Teaching Portfolio Brandy S. Wiegers

2

Foreword In both mentoring research projects and in teaching, my philosophy is that all life experiences have something to teach us. Remembering this is essential to being a commendable instructor. My own career thus far has included a lifetime of experiences with several programs that have taught me much and has inspired my teaching of others. I thank the following programs and people for the contributions they have made to my teaching skills, passion and experience: • CLIMB: Being involved in the CLIMB program has been an unique and amazing experience that has provided opportunities to learn from a diverse set of students and instructors. The work that I have done has been possible through the support of Dr. Carole Hom, Dr. Angela Cheer and Dr. Rick Grosberg. Thank you for providing me the freedom to try new things with these students and teaching me how to not screech the chalk when writing on the board, a lesson that all my future students will surely appreciate. • Explore Math: I am grateful to the many people that have supported and worked with me as I have created, funded and coordinated this program. Now serving over 120 high school students a year and standing out as an amazing program, I thank everyone who has made it possible including the original four graduate students: myself, Yvonne Lai, Dr. Sarah Williams and Spiros Michalakis; our adviser: Dr. Tim Lewis ; the new generation of directors: Eva Strawbridge, Hillel Raz and Rohit Thomas. And, the instructors, the undergrads and the Department of mathematics students, faculty and staff who make it all possible. I look forward to attending the 10 year Explore Math reunion- keep up the good work! • MSRI and the Bay Area Math Circle Leaders: I had no idea when we started Explore Math that we had so many people who would support us. I have appreciated the guidance and support we have received from other Math Circles programs and I thank Jamylle Carter, Tom Davis, Tatiana Shubin, Zvezdelina Stankova, Sam Vandervelde, Paul Zeitz and Joshua Zucker for their contribution to my programs. I also appreciate the chance that MSRI, David Eisenbud, Kathleen O’Hara and Jim Sotiros took with having me start the Oakland/East Bay Math Circle and having me organize the efforts for the National Math Circle organization. • Professors for the Future: I participated in PFTF to gain an understanding of the professorate. Teresa Dillinger, Hector Cuevas and Dean Gibbeling provided a great set of workshops and opportunities that extended the program beyond this original goal by supporting my GSCSC project, a program that has changed my graduate school experience, adding a zest of community outreach experience to my portfolio. I especially thank Chancellor Vanderhoef for believing in me enough to join Campus Compact and recommit our campus to service learning. • UC Davis Teaching Resource Center: The UCD TRC provides inspiring workshops, presented by the best instructors on campus. I appreciate the guidance that the TRC has provided, the exuberance of the teaching consultants and the constant support of TRC Coordinator Mikaela Huntzinger. • Girl Scouts - as a life-long Girl Scout and more recently a Girl Scout trainer, I have had opportunities to teach and work with a diverse and amazingly talented set of instructors, students and adults. I thank everyone for their patience, support and help that always has and always will inspire my overall life presence. Many other people have touched my life that have not been mentioned. I thank you all, for support that I will share with others to inspire them to change the world. I now present my teaching portfolio. Due to the nature of the programs that I have taught in the last several years I have not had any opportunities to have formal individual evaluations. Instead I present this collection of my work as a testament to the passion that I have in my mathematical teaching and the expertise that I would bring to your institution. Sincerely,

Brandy S. Wiegers Brandy S. Wiegers [email protected] http://math.ucdavis.edu/∼wiegers

Contents 1 Statement of Teaching

5

1.1

Ensuring that students learn course material through SMART Course Objectives . . . . . . . . . . . .

5

1.2

Integration of the three components of academia: research, education, and service . . . . . . . . . . . .

6

1.3

Tools for Success . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3.1

Teaching to Different Learning Styles

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3.2

Using Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3.3

Real World Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Learning for Tomorrow’s Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4

2 Training and Other Professional Development

9 9

2.1

Teaching Work Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Professional Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1

On-Campus Professional Development Seminars . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2

Conference Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.3

Education Conferences and Workshops Attended . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 CLIMB: Collaborative Learning at the Interface of Mathematics and Biology

13

3.1

Climb Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2

Group Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3

3.2.1

Using Individual Analysis to Strengthen Group Work . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2

Growth Through Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Using Technology: Online Course Management Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3.1

CLIMB Smartsite Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.2

Smartsite Forum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.3

Smartsite Chatroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Numerical Analysis

