraduate biology students. Those selecting the new approaches should consider the importance of building a strong foundation in mathematics, and the physical and information sciences to prepare students for research that is increasingly interdisciplinary in character. This implementation should be accompanied by a process of assessment.
2" Z" Z":7
2. Concepts, examples, and techniques from mathematics, and the physical and information sciences should be included in biology courses, and biological concepts and examples should be included in other science courses. Faculty must work collaboratively to integrate mathematics and physical sciences into life science courses as well as providing avenues for incorporating life science examples that reflect the emerging nature of the discipline into courses taught in mathematics and physical sciences.,&";8
8"
<9
4. Laboratory courses should be as interdisciplinary as possible, since laboratory experiments confront students with real-world observations do not separate well into conventional disciplines * =:
5. All students should be encouraged to pursue independent research as early as is practical in their education. They should be able to receive academic credit for independent research done in collaboration with faculty or with off-campus researchers * >;
6. Seminar-type courses that highlight cutting-edge developments in biology should be provided on a continual and regular basis throughout the four-year undergraduate education of students. Communicating the excitement of biological research is crucial to attracting, retaining, and sustaining a greater diversity of students to the field. These courses would combine presentations by faculty with student projects on research topics. Z ?<
7. Medical school admissions requirements and the Medical College Admissions Test (MCAT) are hindering change in the undergraduate biology curriculum and should be reexamined in light of the recommendations in this report. , @=
8. Faculty development is a crucial component to improving undergraduate biology education. Efforts must be made on individual campuses and nationally to provide faculty the time necessary to refine their own understanding of how the integrative relationships of biology, mathematics, and the physical sciences can be best melded into either existing courses or new courses in the particular areas of science in which they teach.
<" A>'Summary of Quantitative Recommendations((' 1.5 Biology majors headed for research careers need to be educated in a more quantitative manner, which may require the development of new types of courses. We recommend that all biology majors master the concepts listed below and that life science majors become sufficiently familiar with the elements of programming to carry out simulations of physiological, ecological, and evolutionary processes. They should be adept at using computers to acquire and process data, carry out statistical characterization of the data and perform statistical tests, and graphically display data in a variety of representations. Students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases. (Z"B?.Concepts of Mathematics and Computer Science //"(.Calculus
Complex numbers
Functions
Limits
Continuity
The integral
The derivative and linearization
Elementary functions
Fourier series
Multi-dimensional calculus: linear approximations, integration over multiple variables
D ZZZ""$T}C@
qLinear Algebra
Scalars, vectors, matrices
Linear transformations
Eigenvalues and eigenvectors
Invariant subspaces.c d A&DA
Dynamical Systems
Continuous time dynamics equations of motion and their trajectories
Test points, limit cycles, and stability around them
Phase plane analysis
Cooperativity, positive feedback, and negative feedback
Multistability
Discrete time dynamics mappings, stable points, and stable cycles
Sensitivity to initial conditions and chaosDZGZ""H"6*pEB
Probability and Statistics
Probability distributions
Random numbers and stochastic processes
Covariation, correlation, and independence
Error likelihood@Z~Z" $]0FC
<
Information and Computation
Algorithms (with examples)
Computability
Optimization in mathematics and computation
' Bits : information and mutual information@ZZ" GE
Data Structures
Metrics: generalized 'distance' and sequence comparisons
Clustering
Tree-relationships
Graphics: visualizing and displaying data and models for conceptual understanding < "QNHD?Additional Quantitative Principles Useful to Biology Students @@"(?Rate of change
This can be a specific (e.g., per capita) rate of change or a total rate of change of some system component.
Discrete rates of change arise in difference equations, which have associated with them an inherent time-scale.
Continuous rates of change arise as derivatives or partial derivatives, representing instantaneous (relative to the units in which time is scaled) rates.
Jn
"z"IF"(Modeling
The process of abstracting certain aspects of reality to include in the simplifications we call models.
Scale (spatial and temporal) different questions arise on different scales.
What is included (system variables) depends on the questions addressed, as does the hierarchical level in which the problem is framed (e.g., molecular, cellular, organismal).
There are trade-offs in modeling no one model can address all questions. These trade-offs are between generality, precision, and realism.
Evaluating models depends in part upon the purpose for which the model was constructed. t ZhZNZZ""hN"$`JG
Equilibria and stability
Equilibria arise when a process (or several processes) rate of change is zero.
There can be more than one equilibrium. Multiple stable states are typical of biological systems.
Equilibria can be dynamic, so that a periodic pattern of system response may arise.
There are numerous notions of stability, including not just whether a system that is perturbed from an equilibrium returns to it, but also how the system returns (e.g., how rapidly it does so).
Modifying some system components can lead to destabilization of a previously stable equilibrium, possibly generating entirely new equilibria with differing stability characteristics. BZZ""R
+KH
wStructure
Grouping components of a system affects the kinds of questions addressed and the data required to parameterize the system.
Choosing different aggregated formulations (by sex, age, size, physiological state, activity state) can expand or limit the questions that can be addressed, and data availability can limit the ability to investigate effects of structure.
Geometry of the aggregation can affect the resulting formulation.
Symmetry can be useful in many biological contexts to reduce the complexity of the problem, and situations in which symmetry is lost (symmetry-breaking) can aid in understanding system response.B
ZnZ ""nwLI
(Interactions
There are relatively few ways for system components to interact. Negative feedbacks arise through competitive and predator-prey type interactions, positive feedback through mutualistic or commensal ones.
Some general properties can be derived based upon these (e.g., 2-species competitive interactions), but even relatively few interacting system components can lead to complex dynamics.
Though ultimately everything is hitched to everything else, significant effects are not automatically transferred through a connected system of interacting components locality can matter.
Sequences of interactions can determine outcomes program order matters.B
ZZ""6
MJ
Data and measurement
Only a few basic data types arise (numeric, ordinal, categorical), but these will often be interconnected and expanded (e.g., as vectors or arrays).
Consistency of the units with which one measures a system is important.
A variety of statistical methods exist to characterize single data sets and to make comparisons between data sets. Using such methods with discernment takes practice \ZZ"""NK
WStochasticity
In a stochastic process, individual outcomes cannot be predicted with certainty. Rather, these outcomes are determined randomly according to a probability distribution that arises from the underlying mechanisms of the process. Probabilities for measurements that are continuous (height, weight, etc.), and those that are discrete (sex, cell type) arise in many biological contexts.
Risk can be identified and estimated.
There are ways to determine if an experimental result is significant.
There are instances when stochasticity is significant and averages are not sufficient.ZZ-ZZ
""#,".
0OL
Visualizing
There are diverse methods to display data.
Simple line and bar graphs are often not sufficient.
Non-linear transformations can yield new insights.>""PM
;
Algorithms
These are rules that determine the types of interactions in a system, how decisions are made, and the time course of system response.
These can be thought of as a sequence of actions similar to a computer program, with all the associated options such as assignments, if-then loops, and while-loops.
ZZ,ZZZ"
"""ROPotential curricula@Four examples were given:
First-year math, perhaps a mixture of discrete and calculus
Quantitative emphasis with first year of math, plus probability/biostatistics, DE plus advanced course
Math I plus CS first year, then Math II plus advanced math/CS course later
Calc and DE in year 1, biostatistics and 1 CS course6Z'" Z'6
~
SPMath/CS Panel Comments" There is a distinction between the quantitative biologist, who works at the interface of math/computer science and biology, and the research biologist, who needs familiarity with a range of mathematical and computational ideas without necessarily being expert. Thus, the panel felt that flexibility in offerings is more advisable than a fixed curriculum.
Consider offering quantitatively intensive versions of standard biology courses, with extra credit2Z"g"hTQ
Consider new math courses which condense much of undergrad math into 3-4 semesters
Encourage interdisciplinary modeling courses at both introductory and advanced level, possibly as a first-year and senior seminar
R" $ Key Points:
Success in quantitative life science education requires an integrated approach: formal quantitative courses should be supplemented with explicit quantitative components within life science courses. Z"(
Life science students should be exposed to diverse quantitative concepts: calculus and statistics do not suffice to provide the conceptual quantitative foundations for modern biology."(
H We can t determine a priori who will be the researchers of the future educational initiatives need to be inclusive and not focused just on the elite. Assume all biology students can enhance their quantitative training and proceed to motivate them to realize its importance in real biology.D%Z"$"$"$$URJThe CPA Approach to Quantitative Curriculum Development across DisciplinesKK
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
Understand the Constraints under which your colleagues in other disciplines operate - the limitations on time available in their curriculum for quantitative training.ZZ" Z$$$$$$
WT
z Work with these colleagues to Prioritize the quantitative concepts their students really need, and ensure that your courses include these.
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.
~{Z i$VS
M 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.
.NZI$T
Collaborators$
Drs. Beth Mullin and Otto Schwarz (Botany), Susan Riechert (EEB)
Monica Beals, Susan Harrell - Primer of Quantitative Biology
Drs. Sergey Gavrilets (EEB) and Suzanne Lenhart (Math) NIH Short Courses
Drs. Thomas Hallam (EEB) and Simon Levin (Princeton) International Courses
Society for Mathematical Biology Education Committee www.smb.org4`Z"^"~3> (xProject activities:$Conduct a survey of quantitative course requirements of life science students;
Conduct a workshop with researchers and educators in mathematical and quantitative biology to discuss the quantitative component of the undergraduate life science curriculum;
Develop an entry-level quantitative course sequence based upon recommendations from the workshop;
Implement the course in an hypothesis-formulation and testing framework, coupled to appropriate software;|ZNbbbbbbbkb"
Conduct a workshop for life science faculty to discuss methods to enhance the quantitative component of their own courses;
Develop a set of modules to incorporate within a General Biology course sequence, illustrating the utility of simple mathematical methods in numerous areas of biology;
Develop and evaluate quantitative competency exams in General Biology as a method to encourage quantitative skill development;
Survey quantitative topics within short research communications at life science professional society meetings.
hZZbbqbbB$# < The Entry-level Quantitative Course: Biocalculus Revisited4= :b$b$' In response to workshop recommendations, a new entry-level quantitative course for life science students was constructed and has now become the standard math sequence taken by biology students. The prerequisites assumed are Algebra, Geometry, and Trigonometry.&!`$!Goals: 2 b(!!C 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 DZbe````b```u``C%"Methods: b(a 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 Maple. bZbbbebbbb|bbbbtbbbb$K&#RCourse 1 Content Discrete Math Topics:**b ) Descriptive Statistics - Means, variances, using software, histograms, linear and non-linear regression, allometry
Matrix Algebra - using linear algebra software, matrix models in population biology, eigenvalues, eigenvectors, Markov Chains, compartment models
Discrete Probability - Experiments and sample spaces, probability laws, conditional probability and Bayes' theorem, population genetics models Sequences and difference equations - limits of sequences, limit laws, geometric sequence and Malthusian growth
Zcg_cgcg{cc"gLccHi
T'$RCourse 2 Content Calculus and Modeling:**b )] Linear first and second order difference equations - equilibria, stability, logistic map and chaos, population models
Limits of functions - numerical examples using limits of sequences, basic limit principles, continuity
Derivatives - as rate of growth, use in graphing, basic calculation rules, chain rule, using computer algebra software
Curve sketching - second derivatives, concavity, critical points and inflection points, basic optimization problem Exponentials and logarithms - derivatives, applications to population growth and decay Antiderivatives and integrals - basic properties, numerical computation and computer algebra systems
Trigonometric functions - basic calculus, applications to medical problems
Differential equations and modeling - individual and population growth models, linear compartment models, stability of equilibria
^Zc2gGcgVcgncgdccg<ccgIcg6c#g`cv6
8hD
(%Results: b$XThis sequence is now taken by approximately 150 students per semester, and is taught mostly by math instructors and graduate students in math biology.
