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INTERDISCIPLINARY TEAM APPROACHES TO MATHEMATICAL MODELING Blerta Shtylla Pomona College Mathematics Department

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What, who, why? Pomona College-1700 students, part of Claremont University Consortium a consortium of five undergraduate liberal arts colleges, two graduate institutions Our College: higher numbers of Science Majors (Biology, Chemistry, Molecular Biology, Pre-Health) and. Mathematics(!?). Our Math + Science track: Math 31S-Differential Calculus for Life Sciences (+ODE+Modeling) Math 32S-Multivariate Calculus for Life Sciences (+ODE+Modeling) Math 60-Linear Algebra Math102-Differential Equations with Modeling Math183-Mathematical Modeling (Applied Math Capstone-Advanced Math Methods+ Computation) This pipeline gains us lots of great double majors and or scientists that want to learn more applicable mathematics.

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Mathematical Modeling at Pomona Opportunities to reach outside the classroom: Draper Center: Community Partnership Courses are designed to nurture Student learning through active community engagement. Through purposeful and reflective preparation and community interaction, students enhance their understanding of real world applications of academic content, as well as develop their sense of ethical action, self-awareness and collaboration. Mathematical Modeling is very frequently offered as a community partnership course, where students interact with community members in modeling projects. The CP designation attracts a wide array of students in our course who have a keen interest in applying mathematics to a particular problem/topic. High demand for undergraduate research experiences. Many students may not be aware, or do not think they can participate. Courses that can offer a blended project/research oriented experience can reach a wide range of students who might not discover research on their own. Course Design Philosophy: Critical thinking and participate in an interdisciplinary environment Project driven course + mini research experience through interdisciplinary teams + make students engaged with larger scientific/surrounding community.

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Modeling Course Students Entry surveys give us a good idea of student background (online version is very convenient): Do you have any specific modeling goals or interests that you hope this class will help you attain? What non-math fields appeal to you? List experience and comfort level with mathematical computing or software (e.g., Matlab, C++, R, other programming language etc.). 50% of the class is composed of non-math/double majors. Many students come with pre-conceived idea of what a mathematical model should look like or be able to do. It is important to start an open conversation. Participate in these polls by: PollEv.com/mathbshtylla or Text a CODE to 37607

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www.polleverywhere.com

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Modeling Course Structure 1. Lectures are all module based geared to prepare students to pursue team projects + computation + primary literature readings (3/4 course). 2. Team projects involve collaboration between students, instructor and an outside team consultant; many questions arise from our community partners or science colleagues (1/3 of course).

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Mathematical Modeling Course Structure 4-5 Modules are taught in active lecture settings out of: Counting/Model building and Biochemical Kinetics Transport(traffic) + intro to PDEs Competition/infection + Stochastic Methods/Probability Diffusion Models/Random Walks Parameter Fitting Techniques Individual Based Simulation models etc. Each module is taught through brief lectures or guest lectures, and guided group work where students explore concepts on their own and get to know their classmates. Working groups in class are changed periodically to make sure students are not stuck with one perspective only. Primarily literature readings are introduced with each module using reading questionnaires: students must learn how to read modeling and biology papers.

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Module based active learning All modules are built on few simple principles: Start at a familiar starting point/concept (differential equations, difference equations) Introduce a new analysis technique (that can be understood with some differential equations) Introduce numerical simulation techniques and provide working code examples Self-exploring group worksheet: more advanced methods and review of material learnt. Additional references and related literature given in class. Literature review questions are collected from all students. Module 3: Continuous Time Stochastic Models Intro to stochastic processes and a survey of basic probability concepts Continuous time Markov chains (birth death processes) and generating function techniques-connect with well known ODE models Numerical simulation techniques for CTMC.

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Active learning strategies. Module is concluded with a student directed worksheet that covers new material expanding on previous lectures. Class group work can be used to help students learn new material on their own. Online Discussion forums allow instructor to follow student questions. Derive ODE Model from provided Kinetic Schemes Derive Markov Chain Model Use Generating Functions to derive Expectation ODE

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Modeling projects Course project reports and presentation replace final (significant portion of grade). We have experimented with two models in this course: 1. All projects focused on a particular community partner/problem (smaller tasks) 2. Projects are all different some coming from community partners or through colleagues/primary literature Each team is put together by instructor based on student selections/abilities/background/strengths=>assembled mid-semester. Some project topics can be open ended/not very specific (CP); these require some initial start up work with the team and instructor. We typically do not lecture for 1/3 of the class, while the teams are working on the projects-one-on-one meetings replace lectures.

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How do we encourage/build happy interdisciplinary teams? Team Roles: Each student must assume a specific team role: Computer Person: The computer person must first implement the mathematical model so it is capable of making predictions, then set up the solution methods to tell the mathematical model what we are going to do. Writer: The paper is what your team will be graded on- note that you cannot leave things off the paper because you plan to discuss it in the presentation. The Third: The third team member must look for things that need to be done. The first big hurdle is research. Find out as much as you can about the problem, and possible ways to solve it. Contact person. Goal: All teams contain a healthy mix of math/science/other members. Team Contracts: All team members sign a team contract(conflict resolution) which delineates: Goals Expectations Team policies and procedures Consequences Confidential Peer Evaluation Forms: Incorporated in each student final project report grade (20%) of grade averaged from the following categories: Attendance and participation in team meetings, Respectful and fair Individual contributions to idea generation, research and writing, Promotes team unity and communication. Detailed Project schedules and Paper Grading Rubrics (early!).