21

4.1

Syllabus- Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2

Example: Midterm Solution

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2.1

Solution 1: Use Definition of Natural Cubic Spline . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2.2

Method 2: Formulas in the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.3

Method 3: Linear System of Equations - Checking the Previous Work . . . . . . . . . . . . . . 29 3

4

CONTENTS

5 MME: Math Modeling Experience

31

5.1

Summary of MME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2

SMART Learning Objectives for MME

5.3

MME Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4

MME 2005-2006 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.4.1

Schedule for High School Meetings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.4.2

Schedule for Undergrad TA Meetings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

MME Saturday Lesson

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.5.1

Plant Growth Spreadsheet Lesson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.5.2

Heifer International: A Gift of Rabbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.5.3

Deer Population Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6 Mathematical Outreach

49

6.1

STEM Outreach Organizational Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2

Oakland/East Bay Math Circle (OEBMC) 6.2.1

6.3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

OEBMC Fall 2007 Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Explore Math

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.3.1

Explore Math Program Summary

6.3.2

Explore Math Undergraduate Program Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.3.3

Explore Math Program Successes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7 Conclusion/ Preparing for the Future

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

57

7.1

Course Development: Service Learning and Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2

Course and Program Development: Undergraduate Research and Mathematical Outreach . . . . . . . 57

Chapter 1

Statement of Teaching All life experiences teach us something. Remembering this fact is essential to being an instructor; the experience our students have during class time, during office hours and during out of class mentoring affects our students’ learning. My generalized teaching goals follow directly from this philosophy: 1. Ensure that the students learn the course material. 2. Integrate the three components of academia (research, education, and service) into teaching. 3. Utilize tools for success: variety of teaching methods, technology and real world applications. I approach these goals by utilizing the departmental and campus resources, SMART learning objectives and by using teaching tools that reach out to today’s student.

1.1

Ensuring that students learn course material through SMART Course Objectives

Every facet of our teaching is important to the learning process, and we can leverage this fact with SMART (specific, measurable, attainable, realistic, timely) course learning objectives. Before starting with a new course, I review the approved departmental syllabus to access what material knowledge and skills the students should gain from the course. Using this understanding, I create objectives for the quality and type of work to be achieved by the end of the course, ensuring that the students not only grasp the material, but come to appreciate and be able discuss the material at an appropriate level. I then use these course learning objectives to create a course calendar that will, in turn, define the lesson and assignment learning objectives for the daily classroom lectures. In the past, I’ve taught in a numerical analysis course. For this course, I created the course learning objective that “by the end of the quarter [the] student should be able to list the components of a technical report and to create such a report.” Using the course’s numerical programming assignments, I was able to achieve this goal by ensuring that lectures, office hour sessions, assignments and grading addressed how to write a technical report. At the end of the quarter, students knew how to write a technical report and had gained a more thorough understanding of the course material because they had to move beyond implementing and testing a numerical method to comprehensively appreciate the numerical method in order to complete the assignments. By defining the objectives before planning the course calendar, I ensure that the lectures are relevant and that they include all the key components that need to be covered in the course. In class, I start my lectures with an outline of the learning objectives to which I refer when presenting the lesson. Highlighting the objectives ensures that the students have an understanding of what they will get out of the lesson. At the end of the session, I review the objectives to allow the class, both myself and the students, to assess our progress. The lesson objectives provides context to the students, helping them understand the relevance of the lectures and the assignments in relation to 5

6

CHAPTER 1. STATEMENT OF TEACHING

the course. I’ve also found that defining SMART objectives provide everyone with a shared understanding of the rationale for assigned grades; students must achieve the learning objective in order to obtain the grade, providing an extra incentive and the defined emphasis of the material to be studied.

1.2

Integration of the three components of academia: research, education, and service

The experience of every individual in an academic setting should be infused with the three pillars of research, teaching and service. When creating learning objectives for a mathematics course these three components need to be included. To address research learning, students need to have case studies, real world problems and historical perspectives so that they can move beyond their book studies and develop independent thoughts about the course material. This will show the students that they do math everyday and that math is a necessary component of real world life, not just a class to pass while in college. To integrate teaching, students need to work with each other in a group setting to teach each other the concepts because teaching something requires a much different thought process than learning it. Doing so will ensure that the students have a high quality understanding of the material. To integrate service in my teaching, I use service related examples. When teaching about exponential growth, I discussed the Heifer International program where individuals can buy a family a pair of rabbits to raise, with the condition that the family must give the offspring to other individuals in their community. Through this lesson, a student can understand the impact on the community that a high school graduation gift of rabbits will have by the time they graduate from college.