In many ways the course is more challenging than the standard science calculus sequence, but students are able to assimilate the diversity of concepts.
It is still necessary to review background concepts (exponentials and logs), but this is eased through the use of numerous biological examples.
Despite much experience with word-processing and game software, students have difficulty utilizing mathematical software and developing simple programs.
YZYbX)&RAlternative Routes to Quantitative Literacy for the Life Sciences: General Biology*SAb b R! Determine the utility of alternative methods to enhance the quantitative components of a large-lecture format GB sequence using:
Quantitative competency exams developed specifically to evaluate the quantitative skills of students taking the GB sequence for science majors;
Modules comprising a Primer of Quantitative Biology designed to accompany a GB sequence, providing for each standard section of the course a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. "ZbSbb.bfrbbffbb!*'Quantitative Competency Exams:b(W Multiple choice exams based upon the skills and concepts appropriate for the Organization and Function of the Cell and the Biodiversity (whole organism, ecology and evolutionary) components of GB. Given at beginning and end of the course to track changes in skills. Require only high-school math skills, with questions placed in a GB context.VXZb`x```+(Goals of Competency Exams:b( (i) inform students at the beginning of a course exactly what types of math they are expected to already be able to do;
(ii) help students be informed about exactly what concepts they don't have a grasp of, so they can go back and refresh their memory; and
(iii) ensure that the class is not held back through having to review material that the students should know upon entering. ,bb$},)Pre- and post-testing were done in GB sections taught by collaborators on this project, emphasizing quantitative skills, and other sections taught by faculty in a standard manner, as a control.`
4`bB
Conclusion:( Inclusion of a quantitative emphasis within biology courses can aid students in improving their quantitative skills, if these are made an inherent part of the course and not simply an add-on.
R`b$``@5Do students retain the quantitative skills developed?N We surveyed a sophomore level Genetics class a year after the students had been in the General Biology course, and determined student performance on another quantitative competency exam. We compared exam scores of students who had been in a GB course which emphasized quantitative ideas to those who had been in a standard GB course.OZO !Thus the available evidence suggests that students retain quantitative skills obtained within biology courses through later courses. 6$`!_
-*
Modules in GB(
The objective is to provide, for each standard section of GB, a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. Most examples are at the level of high-school math, though there are some calculus-level and above examples. A standard format for each module was established and a collection of 57 modules have been developed. Zb``?``8``g`.+Use of Modules within GB(TThese modules have been implemented in a variety of ways in GB.
(i) in lectures as a supplement to lecture material.
(ii) assigned to students as outside reading assignments.
(iii) students have been asked to turn in formal reports as homework assignments based around the additional questions to be answered at the end of each module. |UZS``1``@```B$?63"What quantitative topics are used?##@" Surveys were done at annual meetings of the Ecological Society of America and the Society for the Study of Evolution. The most important quantitative topic for each poster was assessed as well as a listing of all quantitative concepts used for each poster.@4152
Some lessons:
!\1. It is entirely feasible to include diverse mathematical and computational approaches in an entry-level quantitative course for life science students. This can be successful, even though it is in many respects more difficult than a standard science and engineering calculus course, if students see the biological context throughout the course.
>]ZY```@[/,
2. Inclusion of a quantitative emphasis within biology courses can aid students to improve their quantitative skills, if these are made an inherent part of the course and not simply an add-on. Evidence suggests that students retain these quantitative skills through later courses.
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3. Instructors can utilize quantitative competency exams to encourage students early in a course to focus on skills they should have mastered and see the connection between these skills and the biological topics in the course.`1.
4. The key quantitative concepts that are used in short scientific communications are basic graphical and statistical ones that are typically covered very little in a formal manner in most undergraduate biology curricula. Visualization/interpretation of data and results are critical to the conceptual foundations of biology training and we should give them higher priority in the curriculum. This might include a formal course on Biological Data Analysis, but needs to be emphasized throughout the science courses students take.
.Zbb2/Future Directions:b$ The BIO2010 Report gives numerous recommendations on quantitative skill development. Accomplishing these above can be aided through:
a. Agreed upon quantitative competency testing across courses.
b. Setting up teaching circles involving the key faculty involved in appropriate groups of courses.
c. Encouraging projects either formally within courses or as part of labs that require quantitative analysis involving the concepts deemed critical for comprehension.
d. Including key quantitative ideas from the beginning in basic entry-level courses - expecting students to utilize skills developed in high school and providing mechanisms to aid those who need remediation. ZbUb/bb@bbfbbbbbbXUImpediments to progressb(~Few math faculty at research universities have any appreciation (or interest) in real applications of math
Few biology faculty (not including many recently hired) have strong quantitative skills except in statistics
Multiculturalism of math departments creates problems for faculty/student interactions
Cultures are different few undergrads in math are expected to work on research with faculty, while it is expected that the better biology undergrads will have some exposure to research in field/lab situations with faculty
Math faculty prefer rigor (proof) over breadth
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l<Quantitative Education for Life Sciences: BIO2010 and Beyond==
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Louis J. Gross
Departments of Ecology and Evolutionary Biology and Mathematics, The Institute for Environmental Modeling, University of Tennessee Knoxville
Financial Support: National Science Foundation (DUE 9150354, DUE 9752339)
National Institutes of Health (GM59924-01)
www.tiem.utk.edu/bioed
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0,R&___PPT968 = O PPQ2R?%ܥ85 Overview:
Summary of BIO2010 quantitative education recommendations
Different viewpoints from the Math/CS Panel summary
Overview of the University of Tennessee programs
Future directions: Suggestions to implement BIO2010 ideas and impediments
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7Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.hZ"H"747Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.Z"")&%96Major BIO2010 Recommendations`Schools should reexamine current approaches to see if they meet the needs of today s undergraduate biology students. Those selecting the new approaches should consider the importance of building a strong foundation in mathematics, and the physical and information sciences to prepare students for research that is increasingly interdisciplinary in character. This implementation should be accompanied by a process of assessment.
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2. Concepts, examples, and techniques from mathematics, and the physical and information sciences should be included in biology courses, and biological concepts and examples should be included in other science courses. Faculty must work collaboratively to integrate mathematics and physical sciences into life science courses as well as providing avenues for incorporating life science examples that reflect the emerging nature of the discipline into courses taught in mathematics and physical sciences.,&";8
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4. Laboratory courses should be as interdisciplinary as possible, since laboratory experiments confront students with real-world observations do not separate well into conventional disciplines * =:
5. All students should be encouraged to pursue independent research as early as is practical in their education. They should be able to receive academic credit for independent research done in collaboration with faculty or with off-campus researchers * >;
6. Seminar-type courses that highlight cutting-edge developments in biology should be provided on a continual and regular basis throughout the four-year undergraduate education of students. Communicating the excitement of biological research is crucial to attracting, retaining, and sustaining a greater diversity of students to the field. These courses would combine presentations by faculty with student projects on research topics. Z ?<
7. Medical school admissions requirements and the Medical College Admissions Test (MCAT) are hindering change in the undergraduate biology curriculum and should be reexamined in light of the recommendations in this report. , @=
8. Faculty development is a crucial component to improving undergraduate biology education. Efforts must be made on individual campuses and nationally to provide faculty the time necessary to refine their own understanding of how the integrative relationships of biology, mathematics, and the physical sciences can be best melded into either existing courses or new courses in the particular areas of science in which they teach.
<" A>'Summary of Quantitative Recommendations((' 1.5 Biology majors headed for research careers need to be educated in a more quantitative manner, which may require the development of new types of courses. We recommend that all biology majors master the concepts listed below and that life science majors become sufficiently familiar with the elements of programming to carry out simulations of physiological, ecological, and evolutionary processes. They should be adept at using computers to acquire and process data, carry out statistical characterization of the data and perform statistical tests, and graphically display data in a variety of representations. Students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases. (Z"B?.Concepts of Mathematics and Computer Science //"(.Calculus
Complex numbers
Functions
Limits
Continuity
The integral
The derivative and linearization
Elementary functions
Fourier series
Multi-dimensional calculus: linear approximations, integration over multiple variables
D ZZZ""$T}C@
qLinear Algebra
Scalars, vectors, matrices
Linear transformations
Eigenvalues and eigenvectors
Invariant subspaces.c d A&DA
Dynamical Systems
Continuous time dynamics equations of motion and their trajectories
Test points, limit cycles, and stability around them
Phase plane analysis
Cooperativity, positive feedback, and negative feedback
Multistability
Discrete time dynamics mappings, stable points, and stable cycles
Sensitivity to initial conditions and chaosDZGZ""H"6*pEB
Probability and Statistics
Probability distributions
Random numbers and stochastic processes
Covariation, correlation, and independence
Error likelihood@Z~Z" $]0FC
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Information and Computation
Algorithms (with examples)
Computability
Optimization in mathematics and computation
' Bits : information and mutual information@ZZ" GE
Data Structures
Metrics: generalized 'distance' and sequence comparisons
Clustering
Tree-relationships
Graphics: visualizing and displaying data and models for conceptual understanding < "QNHD?Additional Quantitative Principles Useful to Biology Students @@"(?Rate of change
This can be a specific (e.g., per capita) rate of change or a total rate of change of some system component.
Discrete rates of change arise in difference equations, which have associated with them an inherent time-scale.
Continuous rates of change arise as derivatives or partial derivatives, representing instantaneous (relative to the units in which time is scaled) rates.