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Team Projects; how do we prepare independent fearless thinkers? Two consultants for each team: course instructor (math person) + non-math consultant. Very important team goal: learn how to talk to the project consultant: Too late to vaccinate-Consultant: Immunologist on campus and Mathematical Immunologist guest visitor in nearby institution. Bee Foraging collective decisions (Community Partnership with our campus farm). Consultant: Community Bee Keeper + Farm Manager. Spatial protein patterning in C. Elegans embryo. Consultant: Cell biologist on campus and Out of town visitor (Cell biologist and mathematical biologist)/colloquium speaker. Math modeling of the Arab Spring movement. Consultant: Political scientist on campus. Modeling of the ventilation system of the campus farm. (Community Partnership with our campus farm). Consultant: Farm Manager. Most consultants asked to be in such role agree enthusiastically. We commit them two up to three meetings with students and attendance of final presentation(tasked to ask hard questions). Guidelines for conversations might need to be given(interview question screening), however, most students take good ownership of project that supervision is very minimal. Try to invite one guest speaker that works on a related modeling project from each team (typically local institutions) Students are encouraged to attend talks from guest speakers and team-speaker meetings.

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Project Example: Bee Foraging and Collective Decisions (CP) Project formulation for the team: Background + Proposed Goals + Initial references. Project expectations: be able to reproduce part of one key modeling reference and either propose expansion or implement expansion/new model. Project Booklet distributed before mid semester break. Bee Foraging and Collective Decisions Background: Nectar foraging is a particularly important and well studied task of the honey bee colony. Colonies require energy in the form of nectar to survive and thrive. Foragers exploit nectar sources (owers) by collecting their nectar and returning it to the hive..An important question for the colony is: how many foragers should we end out of the hive to gather nectar? Goal: One goal of this project can be to work with a differential equation-based mathematical model of nectar foraging by the honey bee Apis mellifera of Edwards and Myerscough (2011). The model focuses on two behavioural classes; nectar foragers and nectar receivers. References: James R. Edwards, Mary R. Myerscough, Intelligent decisions from the hive mind: Foragers and nectar receivers of Apis mellifera collaborate to optimise active forager numbers, Journal of Theoretical Biology, 271- 1, 2011, pp. 64-77

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Out of Class Activities-encouraging collaboration and project ownership. Bee team leads the class for a close bee encounter and lecture with the local bee keeper. Students watched a bee dance video in preparation for the farm class- everyone showed up on a Sunday afternoon: Bee waggle danceBee waggle dance Learn about local bees Bee colony collapse. Bee foraging and local environment on campus. Team made honey and produced an excellent report with a model that proposed some novel ideas.

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Project Example: C. elegans early development patterns Background: In the Caenorhabditis elegans zygote, a conserved network of (PAR) polarity proteins segregates into an anterior and a posterior domain (i.e, front and back), facilitated by flows of the cortical actomyosin meshwork. The physical mechanisms by which stable asymmetric PAR distributions arise from transient cortical flows remain unclear. Another interesting problem has to do with the maintenance of the pattern in these cells. Curiously, cortical actin is not required for the maintenance of these domains. Dawes (2011) developed a model of interactions between the Par proteins that includes both mutual inhibition and PAR-3 oligomerization. Goal: The main goes of this project is to construct models that allow one to understand either pattern establishment in a cell, where a left to right asymmetry arises, or investigate pattern maintenance. Reference: A. Dawes, E. Munro. PAR-3 Oligomerization May Provide an Actin-Independent Mechanism to Maintain Distinct Par Protein Domains in the Early Caenorhabditis elegans Embryo. Biophysical journal 101-6, 2011, pp.1412 1422.

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Bringing C. elegans to us! Use your friends and collaborators: A. Dawes was able to visit us in the spring and gave an excellent colloquium on pattern formation. One on one meetings facilitated between team and visitor. Cell biologist on campus was generous enough to allow our class to be in the lab for one class session to watch C. Elegans under the light microscope and more. Class learned about light and electron microscopy and the types of data collection techniques that the C. elegans team was using for their modeling.

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Data collection? One team focused on working for the Campus Farm with the goal of proposing greenhouse configurations that would allow appropriate temperature control. The team collected temperature data and wrote a predictive model for temperature regulation in the greenhouse.

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What have we learned? Interdisciplinary experience= student teams + non-math consultants. Students are extremely eager to get involved in projects that are relevant. This can be interpreted in three ways: Active research topics that people that are alive are working on Open questions that we do not know the answer to Projects that involve a real life application/community member where work can have a direct impact Active involvement with community members makes a difference. They learn of our existence, new collaborations, great projects: students seem to like them. Skype and technology can bring many researches to our classrooms. We will be experimenting with virtual guest lectures this year. Building project ownership can lead to exiting new things. Two teams are preparing manuscripts for undergraduate journals with their results.

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Thank you! This course has come to its current incarnation thanks to the hard work of many of my colleagues at Pomona College Mathematics Department (A. Radunskaya, A. Rumbos). Pomona College Draper Center for Community Partnerships.