1.3 1.3.1

Tools for Success Teaching to Different Learning Styles

Beyond ensuring that these three pillars of the academic experience are included in classrooms, it is important to provide a variety of teaching techniques to reach out to the varied learning styles. Mathematics is a very traditional discipline which lends itself to a straightforward lecture presentation style. This method does not benefit all students, but radical deviation from this method is not overly successful either because of the expectations that students bring with them to the course. With these two thoughts in mind my class presentation focuses on lecture and also uses group work and technology to add variety. Again, group work allows students to either teach one another or work together in guided inquiry to discover the mathematical concepts together. This has worked particularly well in the CLIMB mathematical biology research course that I am helping to teach, where a group of 9 students (both biologists and mathematicians) are responsible for working together to complete a group report addressing a current mathematical biology research problem. Each member of the research cohort must lead the group report process once throughout the quarter. Also, group members are encouraged to take different roles throughout the quarter, either leading the mathematical derivation, summarizing the biological implications or completing the literature review. I have seen this method result in students gaining a more thorough understanding of the course material because they share their knowledge and questions in order to gain a common-level of understanding among group members.

1.3.2

Using Technology

Technology is useful at all levels of mathematics. While at University of Idaho, I worked as a teaching assistant in the UI POLYA Math Center, where all of the University’s basic algebra courses are taught using a technology- centered method. One of the primary means of teaching they use at the POLYA center are recorded mini-lectures that are available to students via the internet. This allows students to refer back to material that they don’t understand. Additionally, it allows them to rewind and hear something that they missed the first time. My experience with working with this innovative approach to math education has influenced the presentation that I give in my classes. I believe in providing online resources for students to help them outside of class and in showing them during class

1.4. LEARNING FOR TOMORROW’S TEACHING

7

how to use these resources. When working with a numerical analysis course, I show them in class how to create a computer program and interpret the results so that they understand how the graphs in the book were made. Bringing the computing element of the course into the lecture rather than leaving it as homework, makes it a group effort rather than an individual effort. This is especially important with the numerical analysis course because the students come from across disciplines and have a wide variety of computing backgrounds so in-class work helps equalize the students’ programming knowledge. This aspect of teaching also allows for me to incorporate my research into the lectures, something that I have done effectively in guest-lectures for other courses as well as in in the Math Modeling Experience and in CLIMB.

1.3.3

Real World Applications

My double undergraduate degrees in Mathematics and Biological Systems Engineering, my variety of research experiences and my experience with CLIMB, MME, Explore Math and Oakland/East Bay Math Circles have provided opportunities to gain exposure to a wide variety of real-world mathematical applications. In the CLIMB undergraduate course, students are amazed that a simple application of algebra can help them gain understanding of the spread of the Wolbachia microbe in the Drosophila simulans population of California. At the same time, the middle school students that I’ve worked with in Explore Math and the Oakland/East Bay Math Circle are able to grasp the concept of a logistic model through the use of deer population modeling and quick group games. My experience with teaching specialized topic courses has provided a familiarity with real-world applications that give me a wealth of examples and techniques to draw on when teaching all of my courses. present them, taking my lesson plans to higher level that benefited my current and future students.

1.4

Learning for Tomorrow’s Teaching

When preparing for teaching I also create objectives for myself which focus on professionalism and personal growth. My first objective is always to provide a professional presentation that will garner the respect of my students. Establishing mutual respect is the first step to becoming a mentor and I consider being a mentor the most important role that we serve for our students. My second objective is to learn from the course and apply that knowledge to my future courses utilizing personal assessment, student evaluations and campus resources to better my presentation style and student interactions. As I stated at the beginning, all of life’s experiences teach us something and I always seek to learn from my current teaching to improve on future teaching. In addition, when preparing my course, I ensure that students learn the course material through SMART course objectives, I make the most of their learning experience by integrating all aspects of academia and I utilize tools for success to improve their learning experience.