Jn
"z"IF"(Modeling
The process of abstracting certain aspects of reality to include in the simplifications we call models.
Scale (spatial and temporal) different questions arise on different scales.
What is included (system variables) depends on the questions addressed, as does the hierarchical level in which the problem is framed (e.g., molecular, cellular, organismal).
There are trade-offs in modeling no one model can address all questions. These trade-offs are between generality, precision, and realism.
Evaluating models depends in part upon the purpose for which the model was constructed. t ZhZNZZ""hN"$`JG
Equilibria and stability
Equilibria arise when a process (or several processes) rate of change is zero.
There can be more than one equilibrium. Multiple stable states are typical of biological systems.
Equilibria can be dynamic, so that a periodic pattern of system response may arise.
There are numerous notions of stability, including not just whether a system that is perturbed from an equilibrium returns to it, but also how the system returns (e.g., how rapidly it does so).
Modifying some system components can lead to destabilization of a previously stable equilibrium, possibly generating entirely new equilibria with differing stability characteristics. BZZ""R
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wStructure
Grouping components of a system affects the kinds of questions addressed and the data required to parameterize the system.
Choosing different aggregated formulations (by sex, age, size, physiological state, activity state) can expand or limit the questions that can be addressed, and data availability can limit the ability to investigate effects of structure.
Geometry of the aggregation can affect the resulting formulation.
Symmetry can be useful in many biological contexts to reduce the complexity of the problem, and situations in which symmetry is lost (symmetry-breaking) can aid in understanding system response.B
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(Interactions
There are relatively few ways for system components to interact. Negative feedbacks arise through competitive and predator-prey type interactions, positive feedback through mutualistic or commensal ones.
Some general properties can be derived based upon these (e.g., 2-species competitive interactions), but even relatively few interacting system components can lead to complex dynamics.
Though ultimately everything is hitched to everything else, significant effects are not automatically transferred through a connected system of interacting components locality can matter.
Sequences of interactions can determine outcomes program order matters.B
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Data and measurement
Only a few basic data types arise (numeric, ordinal, categorical), but these will often be interconnected and expanded (e.g., as vectors or arrays).
Consistency of the units with which one measures a system is important.
A variety of statistical methods exist to characterize single data sets and to make comparisons between data sets. Using such methods with discernment takes practice \ZZ"""NK
WStochasticity
In a stochastic process, individual outcomes cannot be predicted with certainty. Rather, these outcomes are determined randomly according to a probability distribution that arises from the underlying mechanisms of the process. Probabilities for measurements that are continuous (height, weight, etc.), and those that are discrete (sex, cell type) arise in many biological contexts.
Risk can be identified and estimated.
There are ways to determine if an experimental result is significant.
There are instances when stochasticity is significant and averages are not sufficient.ZZ-ZZ
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Visualizing
There are diverse methods to display data.
Simple line and bar graphs are often not sufficient.
Non-linear transformations can yield new insights.>""PM
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Algorithms
These are rules that determine the types of interactions in a system, how decisions are made, and the time course of system response.
These can be thought of as a sequence of actions similar to a computer program, with all the associated options such as assignments, if-then loops, and while-loops.
ZZ,ZZZ"
"""ROPotential curricula@Four examples were given:
First-year math, perhaps a mixture of discrete and calculus
Quantitative emphasis with first year of math, plus probability/biostatistics, DE plus advanced course
Math I plus CS first year, then Math II plus advanced math/CS course later
Calc and DE in year 1, biostatistics and 1 CS course6Z'" Z'6
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SPMath/CS Panel Comments" There is a distinction between the quantitative biologist, who works at the interface of math/computer science and biology, and the research biologist, who needs familiarity with a range of mathematical and computational ideas without necessarily being expert. Thus, the panel felt that flexibility in offerings is more advisable than a fixed curriculum.
Consider offering quantitatively intensive versions of standard biology courses, with extra credit2Z"g"hTQ
Consider new math courses which condense much of undergrad math into 3-4 semesters
Encourage interdisciplinary modeling courses at both introductory and advanced level, possibly as a first-year and senior seminar
R" $ Key Points:
Success in quantitative life science education requires an integrated approach: formal quantitative courses should be supplemented with explicit quantitative components within life science courses. Z"(
Life science students should be exposed to diverse quantitative concepts: calculus and statistics do not suffice to provide the conceptual quantitative foundations for modern biology."(
H We can t determine a priori who will be the researchers of the future educational initiatives need to be inclusive and not focused just on the elite. Assume all biology students can enhance their quantitative training and proceed to motivate them to realize its importance in real biology.D%Z"$"$"$$URJThe CPA Approach to Quantitative Curriculum Development across DisciplinesKK
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
Understand the Constraints under which your colleagues in other disciplines operate - the limitations on time available in their curriculum for quantitative training.ZZ" Z$$$$$$
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z Work with these colleagues to Prioritize the quantitative concepts their students really need, and ensure that your courses include these.
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.
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M 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.
.NZI$T
Collaborators$
Drs. Beth Mullin and Otto Schwarz (Botany), Susan Riechert (EEB)
Monica Beals, Susan Harrell - Primer of Quantitative Biology
Drs. Sergey Gavrilets (EEB) and Suzanne Lenhart (Math) NIH Short Courses
Drs. Thomas Hallam (EEB) and Simon Levin (Princeton) International Courses
Society for Mathematical Biology Education Committee www.smb.org4`Z"^"~3> (xProject activities:$Conduct a survey of quantitative course requirements of life science students;
Conduct a workshop with researchers and educators in mathematical and quantitative biology to discuss the quantitative component of the undergraduate life science curriculum;
Develop an entry-level quantitative course sequence based upon recommendations from the workshop;
Implement the course in an hypothesis-formulation and testing framework, coupled to appropriate software;|ZNbbbbbbbkb"
Conduct a workshop for life science faculty to discuss methods to enhance the quantitative component of their own courses;
Develop a set of modules to incorporate within a General Biology course sequence, illustrating the utility of simple mathematical methods in numerous areas of biology;
Develop and evaluate quantitative competency exams in General Biology as a method to encourage quantitative skill development;
Survey quantitative topics within short research communications at life science professional society meetings.
hZZbbqbbB$# < The Entry-level Quantitative Course: Biocalculus Revisited4= :b$b$' In response to workshop recommendations, a new entry-level quantitative course for life science students was constructed and has now become the standard math sequence taken by biology students. The prerequisites assumed are Algebra, Geometry, and Trigonometry.&!`$!Goals: 2 b(!!C 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 DZbe````b```u``C%"Methods: b(a 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 Maple. bZbbbebbbb|bbbbtbbbb$K&#RCourse 1 Content Discrete Math Topics:**b ) Descriptive Statistics - Means, variances, using software, histograms, linear and non-linear regression, allometry
Matrix Algebra - using linear algebra software, matrix models in population biology, eigenvalues, eigenvectors, Markov Chains, compartment models
Discrete Probability - Experiments and sample spaces, probability laws, conditional probability and Bayes' theorem, population genetics models Sequences and difference equations - limits of sequences, limit laws, geometric sequence and Malthusian growth
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T'$RCourse 2 Content Calculus and Modeling:**b )] Linear first and second order difference equations - equilibria, stability, logistic map and chaos, population models
Limits of functions - numerical examples using limits of sequences, basic limit principles, continuity
Derivatives - as rate of growth, use in graphing, basic calculation rules, chain rule, using computer algebra software
Curve sketching - second derivatives, concavity, critical points and inflection points, basic optimization problem Exponentials and logarithms - derivatives, applications to population growth and decay Antiderivatives and integrals - basic properties, numerical computation and computer algebra systems
Trigonometric functions - basic calculus, applications to medical problems
Differential equations and modeling - individual and population growth models, linear compartment models, stability of equilibria
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(%Results: b$XThis sequence is now taken by approximately 150 students per semester, and is taught mostly by math instructors and graduate students in math biology.
In many ways the course is more challenging than the standard science calculus sequence, but students are able to assimilate the diversity of concepts.
It is still necessary to review background concepts (exponentials and logs), but this is eased through the use of numerous biological examples.
Despite much experience with word-processing and game software, students have difficulty utilizing mathematical software and developing simple programs.
YZYbX)&RAlternative Routes to Quantitative Literacy for the Life Sciences: General Biology*SAb b R! Determine the utility of alternative methods to enhance the quantitative components of a large-lecture format GB sequence using:
Quantitative competency exams developed specifically to evaluate the quantitative skills of students taking the GB sequence for science majors;
Modules comprising a Primer of Quantitative Biology designed to accompany a GB sequence, providing for each standard section of the course a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. "ZbSbb.bfrbbffbb!*'Quantitative Competency Exams:b(W Multiple choice exams based upon the skills and concepts appropriate for the Organization and Function of the Cell and the Biodiversity (whole organism, ecology and evolutionary) components of GB. Given at beginning and end of the course to track changes in skills. Require only high-school math skills, with questions placed in a GB context.VXZb`x```+(Goals of Competency Exams:b( (i) inform students at the beginning of a course exactly what types of math they are expected to already be able to do;
(ii) help students be informed about exactly what concepts they don't have a grasp of, so they can go back and refresh their memory; and
(iii) ensure that the class is not held back through having to review material that the students should know upon entering. ,bb$},)Pre- and post-testing were done in GB sections taught by collaborators on this project, emphasizing quantitative skills, and other sections taught by faculty in a standard manner, as a control.`
4`bB
Conclusion:( Inclusion of a quantitative emphasis within biology courses can aid students in improving their quantitative skills, if these are made an inherent part of the course and not simply an add-on.
R`b$``@5Do students retain the quantitative skills developed?N We surveyed a sophomore level Genetics class a year after the students had been in the General Biology course, and determined student performance on another quantitative competency exam. We compared exam scores of students who had been in a GB course which emphasized quantitative ideas to those who had been in a standard GB course.OZO !Thus the available evidence suggests that students retain quantitative skills obtained within biology courses through later courses. 6$`!_
-*
Modules in GB(
The objective is to provide, for each standard section of GB, a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. Most examples are at the level of high-school math, though there are some calculus-level and above examples. A standard format for each module was established and a collection of 57 modules have been developed. Zb``?``8``g`.+Use of Modules within GB(TThese modules have been implemented in a variety of ways in GB.
(i) in lectures as a supplement to lecture material.
(ii) assigned to students as outside reading assignments.
(iii) students have been asked to turn in formal reports as homework assignments based around the additional questions to be answered at the end of each module. |UZS``1``@```B$?63"What quantitative topics are used?##@" Surveys were done at annual meetings of the Ecological Society of America and the Society for the Study of Evolution. The most important quantitative topic for each poster was assessed as well as a listing of all quantitative concepts used for each poster.@4152
Some lessons:
!\1. It is entirely feasible to include diverse mathematical and computational approaches in an entry-level quantitative course for life science students. This can be successful, even though it is in many respects more difficult than a standard science and engineering calculus course, if students see the biological context throughout the course.
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2. Inclusion of a quantitative emphasis within biology courses can aid students to improve their quantitative skills, if these are made an inherent part of the course and not simply an add-on. Evidence suggests that students retain these quantitative skills through later courses.
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3. Instructors can utilize quantitative competency exams to encourage students early in a course to focus on skills they should have mastered and see the connection between these skills and the biological topics in the course.`1.
4. The key quantitative concepts that are used in short scientific communications are basic graphical and statistical ones that are typically covered very little in a formal manner in most undergraduate biology curricula. Visualization/interpretation of data and results are critical to the conceptual foundations of biology training and we should give them higher priority in the curriculum. This might include a formal course on Biological Data Analysis, but needs to be emphasized throughout the science courses students take.
.Zbb2/Future Directions:b$ The BIO2010 Report gives numerous recommendations on quantitative skill development. Accomplishing these above can be aided through:
a. Agreed upon quantitative competency testing across courses.
b. Setting up teaching circles involving the key faculty involved in appropriate groups of courses.
c. Encouraging projects either formally within courses or as part of labs that require quantitative analysis involving the concepts deemed critical for comprehension.
d. Including key quantitative ideas from the beginning in basic entry-level courses - expecting students to utilize skills developed in high school and providing mechanisms to aid those who need remediation. ZbUb/bb@bbfbbbbbbXUImpediments to progressb(~Few math faculty at research universities have any appreciation (or interest) in real applications of math
Few biology faculty (not including many recently hired) have strong quantitative skills except in statistics
Multiculturalism of math departments creates problems for faculty/student interactions
Cultures are different few undergrads in math are expected to work on research with faculty, while it is expected that the better biology undergrads will have some exposure to research in field/lab situations with faculty
Math faculty prefer rigor (proof) over breadth
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0,R&___PPT96O Q&?%85 Overview:
Summary of BIO2010 quantitative education recommendations
Different viewpoints from the Math/CS Panel summary
Overview of the University of Tennessee projects
Future directions: Suggestions to implement BIO2010 ideas and impediments
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7Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.hZ"H"747Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.Z"")&%96Major BIO2010 Recommendations`Schools should reexamine current approaches to see if they meet the needs of today s undergraduate biology students. Those selecting the new approaches should consider the importance of building a strong foundation in mathematics, and the physical and information sciences to prepare students for research that is increasingly interdisciplinary in character. This implementation should be accompanied by a process of assessment.
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2. Concepts, examples, and techniques from mathematics, and the physical and information sciences should be included in biology courses, and biological concepts and examples should be included in other science courses. Faculty must work collaboratively to integrate mathematics and physical sciences into life science courses as well as providing avenues for incorporating life science examples that reflect the emerging nature of the discipline into courses taught in mathematics and physical sciences.";8
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4. Laboratory courses should be as interdisciplinary as possible, since laboratory experiments confront students with real-world observations do not separate well into conventional disciplines * =:
5. All students should be encouraged to pursue independent research as early as is practical in their education. They should be able to receive academic credit for independent research done in collaboration with faculty or with off-campus researchers * >;
6. Seminar-type courses that highlight cutting-edge developments in biology should be provided on a continual and regular basis throughout the four-year undergraduate education of students. Communicating the excitement of biological research is crucial to attracting, retaining, and sustaining a greater diversity of students to the field. These courses would combine presentations by faculty with student projects on research topics. Z ?<
7. Medical school admissions requirements and the Medical College Admissions Test (MCAT) are hindering change in the undergraduate biology curriculum and should be reexamined in light of the recommendations in this report. , @=
8. Faculty development is a crucial component to improving undergraduate biology education. Efforts must be made on individual campuses and nationally to provide faculty the time necessary to refine their own understanding of how the integrative relationships of biology, mathematics, and the physical sciences can be best melded into either existing courses or new courses in the particular areas of science in which they teach.
,A>'Summary of Quantitative Recommendations((' 1.5 Biology majors headed for research careers need to be educated in a more quantitative manner, which may require the development of new types of courses. We recommend that all biology majors master the concepts listed below and that life science majors become sufficiently familiar with the elements of programming to carry out simulations of physiological, ecological, and evolutionary processes. They should be adept at using computers to acquire and process data, carry out statistical characterization of the data and perform statistical tests, and graphically display data in a variety of representations. Students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases. (Z"B?.Concepts of Mathematics and Computer Science //"(.Calculus
Complex numbers
Functions
Limits
Continuity
The integral
The derivative and linearization
Elementary functions
Fourier series
Multi-dimensional calculus: linear approximations, integration over multiple variables
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qLinear Algebra
Scalars, vectors, matrices
Linear transformations
Eigenvalues and eigenvectors
Invariant subspaces.c d A&DA
Dynamical Systems
Continuous time dynamics equations of motion and their trajectories
Test points, limit cycles, and stability around them
Phase plane analysis
Cooperativity, positive feedback, and negative feedback
Multistability
Discrete time dynamics mappings, stable points, and stable cycles
Sensitivity to initial conditions and chaosDZGZ""H"6*pEB
Probability and Statistics
Probability distributions
Random numbers and stochastic processes
Covariation, correlation, and independence
Error likelihood@Z~Z" $]0FC
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Information and Computation
Algorithms (with examples)
Computability
Optimization in mathematics and computation
' Bits : information and mutual information@ZZ" GE
Data Structures
Metrics: generalized 'distance' and sequence comparisons
Clustering
Tree-relationships
Graphics: visualizing and displaying data and models for conceptual understanding < "QNHD?Additional Quantitative Principles Useful to Biology Students @@"(?Rate of change
This can be a specific (e.g., per capita) rate of change or a total rate of change of some system component.
Discrete rates of change arise in difference equations, which have associated with them an inherent time-scale.
Continuous rates of change arise as derivatives or partial derivatives, representing instantaneous (relative to the units in which time is scaled) rates.
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The process of abstracting certain aspects of reality to include in the simplifications we call models.
Scale (spatial and temporal) different questions arise on different scales.
What is included (system variables) depends on the questions addressed, as does the hierarchical level in which the problem is framed (e.g., molecular, cellular, organismal).
There are trade-offs in modeling no one model can address all questions. These trade-offs are between generality, precision, and realism.
Evaluating models depends in part upon the purpose for which the model was constructed. \ ZHZ""hN"$`JG
Equilibria and stability
Equilibria arise when a process (or several processes) rate of change is zero.
There can be more than one equilibrium. Multiple stable states are typical of biological systems.
Equilibria can be dynamic, so that a periodic pattern of system response may arise.
There are numerous notions of stability, including not just whether a system that is perturbed from an equilibrium returns to it, but also how the system returns (e.g., how rapidly it does so).
Modifying some system components can lead to destabilization of a previously stable equilibrium, possibly generating entirely new equilibria with differing stability characteristics. BZZ""R
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wStructure
Grouping components of a system affects the kinds of questions addressed and the data required to parameterize the system.
Choosing different aggregated formulations (by sex, age, size, physiological state, activity state) can expand or limit the questions that can be addressed, and data availability can limit the ability to investigate effects of structure.
Geometry of the aggregation can affect the resulting formulation.
Symmetry can be useful in many biological contexts to reduce the complexity of the problem, and situations in which symmetry is lost (symmetry-breaking) can aid in understanding system response.B
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(Interactions
There are relatively few ways for system components to interact. Negative feedbacks arise through competitive and predator-prey type interactions, positive feedback through mutualistic or commensal ones.
Some general properties can be derived based upon these (e.g., 2-species competitive interactions), but even relatively few interacting system components can lead to complex dynamics.
Though ultimately everything is hitched to everything else, significant effects are not automatically transferred through a connected system of interacting components locality can matter.
Sequences of interactions can determine outcomes program order matters.B
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Data and measurement
Only a few basic data types arise (numeric, ordinal, categorical), but these will often be interconnected and expanded (e.g., as vectors or arrays).
Consistency of the units with which one measures a system is important.
A variety of statistical methods exist to characterize single data sets and to make comparisons between data sets. Using such methods with discernment takes practice \ZZ"""NK
WStochasticity
In a stochastic process, individual outcomes cannot be predicted with certainty. Rather, these outcomes are determined randomly according to a probability distribution that arises from the underlying mechanisms of the process. Probabilities for measurements that are continuous (height, weight, etc.), and those that are discrete (sex, cell type) arise in many biological contexts.
Risk can be identified and estimated.
There are ways to determine if an experimental result is significant.
There are instances when stochasticity is significant and averages are not sufficient.ZZ-ZZ
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Visualizing
There are diverse methods to display data.
Simple line and bar graphs are often not sufficient.
Non-linear transformations can yield new insights.>""PM
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Algorithms
These are rules that determine the types of interactions in a system, how decisions are made, and the time course of system response.
These can be thought of as a sequence of actions similar to a computer program, with all the associated options such as assignments, if-then loops, and while-loops.
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"""ROPotential curricula@Four examples were given:
First-year math, perhaps a mixture of discrete and calculus
Quantitative emphasis with first year of math, plus probability/biostatistics, DE plus advanced course
Math I plus CS first year, then Math II plus advanced math/CS course later
Calc and DE in year 1, biostatistics and 1 CS course6Z'" Z'6
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SPMath/CS Panel Comments There is a distinction between the quantitative biologist, who works at the interface of math/computer science and biology, and the research biologist, who needs familiarity with a range of mathematical and computational ideas without necessarily being expert. Thus, the panel felt that flexibility in offerings is more advisable than a fixed curriculum.
Consider offering quantitatively intensive versions of standard biology courses, with extra credit2Z"g"hTQ
Consider new math courses which condense much of undergrad math into 3-4 semesters
Encourage interdisciplinary modeling courses at both introductory and advanced level, possibly as a first-year and senior seminar
N" $ Key Points:
Success in quantitative life science education requires an integrated approach: formal quantitative courses should be supplemented with explicit quantitative components within life science courses. Z"(
Life science students should be exposed to diverse quantitative concepts: calculus and statistics do not suffice to provide the conceptual quantitative foundations for modern biology."(
H We can t determine a priori who will be the researchers of the future educational initiatives need to be inclusive and not focused just on the elite. Assume all biology students can enhance their quantitative training and proceed to motivate them to realize its importance in real biology.D%Z"$"$"$$URJThe CPA Approach to Quantitative Curriculum Development across DisciplinesKK
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
Understand the Constraints under which your colleagues in other disciplines operate - the limitations on time available in their curriculum for quantitative training.ZZ$$$$$$
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z Work with these colleagues to Prioritize the quantitative concepts their students really need, and ensure that your courses include these.