8

CHAPTER 1. STATEMENT OF TEACHING

Chapter 2

Training and Other Professional Development 2.1

Teaching Work Experience

Assistant, UC Davis CLIMB Fall 2007, Fall 2006 Collaborative Learning at the Interface of Mathematics and Biology (CLIMB), UC Davis. Program emphasizes hands-on training using mathematics and computation to answer state-of-the-art questions in biology. Taught an overview of mathematical biology and guided the undergraduates in completing research exploration with 10 different UC Davis researchers. (http://climb.ucdavis.edu) Instructor, Math Modeling Experience (MME) Aug 2005-Feb 2006 Explore Math, Department of Mathematics, UC Davis. Organized a series of workshops for high school and undergraduate students, preparing them for the COMAP (Consortium of Mathematics and its Applications) MCM (Mathematical Contest in Modeling) and HiMCM (High School Mathematical Contest in Modeling) competitions. Prepared a series of 2 hour lesson plans to introduce students to math modeling topics including disease and population growth models while providing them with the tools to develop their own models for competition modeling problems. The result of the HiMCM is testament to the success of the 2005 program, with one regional Outstanding, two Meritorious and two Honorable Mention awards. In addition, one of the competing undergraduate teams received National Outstanding. Teaching Assistant Department of Mathematics, UC Davis. Held office hours, graded technical reports & exams, and did substitute lectures for Numerical Analysis.

Fall 2005

Girl Scout Adult and Older Girl Trainer Spring 2004 - Spring 2007 Tierra del Oro Girl Scout Council, Northern California, CA. While on fellowship at UC Davis I took time to develop my teaching skills by working as a Girl Scout trainer. In this position I strengthened my skills in teaching a diverse range of individuals (age, experience, socioeconomic) in a range of topics and over a range of times (2 hours - 3 days). I developed course materials, led six person training teams and received the Girl Scout Outstanding Volunteer pin for my work. Mathematics Tutor Jan 2001-Dec 2002 POLYA Mathematics Center, University of Idaho. The POLYA Center is an active learning environment with traditional courses offerings combined with online video lessons, quizzes and an interactive tutoring. I worked with students individually and in groups to compliment their other work.

9

10

2.2 2.2.1

CHAPTER 2. TRAINING AND OTHER PROFESSIONAL DEVELOPMENT

Professional Development On-Campus Professional Development Seminars

UCD Graduate Studies Professors for the Future (PFTF) Fall 2005 - Spring 2006 UC Davis, Davis, CA. Workshop participant. PFTF is a year-long competitive professional development fellowship that strengthens the leadership skills of 13 graduate student scholars. The program includes participation in the Seminar of College Teaching, Seminar on Ethics and Professionalism in the University and the PFTF professional development workshop. In addition, the PFTF project requited completion of an individual project. My project focused on graduate student community service involvement (GSCSC). Seminar on Ethics and Professionalism in the University Winter 2005 UC Davis, Davis, CA. Workshop participant. A reading and discussion seminar that reviewed the most contemporary research on academic professionalism and ethical issues. Seminar on College Teaching Fall 2005 UC Davis, Davis, CA. Workshop participant. Interactive course sessions discussed creating course goals and objectives, lesson plans, communication strategies, learning assessment, course management, and student diversity. Teaching Resources Center Basic Skills & Complex Issues Series Fall 2005 UC Davis, Davis, CA. Workshop participant. Topics included student disabilities, encouraging student interaction, sexual harassment, gender equity, student academic honesty, and using writing in the classroom. Mathematics Teaching- Math 390 Course Fall 2003 This course is intended to prepare graduate students to become successful teaching assistants. Topics included preparing a discussion session, grading, preparing exams and overview of what it means to teach and learn mathematics.

2.2.2

Conference Talks

Special Session on Mathematics for Teaching: Educating Elementary and Middle School Teachers for Success Jan 7, 2008 (scheduled) 2008 Joint Mathematics Meetings (JMM), San Diego, CA. Presenter. “Oakland/East Bay Middle School Math Circle.” More Thoughtful Teaching Mini-Symposium UC Davis, Davis, CA. Presenter. “Community Outreach and Working with Community Partners.”