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.
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M 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.
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Collaborators$
Drs. Beth Mullin and Otto Schwarz (Botany), Susan Riechert (EEB)
Monica Beals, Susan Harrell - Primer of Quantitative Biology
Drs. Sergey Gavrilets (EEB) and Suzanne Lenhart (Math) NIH Short Courses
Drs. Thomas Hallam (EEB) and Simon Levin (Princeton) International Courses
Society for Mathematical Biology Education Committee www.smb.org4`Z"^"~3> (xProject activities:$Conduct a survey of quantitative course requirements of life science students;
Conduct a workshop with researchers and educators in mathematical and quantitative biology to discuss the quantitative component of the undergraduate life science curriculum;
Develop an entry-level quantitative course sequence based upon recommendations from the workshop;
Implement the course in an hypothesis-formulation and testing framework, coupled to appropriate software;|ZNbbbbbbbkb"
Conduct a workshop for life science faculty to discuss methods to enhance the quantitative component of their own courses;
Develop a set of modules to incorporate within a General Biology course sequence, illustrating the utility of simple mathematical methods in numerous areas of biology;
Develop and evaluate quantitative competency exams in General Biology as a method to encourage quantitative skill development;
Survey quantitative topics within short research communications at life science professional society meetings.
hZZbbqbbB$# < The Entry-level Quantitative Course: Biocalculus Revisited4= :b$b$' In response to workshop recommendations, a new entry-level quantitative course for life science students was constructed and has now become the standard math sequence taken by biology students. The prerequisites assumed are Algebra, Geometry, and Trigonometry.&!`$!Goals: 2 b(!!C 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 DZbe````b```u``C%"Methods: b(a 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 Maple. bZbbbebbbb|bbbbtbbbb$K&#RCourse 1 Content Discrete Math Topics:**b ) Descriptive Statistics - Means, variances, using software, histograms, linear and non-linear regression, allometry
Matrix Algebra - using linear algebra software, matrix models in population biology, eigenvalues, eigenvectors, Markov Chains, compartment models
Discrete Probability - Experiments and sample spaces, probability laws, conditional probability and Bayes' theorem, population genetics models Sequences and difference equations - limits of sequences, limit laws, geometric sequence and Malthusian growth
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T'$RCourse 2 Content Calculus and Modeling:**b )] Linear first and second order difference equations - equilibria, stability, logistic map and chaos, population models
Limits of functions - numerical examples using limits of sequences, basic limit principles, continuity
Derivatives - as rate of growth, use in graphing, basic calculation rules, chain rule, using computer algebra software
Curve sketching - second derivatives, concavity, critical points and inflection points, basic optimization problem Exponentials and logarithms - derivatives, applications to population growth and decay Antiderivatives and integrals - basic properties, numerical computation and computer algebra systems
Trigonometric functions - basic calculus, applications to medical problems
Differential equations and modeling - individual and population growth models, linear compartment models, stability of equilibria
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(%Results: b$XThis sequence is now taken by approximately 150 students per semester, and is taught mostly by math instructors and graduate students in math biology.
In many ways the course is more challenging than the standard science calculus sequence, but students are able to assimilate the diversity of concepts.
It is still necessary to review background concepts (exponentials and logs), but this is eased through the use of numerous biological examples.
Despite much experience with word-processing and game software, students have difficulty utilizing mathematical software and developing simple programs.
YZYbX)&RAlternative Routes to Quantitative Literacy for the Life Sciences: General Biology*SAb b R! Determine the utility of alternative methods to enhance the quantitative components of a large-lecture format GB sequence using:
Quantitative competency exams developed specifically to evaluate the quantitative skills of students taking the GB sequence for science majors;
Modules comprising a Primer of Quantitative Biology designed to accompany a GB sequence, providing for each standard section of the course a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. "ZbSbb.bfrbbffbb!*'Quantitative Competency Exams:b(W Multiple choice exams based upon the skills and concepts appropriate for the Organization and Function of the Cell and the Biodiversity (whole organism, ecology and evolutionary) components of GB. Given at beginning and end of the course to track changes in skills. Require only high-school math skills, with questions placed in a GB context.VXZb`x```+(Goals of Competency Exams:b( (i) inform students at the beginning of a course exactly what types of math they are expected to already be able to do;
(ii) help students be informed about exactly what concepts they don't have a grasp of, so they can go back and refresh their memory; and
(iii) ensure that the class is not held back through having to review material that the students should know upon entering. ,bb$},)Pre- and post-testing were done in GB sections taught by collaborators on this project, emphasizing quantitative skills, and other sections taught by faculty in a standard manner, as a control.`
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Conclusion:( Inclusion of a quantitative emphasis within biology courses can aid students in improving their quantitative skills, if these are made an inherent part of the course and not simply an add-on.
R`b$``@5Do students retain the quantitative skills developed?N We surveyed a sophomore level Genetics class a year after the students had been in the General Biology course, and determined student performance on another quantitative competency exam. We compared exam scores of students who had been in a GB course which emphasized quantitative ideas to those who had been in a standard GB course.OZO !Thus the available evidence suggests that students retain quantitative skills obtained within biology courses through later courses. 6$`!_
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Modules in GB(
The objective is to provide, for each standard section of GB, a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. Most examples are at the level of high-school math, though there are some calculus-level and above examples. A standard format for each module was established and a collection of 57 modules have been developed. Zb``?``8``g`.+Use of Modules within GB(TThese modules have been implemented in a variety of ways in GB.
(i) in lectures as a supplement to lecture material.
(ii) assigned to students as outside reading assignments.
(iii) students have been asked to turn in formal reports as homework assignments based around the additional questions to be answered at the end of each module. |UZS``1``@```B$?63"What quantitative topics are used?##@" Surveys were done at annual meetings of the Ecological Society of America and the Society for the Study of Evolution. The most important quantitative topic for each poster was assessed as well as a listing of all quantitative concepts used for each poster.@4152
Some lessons:
!\1. It is entirely feasible to include diverse mathematical and computational approaches in an entry-level quantitative course for life science students. This can be successful, even though it is in many respects more difficult than a standard science and engineering calculus course, if students see the biological context throughout the course.
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2. Inclusion of a quantitative emphasis within biology courses can aid students to improve their quantitative skills, if these are made an inherent part of the course and not simply an add-on. Evidence suggests that students retain these quantitative skills through later courses.
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3. Instructors can utilize quantitative competency exams to encourage students early in a course to focus on skills they should have mastered and see the connection between these skills and the biological topics in the course.`1.
4. The key quantitative concepts that are used in short scientific communications are basic graphical and statistical ones that are typically covered very little in a formal manner in most undergraduate biology curricula. Visualization/interpretation of data and results are critical to the conceptual foundations of biology training and we should give them higher priority in the curriculum. This might include a formal course on Biological Data Analysis, but needs to be emphasized throughout the science courses students take.
.Zbb2/Future Directions:b$ The BIO2010 Report gives numerous recommendations on quantitative skill development. Accomplishing these above can be aided through:
a. Agreed upon quantitative competency testing across courses.
b. Setting up teaching circles involving the key faculty involved in appropriate groups of courses.
c. Encouraging projects either formally within courses or as part of labs that require quantitative analysis involving the concepts deemed critical for comprehension.
d. Including key quantitative ideas from the beginning in basic entry-level courses - expecting students to utilize skills developed in high school and providing mechanisms to aid those who need remediation. ZbUb/bb@bbfbbbbbbXUImpediments to progressb(~Few math faculty at research universities have any appreciation (or interest) in real applications of math
Few biology faculty (not including many recently hired) have strong quantitative skills except in statistics
Multiculturalism of math departments creates problems for faculty/student interactions
Cultures are different few undergrads in math are expected to work on research with faculty, while it is expected that the better biology undergrads will have some exposure to research in field/lab situations with faculty
Math faculty prefer rigor (proof) over breadth
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0,R&___PPT96O Q&?%,85 Overview:
Summary of BIO2010 quantitative education recommendations
Different viewpoints from the Math/CS Panel summary
Overview of the University of Tennessee projects
Future directions: Suggestions to implement BIO2010 ideas and impediments
0
7Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.hZ"H"747Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.Z"")&%96Major BIO2010 Recommendations`Schools should reexamine current approaches to see if they meet the needs of today s undergraduate biology students. Those selecting the new approaches should consider the importance of building a strong foundation in mathematics, and the physical and information sciences to prepare students for research that is increasingly interdisciplinary in character. This implementation should be accompanied by a process of assessment.
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2. Concepts, examples, and techniques from mathematics, and the physical and information sciences should be included in biology courses, and biological concepts and examples should be included in other science courses. Faculty must work collaboratively to integrate mathematics and physical sciences into life science courses as well as providing avenues for incorporating life science examples that reflect the emerging nature of the discipline into courses taught in mathematics and physical sciences.";8
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4. Laboratory courses should be as interdisciplinary as possible, since laboratory experiments confront students with real-world observations do not separate well into conventional disciplines * =:
5. All students should be encouraged to pursue independent research as early as is practical in their education. They should be able to receive academic credit for independent research done in collaboration with faculty or with off-campus researchers * >;
6. Seminar-type courses that highlight cutting-edge developments in biology should be provided on a continual and regular basis throughout the four-year undergraduate education of students. Communicating the excitement of biological research is crucial to attracting, retaining, and sustaining a greater diversity of students to the field. These courses would combine presentations by faculty with student projects on research topics. 0""?<
7. Medical school admissions requirements and the Medical College Admissions Test (MCAT) are hindering change in the undergraduate biology curriculum and should be reexamined in light of the recommendations in this report. , @=
8. Faculty development is a crucial component to improving undergraduate biology education. Efforts must be made on individual campuses and nationally to provide faculty the time necessary to refine their own understanding of how the integrative relationships of biology, mathematics, and the physical sciences can be best melded into either existing courses or new courses in the particular areas of science in which they teach.
,A>'Summary of Quantitative Recommendations((' 1.5 Biology majors headed for research careers need to be educated in a more quantitative manner, which may require the development of new types of courses. We recommend that all biology majors master the concepts listed below and that life science majors become sufficiently familiar with the elements of programming to carry out simulations of physiological, ecological, and evolutionary processes. They should be adept at using computers to acquire and process data, carry out statistical characterization of the data and perform statistical tests, and graphically display data in a variety of representations. Students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases. (Z"B?.Concepts of Mathematics and Computer Science //"(.Calculus
Complex numbers
Functions
Limits
Continuity
The integral
The derivative and linearization
Elementary functions
Fourier series
Multi-dimensional calculus: linear approximations, integration over multiple variables
D ZZZ""$T}C@
qLinear Algebra
Scalars, vectors, matrices
Linear transformations
Eigenvalues and eigenvectors
Invariant subspaces.c d A&DA
Dynamical Systems
Continuous time dynamics equations of motion and their trajectories
Test points, limit cycles, and stability around them
Phase plane analysis
Cooperativity, positive feedback, and negative feedback
Multistability
Discrete time dynamics mappings, stable points, and stable cycles
Sensitivity to initial conditions and chaosDZGZ""H"6*pEB
Probability and Statistics
Probability distributions
Random numbers and stochastic processes
Covariation, correlation, and independence
Error likelihood@Z~Z" $]0FC
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Information and Computation
Algorithms (with examples)
Computability
Optimization in mathematics and computation
' Bits : information and mutual information@ZZ" GE
Data Structures
Metrics: generalized 'distance' and sequence comparisons
Clustering
Tree-relationships
Graphics: visualizing and displaying data and models for conceptual understanding < "QNHD?Additional Quantitative Principles Useful to Biology Students @@"(?Rate of change
This can be a specific (e.g., per capita) rate of change or a total rate of change of some system component.
Discrete rates of change arise in difference equations, which have associated with them an inherent time-scale.
Continuous rates of change arise as derivatives or partial derivatives, representing instantaneous (relative to the units in which time is scaled) rates.
>}
"z"IF"(Modeling
The process of abstracting certain aspects of reality to include in the simplifications we call models.
Scale (spatial and temporal) different questions arise on different scales.
What is included (system variables) depends on the questions addressed, as does the hierarchical level in which the problem is framed (e.g., molecular, cellular, organismal).
There are trade-offs in modeling no one model can address all questions. These trade-offs are between generality, precision, and realism.
Evaluating models depends in part upon the purpose for which the model was constructed. \ ZHZ""hN"$`JG
Equilibria and stability
Equilibria arise when a process (or several processes) rate of change is zero.
There can be more than one equilibrium. Multiple stable states are typical of biological systems.
Equilibria can be dynamic, so that a periodic pattern of system response may arise.
There are numerous notions of stability, including not just whether a system that is perturbed from an equilibrium returns to it, but also how the system returns (e.g., how rapidly it does so).
Modifying some system components can lead to destabilization of a previously stable equilibrium, possibly generating entirely new equilibria with differing stability characteristics. BZZ""R
+KH
wStructure
Grouping components of a system affects the kinds of questions addressed and the data required to parameterize the system.
Choosing different aggregated formulations (by sex, age, size, physiological state, activity state) can expand or limit the questions that can be addressed, and data availability can limit the ability to investigate effects of structure.
Geometry of the aggregation can affect the resulting formulation.
Symmetry can be useful in many biological contexts to reduce the complexity of the problem, and situations in which symmetry is lost (symmetry-breaking) can aid in understanding system response.B
ZnZ ""nwLI
(Interactions
There are relatively few ways for system components to interact. Negative feedbacks arise through competitive and predator-prey type interactions, positive feedback through mutualistic or commensal ones.
Some general properties can be derived based upon these (e.g., 2-species competitive interactions), but even relatively few interacting system components can lead to complex dynamics.
Though ultimately everything is hitched to everything else, significant effects are not automatically transferred through a connected system of interacting components locality can matter.
Sequences of interactions can determine outcomes program order matters.B
ZZ""6
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Data and measurement
Only a few basic data types arise (numeric, ordinal, categorical), but these will often be interconnected and expanded (e.g., as vectors or arrays).
Consistency of the units with which one measures a system is important.
A variety of statistical methods exist to characterize single data sets and to make comparisons between data sets. Using such methods with discernment takes practice \ZZ"""NK
WStochasticity
In a stochastic process, individual outcomes cannot be predicted with certainty. Rather, these outcomes are determined randomly according to a probability distribution that arises from the underlying mechanisms of the process. Probabilities for measurements that are continuous (height, weight, etc.), and those that are discrete (sex, cell type) arise in many biological contexts.
Risk can be identified and estimated.
There are ways to determine if an experimental result is significant.
There are instances when stochasticity is significant and averages are not sufficient.ZZ-ZZ
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Visualizing
There are diverse methods to display data.
Simple line and bar graphs are often not sufficient.
Non-linear transformations can yield new insights.>""PM
;
Algorithms
These are rules that determine the types of interactions in a system, how decisions are made, and the time course of system response.
These can be thought of as a sequence of actions similar to a computer program, with all the associated options such as assignments, if-then loops, and while-loops.
Z,ZZZ"
"""ROPotential curricula@Four examples were given:
First-year math, perhaps a mixture of discrete and calculus
Quantitative emphasis with first year of math, plus probability/biostatistics, DE plus advanced course
Math I plus CS first year, then Math II plus advanced math/CS course later
Calc and DE in year 1, biostatistics and 1 CS course6Z'" Z'6
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SPMath/CS Panel Comments There is a distinction between the quantitative biologist, who works at the interface of math/computer science and biology, and the research biologist, who needs familiarity with a range of mathematical and computational ideas without necessarily being expert. Thus, the panel felt that flexibility in offerings is more advisable than a fixed curriculum.
Consider offering quantitatively intensive versions of standard biology courses, with extra credit2Z"g"hTQ
Consider new math courses which condense much of undergrad math into 3-4 semesters
Encourage interdisciplinary modeling courses at both introductory and advanced level, possibly as a first-year and senior seminar
N" $ Key Points:
Success in quantitative life science education requires an integrated approach: formal quantitative courses should be supplemented with explicit quantitative components within life science courses. Z"(
Life science students should be exposed to diverse quantitative concepts: calculus and statistics do not suffice to provide the conceptual quantitative foundations for modern biology."(
H We can t determine a priori who will be the researchers of the future educational initiatives need to be inclusive and not focused just on the elite. Assume all biology students can enhance their quantitative training and proceed to motivate them to realize its importance in real biology.D%Z"$"$"$$URJThe CPA Approach to Quantitative Curriculum Development across DisciplinesKK
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
Understand the Constraints under which your colleagues in other disciplines operate - the limitations on time available in their curriculum for quantitative training.ZZ$$$$$$
WT
z Work with these colleagues to Prioritize the quantitative concepts their students really need, and ensure that your courses include these.
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.
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M 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.
.NZI$T
Collaborators$
Drs. Beth Mullin and Otto Schwarz (Botany), Susan Riechert (EEB)
Monica Beals, Susan Harrell - Primer of Quantitative Biology
Drs. Sergey Gavrilets (EEB) and Suzanne Lenhart (Math) NIH Short Courses
Drs. Thomas Hallam (EEB) and Simon Levin (Princeton) International Courses
Society for Mathematical Biology Education Committee www.smb.org4`Z"^"~3> (xProject activities:$Conduct a survey of quantitative course requirements of life science students;
Conduct a workshop with researchers and educators in mathematical and quantitative biology to discuss the quantitative component of the undergraduate life science curriculum;
Develop an entry-level quantitative course sequence based upon recommendations from the workshop;
Implement the course in an hypothesis-formulation and testing framework, coupled to appropriate software;|ZNbbbbbbbkb"
Conduct a workshop for life science faculty to discuss methods to enhance the quantitative component of their own courses;
Develop a set of modules to incorporate within a General Biology course sequence, illustrating the utility of simple mathematical methods in numerous areas of biology;
Develop and evaluate quantitative competency exams in General Biology as a method to encourage quantitative skill development;
Survey quantitative topics within short research communications at life science professional society meetings.
hZZbbqbbB$# < The Entry-level Quantitative Course: Biocalculus Revisited4= :b$b$' In response to workshop recommendations, a new entry-level quantitative course for life science students was constructed and has now become the standard math sequence taken by biology students. The prerequisites assumed are Algebra, Geometry, and Trigonometry.&!`$!Goals: 2 b(!!C 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 DZbe````b```u``C%"Methods: b(a 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 Maple. bZbbbebbbb|bbbbtbbbb$K&#RCourse 1 Content Discrete Math Topics:**b ) Descriptive Statistics - Means, variances, using software, histograms, linear and non-linear regression, allometry
Matrix Algebra - using linear algebra software, matrix models in population biology, eigenvalues, eigenvectors, Markov Chains, compartment models
Discrete Probability - Experiments and sample spaces, probability laws, conditional probability and Bayes' theorem, population genetics models Sequences and difference equations - limits of sequences, limit laws, geometric sequence and Malthusian growth
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T'$RCourse 2 Content Calculus and Modeling:**b )] Linear first and second order difference equations - equilibria, stability, logistic map and chaos, population models
Limits of functions - numerical examples using limits of sequences, basic limit principles, continuity
Derivatives - as rate of growth, use in graphing, basic calculation rules, chain rule, using computer algebra software
Curve sketching - second derivatives, concavity, critical points and inflection points, basic optimization problem Exponentials and logarithms - derivatives, applications to population growth and decay Antiderivatives and integrals - basic properties, numerical computation and computer algebra systems
Trigonometric functions - basic calculus, applications to medical problems
Differential equations and modeling - individual and population growth models, linear compartment models, stability of equilibria
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(%Results: b$XThis sequence is now taken by approximately 150 students per semester, and is taught mostly by math instructors and graduate students in math biology.
In many ways the course is more challenging than the standard science calculus sequence, but students are able to assimilate the diversity of concepts.
It is still necessary to review background concepts (exponentials and logs), but this is eased through the use of numerous biological examples.
Despite much experience with word-processing and game software, students have difficulty utilizing mathematical software and developing simple programs.