May 29, 2007

Special Session on Math Circles and Similar Programs Jan 5, 2007 2007 Joint Mathematics Meetings (JMM), New Orleans, LA. Presenter. “University of California, Davis’s Explore Math Program: Graduate students bringing cutting-edge research into the classroom to share with undergraduate and high school students.”

2.2. PROFESSIONAL DEVELOPMENT

2.2.3

11

Education Conferences and Workshops Attended

California Girls Collaborative Mini-Grant Conference Oct 10, 2007 Berkeley, CA. Workshop participant. This one-day conference targeted organizations with programs that support girls in science, technology, engineering and math (STEM). Attendees included representatives from K-12 schools, higher education, professional organizations, business, government and community-based organizations. Math Fest Women Count Conference Aug 2, 2007 San Jose, CA. Workshop participant. A Conference for Directors of Mathematics Outreach Programs for Young Women. Workshop Participant. Critical Issues in Education: Teaching Teachers Mathematics Mathematical Sciences Research Institute (MSRI), Berkeley, CA. Workshop participant.

May 30- June 1, 2007

Changing the Culture of the Academy: Toward a More Inclusive Practice March 22, 2007 University of California System-wide Conference, UC Berkeley, Berkeley, CA. Workshop participant. ‘Continuums of Service,’ Western Regional Colloquium on Civic Engagement and Graduate Study at Research Universities April 12, 2007 San Jose, CA, Workshop participant. Transforming the Culture of the Academy: Undergraduate Education and the Multiple Functions of the Research University Fall 2006 Sponsored by The Reinvention Center at Stony Brook. Recorder for conference activities. Published special sessions summary in conference proceedings. Northern California Trainers Consortium (NCTC) Track II Training Marin, CA. Focused on group work and conflict management while teaching and organizing activities. Symposium on Civic Engagement and Graduate Education Stanford University, Palo Alto, CA. Recorder.

September 17-18, 2006

April 24, 2006

Colloquium on Civic Engagement and Graduate Education at Research I Universities in Northern California Oct 21, 2005 California Campus Compact, Daly City, CA. Workshop participant. Northern California Trainers Consortium (NCTC) Track II Training September 16-17, 2006 Los Gatos, CA. Focused on bringing diverse teaching styles into the classroom, incorporating service learning and networking with other trainers. Northern California Trainers Consortium (NCTC) Track I Training Spring 2004 Placerville, CA. Focused on teaching to diverse learning styles, creating SMART learning objectives and organizing lesson plans.

12

CHAPTER 2. TRAINING AND OTHER PROFESSIONAL DEVELOPMENT

Chapter 3

CLIMB: Collaborative Learning at the Interface of Mathematics and Biology The concept behind CLIMB is simple, in order to create successful mathematical biology collaborations universities should start training undergraduate mathematics and biology students how to collaborate. This year-long program brings together 7-9 undergraduate students and orchestrates their learning process, group dynamics and research skill set to allow for a collaborative summer research project. As the Fall CLIMB assistant, I have first hand experience with the difficulty and reward of bringing together such a group of students with diverse educational and research backgrounds and helping them complete collaborative projects. In this program I have worked with students on over ten different research projects, learned how to manage undergraduate research collaboration and gained insight into student research process that will influence and provide the necessary experience to successful coordinate my own student research projects. CLIMB Research Topics • Evolution of recognition systems • Models of behavioral evolution • Biofluids of fish feeding • Models of marine reserves • Quantitate and molecular analysis of mitosis and intraflagellar transport • Models of cell motility • Population genetics • Models in neurobiology • Computational models of decision making • Growth kinematics • Phylogenomics and the study of microbial diversity • Wollbachia, population genetics, disease control, spatial spread • Source-sink population dynamics

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14CHAPTER 3. CLIMB: COLLABORATIVE LEARNING AT THE INTERFACE OF MATHEMATICS AND BIOLOGY

3.1

Climb Syllabus

MAT 180

CLIMB Core Course

Fall 2007

TTh 12:10-1:30, PHYGEO 148 Instructor: Coordinator: TA:

Angela Cheer,[email protected], Mathematical Sciences Building 2138, office hours TBA Carole Hom, [email protected], 754-9733, 4314 Storer Hall Brandy Wiegers, [email protected] 2139 Mathematical Sciences Building, office hours TBA

Welcome to the CLIMB core course! In this course, you will study emerging research problems that lie at the interface between mathematics and biology. Over the course of the quarter, you will be introduced to five different research areas, each presented by a CLIMB faculty member at the forefront of his or her discipline. Each two-week module will include readings, discussions, and lectures. Readings will include background material and original literature, and will be selected to provide an entré into the literature appropriate to a faculty member’s research talk. Grades in this course will be based on participation in class discussions (30%) and on problem sets (70%). Tentative schedule Weeks

dates

faculty presenter

topic

0

27 Sep

Angela Cheer

intro to reading math biology papers

1-2

2,4,9,11 Oct

Wendy Silk

growth kinematics

3-4

16,18,23,25 Oct Jonathan Eisen

5-6

30 Oct 1,6,8 Nov

Michael Turelli

Wollbachia, population genetics, disease control, spatial spread

7-8

13,15,20 Nov

Sebastian Schreiber

Source-sink population dynamics

9-10

27, 29 Nov 4,6 Dec

Jonathan Scholey

Mitosis, molecular motors and cell biophysics

Phylogenomics and the study of microbial diversity

Format for most modules: Day 1: overview lecture by faculty guest speaker, distribute readings and problem set Day 2: discuss readings and initial student work on problem set Day 3: second lecture by faculty guest speaker, questions and clarification on problem set Day 4: wrap-up discussion on papers, problem set Problems will be due at the start of the next module. We will adjust this schedule for the last two modules, both of which are slightly shorter. In general, the discussions of problem sets will be very important to your understanding. To get the most out of them, you will need to start on the

3.1. CLIMB SYLLABUS

problem set after doing the readings and before Day 2. The format for the last two modules will differ slightly; we’ll fill you in later in the term. Problem sets Problem sets will be assigned at the start of each module and due at the start of the subsequent module. Problem sets will require both mathematical and biological literacy to solve, and we expect you to collaborate with each other in working on these. We would like each of you to hand in solutions to the problems (which you may solve as a group). In addition to the solutions, at the end, please write two paragraphs: (a) who did what, how did you contribute; (b) what you learned from the problem set. See the handout for details. Final exam: Thursday, 13 Dec, 10:30am-12:30pm We will meet during the final exam slot to discuss winter quarter projects. Readings and miscellaneous references Readings will be assigned weekly and can be downloaded from the course web site on smartsite.ucdavis.edu. Readings not available in electronic form will be indicated clearly. Do not wait until the last minute to do the week’s reading. The CLIMB room, 1450 Storer Hall, also has the general reference books listed below – please keep them there. General Edelstein-Keshet, L. 2005. Mathematical Models in Biology. SIAM. Ellner and Guckenheimer, 2006. Dynamic Models in Biology. Princeton Univ. Press. Murray, 2002. Mathematical Biology, Vol 1 and 2. Springer, NY. Cell/physiology Fall, Wagner, Marland, and Tyson, 2002. Computational Cell Biology. Springer, NY. Keener & Sneyd. 1998. Mathematical Physiology. Springer, NY. Biofluids/biomechanics Vogel, Steven. 1988. Life’s Devices. Princeton Vogel, 2005. Comparative Biomechanics. Princeton Vogel, S. 1994. Life in moving fluids, Princeton.

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16CHAPTER 3. CLIMB: COLLABORATIVE LEARNING AT THE INTERFACE OF MATHEMATICS AND BIOLOGY

3.2

Group Work

A key aspect of the CLIMB program is developing a team of collaborators. In order to achieve this goal, students work on group projects throughout the quarter, developing a synergestic group dynamic in preparation for a summer research project. The students complete the projects and the reports as a group and then turn in individual analysis of the assignment and group process, providing an opportunity to learn from each other.