YZYbX)&RAlternative Routes to Quantitative Literacy for the Life Sciences: General Biology*SAb b R! Determine the utility of alternative methods to enhance the quantitative components of a large-lecture format GB sequence using:
Quantitative competency exams developed specifically to evaluate the quantitative skills of students taking the GB sequence for science majors;
Modules comprising a Primer of Quantitative Biology designed to accompany a GB sequence, providing for each standard section of the course a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. "ZbSbb.bfrbbffbb!*'Quantitative Competency Exams:b(W Multiple choice exams based upon the skills and concepts appropriate for the Organization and Function of the Cell and the Biodiversity (whole organism, ecology and evolutionary) components of GB. Given at beginning and end of the course to track changes in skills. Require only high-school math skills, with questions placed in a GB context.VXZb`x```+(Goals of Competency Exams:b( (i) inform students at the beginning of a course exactly what types of math they are expected to already be able to do;
(ii) help students be informed about exactly what concepts they don't have a grasp of, so they can go back and refresh their memory; and
(iii) ensure that the class is not held back through having to review material that the students should know upon entering. ,bb$},)Pre- and post-testing were done in GB sections taught by collaborators on this project, emphasizing quantitative skills, and other sections taught by faculty in a standard manner, as a control.`
4`bB
Conclusion:( Inclusion of a quantitative emphasis within biology courses can aid students in improving their quantitative skills, if these are made an inherent part of the course and not simply an add-on.
R`b$``@5Do students retain the quantitative skills developed?N We surveyed a sophomore level Genetics class a year after the students had been in the General Biology course, and determined student performance on another quantitative competency exam. We compared exam scores of students who had been in a GB course which emphasized quantitative ideas to those who had been in a standard GB course.OZO !Thus the available evidence suggests that students retain quantitative skills obtained within biology courses through later courses. 6$`!_
-*
Modules in GB(
The objective is to provide, for each standard section of GB, a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. Most examples are at the level of high-school math, though there are some calculus-level and above examples. A standard format for each module was established and a collection of 57 modules have been developed. Zb``?``8``g`.+Use of Modules within GB(TThese modules have been implemented in a variety of ways in GB.
(i) in lectures as a supplement to lecture material.
(ii) assigned to students as outside reading assignments.
(iii) students have been asked to turn in formal reports as homework assignments based around the additional questions to be answered at the end of each module. |UZS``1``@```B$?63"What quantitative topics are used?##@" Surveys were done at annual meetings of the Ecological Society of America and the Society for the Study of Evolution. The most important quantitative topic for each poster was assessed as well as a listing of all quantitative concepts used for each poster.@4152
Some lessons:
!\1. It is entirely feasible to include diverse mathematical and computational approaches in an entry-level quantitative course for life science students. This can be successful, even though it is in many respects more difficult than a standard science and engineering calculus course, if students see the biological context throughout the course.
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2. Inclusion of a quantitative emphasis within biology courses can aid students to improve their quantitative skills, if these are made an inherent part of the course and not simply an add-on. Evidence suggests that students retain these quantitative skills through later courses.
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3. Instructors can utilize quantitative competency exams to encourage students early in a course to focus on skills they should have mastered and see the connection between these skills and the biological topics in the course.`1.
4. The key quantitative concepts that are used in short scientific communications are basic graphical and statistical ones that are typically covered very little in a formal manner in most undergraduate biology curricula. Visualization/interpretation of data and results are critical to the conceptual foundations of biology training and we should give them higher priority in the curriculum. This might include a formal course on Biological Data Analysis, but needs to be emphasized throughout the science courses students take.
.Zbb2/Future Directions:b$ The BIO2010 Report gives numerous recommendations on quantitative skill development. Accomplishing these above can be aided through:
a. Agreed upon quantitative competency testing across courses.
b. Setting up teaching circles involving the key faculty involved in appropriate groups of courses.
c. Encouraging projects either formally within courses or as part of labs that require quantitative analysis involving the concepts deemed critical for comprehension.
d. Including key quantitative ideas from the beginning in basic entry-level courses - expecting students to utilize skills developed in high school and providing mechanisms to aid those who need remediation. ZbUb/bb@bbfbbbbbbXUImpediments to progressb(~Few math faculty at research universities have any appreciation (or interest) in real applications of math
Few biology faculty (not including many recently hired) have strong quantitative skills except in statistics
Multiculturalism of math departments creates problems for faculty/student interactions
Cultures are different few undergrads in math are expected to work on research with faculty, while it is expected that the better biology undergrads will have some exposure to research in field/lab situations with faculty
Math faculty prefer rigor (proof) over breadth
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0,R&___PPT96O Q&?%.85 Overview:
Summary of BIO2010 quantitative education recommendations
Different viewpoints from the Math/CS Panel summary
Overview of the University of Tennessee projects
Future directions: Suggestions to implement BIO2010 ideas and impediments
0
7Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.hZ"H"747Main components of quantitative life science education:"87$ (i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative bio,
bio quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.Z"")&%96Major BIO2010 Recommendations`Schools should reexamine current approaches to see if they meet the needs of today s undergraduate biology students. Those selecting the new approaches should consider the importance of building a strong foundation in mathematics, and the physical and information sciences to prepare students for research that is increasingly interdisciplinary in character. This implementation should be accompanied by a process of assessment.
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2. Concepts, examples, and techniques from mathematics, and the physical and information sciences should be included in biology courses, and biological concepts and examples should be included in other science courses. Faculty must work collaboratively to integrate mathematics and physical sciences into life science courses as well as providing avenues for incorporating life science examples that reflect the emerging nature of the discipline into courses taught in mathematics and physical sciences.";8
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4. Laboratory courses should be as interdisciplinary as possible, since laboratory experiments confront students with real-world observations do not separate well into conventional disciplines * =:
5. All students should be encouraged to pursue independent research as early as is practical in their education. They should be able to receive academic credit for independent research done in collaboration with faculty or with off-campus researchers * >;
6. Seminar-type courses that highlight cutting-edge developments in biology should be provided on a continual and regular basis throughout the four-year undergraduate education of students. Communicating the excitement of biological research is crucial to attracting, retaining, and sustaining a greater diversity of students to the field. These courses would combine presentations by faculty with student projects on research topics. 0""?<
7. Medical school admissions requirements and the Medical College Admissions Test (MCAT) are hindering change in the undergraduate biology curriculum and should be reexamined in light of the recommendations in this report. , @=
8. Faculty development is a crucial component to improving undergraduate biology education. Efforts must be made on individual campuses and nationally to provide faculty the time necessary to refine their own understanding of how the integrative relationships of biology, mathematics, and the physical sciences can be best melded into either existing courses or new courses in the particular areas of science in which they teach.
,A>'Summary of Quantitative Recommendations((' 1.5 Biology majors headed for research careers need to be educated in a more quantitative manner, which may require the development of new types of courses. We recommend that all biology majors master the concepts listed below and that life science majors become sufficiently familiar with the elements of programming to carry out simulations of physiological, ecological, and evolutionary processes. They should be adept at using computers to acquire and process data, carry out statistical characterization of the data and perform statistical tests, and graphically display data in a variety of representations. Students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases. (Z"B?.Concepts of Mathematics and Computer Science //"(.Calculus
Complex numbers
Functions
Limits
Continuity
The integral
The derivative and linearization
Elementary functions
Fourier series
Multi-dimensional calculus: linear approximations, integration over multiple variables
D ZZZ""$T}C@
qLinear Algebra
Scalars, vectors, matrices
Linear transformations
Eigenvalues and eigenvectors
Invariant subspaces.c d A&DA
Dynamical Systems
Continuous time dynamics equations of motion and their trajectories
Test points, limit cycles, and stability around them
Phase plane analysis
Cooperativity, positive feedback, and negative feedback
Multistability
Discrete time dynamics mappings, stable points, and stable cycles
Sensitivity to initial conditions and chaosDZGZ""H"6*pEB
Probability and Statistics
Probability distributions
Random numbers and stochastic processes
Covariation, correlation, and independence
Error likelihood@Z~Z" $]0FC
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Information and Computation
Algorithms (with examples)
Computability
Optimization in mathematics and computation
' Bits : information and mutual information@ZZ" GE
Data Structures
Metrics: generalized 'distance' and sequence comparisons
Clustering
Tree-relationships
Graphics: visualizing and displaying data and models for conceptual understanding < "QNHD?Additional Quantitative Principles Useful to Biology Students @@"(?Rate of change
This can be a specific (e.g., per capita) rate of change or a total rate of change of some system component.
Discrete rates of change arise in difference equations, which have associated with them an inherent time-scale.
Continuous rates of change arise as derivatives or partial derivatives, representing instantaneous (relative to the units in which time is scaled) rates.
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The process of abstracting certain aspects of reality to include in the simplifications we call models.
Scale (spatial and temporal) different questions arise on different scales.
What is included (system variables) depends on the questions addressed, as does the hierarchical level in which the problem is framed (e.g., molecular, cellular, organismal).
There are trade-offs in modeling no one model can address all questions. These trade-offs are between generality, precision, and realism.
Evaluating models depends in part upon the purpose for which the model was constructed. \ ZHZ""hN"$`JG
Equilibria and stability
Equilibria arise when a process (or several processes) rate of change is zero.
There can be more than one equilibrium. Multiple stable states are typical of biological systems.
Equilibria can be dynamic, so that a periodic pattern of system response may arise.
There are numerous notions of stability, including not just whether a system that is perturbed from an equilibrium returns to it, but also how the system returns (e.g., how rapidly it does so).
Modifying some system components can lead to destabilization of a previously stable equilibrium, possibly generating entirely new equilibria with differing stability characteristics. BZZ""R
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wStructure
Grouping components of a system affects the kinds of questions addressed and the data required to parameterize the system.
Choosing different aggregated formulations (by sex, age, size, physiological state, activity state) can expand or limit the questions that can be addressed, and data availability can limit the ability to investigate effects of structure.
Geometry of the aggregation can affect the resulting formulation.
Symmetry can be useful in many biological contexts to reduce the complexity of the problem, and situations in which symmetry is lost (symmetry-breaking) can aid in understanding system response.B
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(Interactions
There are relatively few ways for system components to interact. Negative feedbacks arise through competitive and predator-prey type interactions, positive feedback through mutualistic or commensal ones.
Some general properties can be derived based upon these (e.g., 2-species competitive interactions), but even relatively few interacting system components can lead to complex dynamics.
Though ultimately everything is hitched to everything else, significant effects are not automatically transferred through a connected system of interacting components locality can matter.
Sequences of interactions can determine outcomes program order matters.B
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Data and measurement
Only a few basic data types arise (numeric, ordinal, categorical), but these will often be interconnected and expanded (e.g., as vectors or arrays).
Consistency of the units with which one measures a system is important.
A variety of statistical methods exist to characterize single data sets and to make comparisons between data sets. Using such methods with discernment takes practice \ZZ"""NK
WStochasticity
In a stochastic process, individual outcomes cannot be predicted with certainty. Rather, these outcomes are determined randomly according to a probability distribution that arises from the underlying mechanisms of the process. Probabilities for measurements that are continuous (height, weight, etc.), and those that are discrete (sex, cell type) arise in many biological contexts.
Risk can be identified and estimated.
There are ways to determine if an experimental result is significant.
There are instances when stochasticity is significant and averages are not sufficient.ZZ-ZZ
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Visualizing
There are diverse methods to display data.
Simple line and bar graphs are often not sufficient.
Non-linear transformations can yield new insights.>""PM
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Algorithms
These are rules that determine the types of interactions in a system, how decisions are made, and the time course of system response.
These can be thought of as a sequence of actions similar to a computer program, with all the associated options such as assignments, if-then loops, and while-loops.
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"""ROPotential curricula@Four examples were given:
First-year math, perhaps a mixture of discrete and calculus
Quantitative emphasis with first year of math, plus probability/biostatistics, DE plus advanced course
Math I plus CS first year, then Math II plus advanced math/CS course later
Calc and DE in year 1, biostatistics and 1 CS course8Z'" Z'6
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SPMath/CS Panel Comments There is a distinction between the quantitative biologist, who works at the interface of math/computer science and biology, and the research biologist, who needs familiarity with a range of mathematical and computational ideas without necessarily being expert. Thus, the panel felt that flexibility in offerings is more advisable than a fixed curriculum.
Consider offering quantitatively intensive versions of standard biology courses, with extra credit2Z"g"hTQ
Consider new math courses which condense much of undergrad math into 3-4 semesters
Encourage interdisciplinary modeling courses at both introductory and advanced level, possibly as a first-year and senior seminar
N" $ Key Points:
Success in quantitative life science education requires an integrated approach: formal quantitative courses should be supplemented with explicit quantitative components within life science courses. Z"(
Life science students should be exposed to diverse quantitative concepts: calculus and statistics do not suffice to provide the conceptual quantitative foundations for modern biology."(
H We can t determine a priori who will be the researchers of the future educational initiatives need to be inclusive and not focused just on the elite. Assume all biology students can enhance their quantitative training and proceed to motivate them to realize its importance in real biology.D%Z"$"$"$$URJThe CPA Approach to Quantitative Curriculum Development across DisciplinesKK
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
Understand the Constraints under which your colleagues in other disciplines operate - the limitations on time available in their curriculum for quantitative training.ZZ$$$$$$
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z Work with these colleagues to Prioritize the quantitative concepts their students really need, and ensure that your courses include these.
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.
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M 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.
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Collaborators$
Drs. Beth Mullin and Otto Schwarz (Botany), Susan Riechert (EEB)
Monica Beals, Susan Harrell - Primer of Quantitative Biology
Drs. Sergey Gavrilets (EEB) and Suzanne Lenhart (Math) NIH Short Courses
Drs. Thomas Hallam (EEB) and Simon Levin (Princeton) International Courses
Society for Mathematical Biology Education Committee www.smb.org4`Z"^"~3> (xProject activities:$Conduct a survey of quantitative course requirements of life science students;
Conduct a workshop with researchers and educators in mathematical and quantitative biology to discuss the quantitative component of the undergraduate life science curriculum;
Develop an entry-level quantitative course sequence based upon recommendations from the workshop;
Implement the course in an hypothesis-formulation and testing framework, coupled to appropriate software;|ZNbbbbbbbkb"
Conduct a workshop for life science faculty to discuss methods to enhance the quantitative component of their own courses;
Develop a set of modules to incorporate within a General Biology course sequence, illustrating the utility of simple mathematical methods in numerous areas of biology;
Develop and evaluate quantitative competency exams in General Biology as a method to encourage quantitative skill development;
Survey quantitative topics within short research communications at life science professional society meetings.
hZZbbqbbB$# < The Entry-level Quantitative Course: Biocalculus Revisited4= :b$b$' In response to workshop recommendations, a new entry-level quantitative course for life science students was constructed and has now become the standard math sequence taken by biology students. The prerequisites assumed are Algebra, Geometry, and Trigonometry.&!`$!Goals: 2 b(!!C Develop a Student's ability to Quantitatively Analyze Problems arising in their own Biological Field.Illustrate the Great Utility of Mathematical Models to provide answers to Key Biological Problems.Develop a Student's Appreciation of the Diversity of Mathematical Approaches potentially useful in the Life Sciences DZbe````b```u``C%"Methods: b(a Encourage hypothesis formulation and testing for both the biological and mathematical topics covered. Encourage investigation of real-world biological problems through the use of data in class, for homework, and examinations. Reduce rote memorization of mathematical formulae and rules through the use of software such as Matlab and Maple. bZbbbebbbb|bbbbtbbbb$K&#RCourse 1 Content Discrete Math Topics:**b ) Descriptive Statistics - Means, variances, using software, histograms, linear and non-linear regression, allometry
Matrix Algebra - using linear algebra software, matrix models in population biology, eigenvalues, eigenvectors, Markov Chains, compartment models
Discrete Probability - Experiments and sample spaces, probability laws, conditional probability and Bayes' theorem, population genetics models Sequences and difference equations - limits of sequences, limit laws, geometric sequence and Malthusian growth
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T'$RCourse 2 Content Calculus and Modeling:**b )] Linear first and second order difference equations - equilibria, stability, logistic map and chaos, population models
Limits of functions - numerical examples using limits of sequences, basic limit principles, continuity
Derivatives - as rate of growth, use in graphing, basic calculation rules, chain rule, using computer algebra software
Curve sketching - second derivatives, concavity, critical points and inflection points, basic optimization problem Exponentials and logarithms - derivatives, applications to population growth and decay Antiderivatives and integrals - basic properties, numerical computation and computer algebra systems
Trigonometric functions - basic calculus, applications to medical problems
Differential equations and modeling - individual and population growth models, linear compartment models, stability of equilibria
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(%Results: b$XThis sequence is now taken by approximately 150 students per semester, and is taught mostly by math instructors and graduate students in math biology.
In many ways the course is more challenging than the standard science calculus sequence, but students are able to assimilate the diversity of concepts.
It is still necessary to review background concepts (exponentials and logs), but this is eased through the use of numerous biological examples.
Despite much experience with word-processing and game software, students have difficulty utilizing mathematical software and developing simple programs.
YZYbX)&RAlternative Routes to Quantitative Literacy for the Life Sciences: General Biology*SAb b R! Determine the utility of alternative methods to enhance the quantitative components of a large-lecture format GB sequence using:
Quantitative competency exams developed specifically to evaluate the quantitative skills of students taking the GB sequence for science majors;
Modules comprising a Primer of Quantitative Biology designed to accompany a GB sequence, providing for each standard section of the course a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. "ZbSbb.bfrbbffbb!*'Quantitative Competency Exams:b(W Multiple choice exams based upon the skills and concepts appropriate for the Organization and Function of the Cell and the Biodiversity (whole organism, ecology and evolutionary) components of GB. Given at beginning and end of the course to track changes in skills. Require only high-school math skills, with questions placed in a GB context.VXZb`x```+(Goals of Competency Exams:b( (i) inform students at the beginning of a course exactly what types of math they are expected to already be able to do;
(ii) help students be informed about exactly what concepts they don't have a grasp of, so they can go back and refresh their memory; and
(iii) ensure that the class is not held back through having to review material that the students should know upon entering. ,bb$},)Pre- and post-testing were done in GB sections taught by collaborators on this project, emphasizing quantitative skills, and other sections taught by faculty in a standard manner, as a control.`
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Conclusion:( Inclusion of a quantitative emphasis within biology courses can aid students in improving their quantitative skills, if these are made an inherent part of the course and not simply an add-on.
R`b$``@5Do students retain the quantitative skills developed?N We surveyed a sophomore level Genetics class a year after the students had been in the General Biology course, and determined student performance on another quantitative competency exam. We compared exam scores of students who had been in a GB course which emphasized quantitative ideas to those who had been in a standard GB course.OZO !Thus the available evidence suggests that students retain quantitative skills obtained within biology courses through later courses. 6$`!_
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Modules in GB(
The objective is to provide, for each standard section of GB, a set of short, self-contained examples of how quantitative approaches have taught us something new in that area of biology. Most examples are at the level of high-school math, though there are some calculus-level and above examples. A standard format for each module was established and a collection of 57 modules have been developed. Zb``?``8``g`.+Use of Modules within GB(TThese modules have been implemented in a variety of ways in GB.
(i) in lectures as a supplement to lecture material.
(ii) assigned to students as outside reading assignments.
(iii) students have been asked to turn in formal reports as homework assignments based around the additional questions to be answered at the end of each module. |UZS``1``@```B$?63"What quantitative topics are used?##@" Surveys were done at annual meetings of the Ecological Society of America and the Society for the Study of Evolution. The most important quantitative topic for each poster was assessed as well as a listing of all quantitative concepts used for each poster.@4152
Some lessons:
!\1. It is entirely feasible to include diverse mathematical and computational approaches in an entry-level quantitative course for life science students. This can be successful, even though it is in many respects more difficult than a standard science and engineering calculus course, if students see the biological context throughout the course.
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2. Inclusion of a quantitative emphasis within biology courses can aid students to improve their quantitative skills, if these are made an inherent part of the course and not simply an add-on. Evidence suggests that students retain these quantitative skills through later courses.
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3. Instructors can utilize quantitative competency exams to encourage students early in a course to focus on skills they should have mastered and see the connection between these skills and the biological topics in the course.`1.
4. The key quantitative concepts that are used in short scientific communications are basic graphical and statistical ones that are typically covered very little in a formal manner in most undergraduate biology curricula. Visualization/interpretation of data and results are critical to the conceptual foundations of biology training and we should give them higher priority in the curriculum. This might include a formal course on Biological Data Analysis, but needs to be emphasized throughout the science courses students take.
.Zbb2/Future Directions:b$ The BIO2010 Report gives numerous recommendations on quantitative skill development. Accomplishing these above can be aided through:
a. Agreed upon quantitative competency testing across courses.
b. Setting up teaching circles involving the key faculty involved in appropriate groups of courses.
c. Encouraging projects either formally within courses or as part of labs that require quantitative analysis involving the concepts deemed critical for comprehension.
d. Including key quantitative ideas from the beginning in basic entry-level courses - expecting students to utilize skills developed in high school and providing mechanisms to aid those who need remediation. ZbUb/bb@bbfbbbbbbXUImpediments to progressb(~Few math faculty at research universities have any appreciation (or interest) in real applications of math
Few biology faculty (not including many recently hired) have strong quantitative skills except in statistics
Multiculturalism of math departments creates problems for faculty/student interactions
Cultures are different few undergrads in math are expected to work on research with faculty, while it is expected that the better biology undergrads will have some exposure to research in field/lab situations with faculty
Math faculty prefer rigor (proof) over breadth
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