3.2.1

Using Individual Analysis to Strengthen Group Work

We started the quarter by explaining the group work dynamic of the CLIMB programming and sharing with the students the comments that previous CLIMB cohort students had: It is not easy to collaborate, many of us have been successful independent workers and sometimes our abilities, styles of study, etc dont mesh. Also, group work can be intimidating because each trainee is trying to prove their value and worth among each other and there is the additional pressure where not all students have enough time outside of call to prepare for the readings and assignments. This is compounded by the fact that we all have work outside of the class to do, and its hard to find a time to meet that is convenient for everyone. At the same time, group meetings are fun and interesting because we are enjoying the work and each others company. Our team chemistry is improving and its easier to relax and act more like ourselves, allowing our strengths and weakness to compliment each other and bring us together as a team. The workload is more evenly distributed with everyone working together on the math portion by taking turns writing on the board and everyone has chipped in on the discussions and write-up. It is especially helpful that if someone doesnt make it to a meeting when a discussion was held, the other people will explain the work that was done and how the answer was achieved. This is an excellent summary of group work. It is a lot of work to successful facilitate a group project and to have a group come together. We facilitate this process by asking the students to complete an individual analysis of each project. At a sum total of three paragraphs, this assignment isn’t a huge commitment but it provides necessary insight into the group that can shared. The components of the analysis include: 1. Statement of Authorship: Who did what? How did you contribute? This is not an evaluation of your cohort members but a simple statement of what happened. 2. Commentary: Comment on the groups solution. Did you disagree with anything? Would you have taken a different approach? Also include suggestions about what worked and didnt work for the group. If you feel comfortable with us sharing these suggestions anonymously with the group, let us know. 3. What did you learn?: Lastly, include a brief statement about what you learned. This might also be a good place to comment on a new research idea that you thought of while working on this project that you thought of that you might want the group to work on next summer. I find this analysis very useful in grading and monitoring group issues. It is understood that students have outside commitments but we can read through the quarter to ensure that students are still meeting their commitment to the CLIMB program and their fellow students. Students are encouraged to try different tasks throughout the quarter, as writer or programmer or literature reviewer, and reading the analysis we can see how students develop in these different tasks. Also, by providing an anonymous third party summary of the suggestions and sharing it the students, the group can grow with the suggestions that they aren’t ready to share with each other including, “Make sure that everyone is at the same place in the problems and the readings so we can work together as a team without one person taking over; It would be helpful if the teams effort was more focused or if we divided the work up to the people so we dont have to meet on weekends. Specifically, its helpful when people that cant make meetings volunteer to do other work. That way everyone has the same time commitment to the class”. These comments provide valuable direction for the group and this analysis was especially useful when the students failed mid-quarter at completing a successful model.

3.2. GROUP WORK

3.2.2

17

Growth Through Failure

The most successful group project in regards to the project that secured the group dynamic was the project that the students failed. At the beginning of the project they misused the quadratic equation and introduced an incorrect equilibrium point. This error propagated when no one checked the original calculations and instead they went forward with the calculated equilibrium, simulating a negative population. The students didn’t stop to think of the biological implications of this mistake and as a result half of the report was wrong. Instead of accepting the incorrect work, we had the group members re-write the group report individually. The resulting report (and corresponding changes made) provided much insight into what each member of the group focused on in terms of the project, seeing who understood the biology and who focused on the mathematics. At the same time the group refocused themselves, using their own group work suggestions to improve the group dynamic. They now have created project schedules to ensure that they aren’t attempting to complete the project the day before it is due. They are checking each others’ work and taking time to understand and discuss both the biological and mathematical implications of their modeling. The results of this project can be seen in the 26 page report that they completed for their next group project (see samples below). The report included 4 different computational models, beautiful plots that had been accomplished through collaborative programming efforts and a complete biological and mathematical analysis. It was truly a success that speaks to what they are going to be able to accomplish this summer as a group. CLIMB

Source-Sink Population Dynamics

6 DEC 2007

CLIMB

Source-Sink Population Dynamics

6 DEC 2007

Source-Sink Population Dynamics Andy Huang, Chris Mosser, Daniel Suderow, Mary Jacklin, Matthew Reed, Michelle Jensen, Tania Gonzalez, Tushar Rawat, Ying Wu Collaborative Learning at the Interface of Mathematics and Biology, University of California, Davis, California USA1 POPULATION PERSISTENCE Ecologists study many aspects of species dispersal patterns in heterogeneous environments where some habitats are more suitable than others. These different habitats can be described in terms of the following demographic parameters: birth, death, immigration, and emigration. Source habitats are defined to have birth rates greater than death rates and emigration rates greater than immigration rates. This implies that the source is a net exporter of individuals. Sink patches, on the other hand, have higher death rates than birth rates, suggesting that the sink population declines toward extinction unless individuals emigrate from a source patch (Pulliam 1988). Habitat patches can therefore be described in terms of their ability to maintain a population without immigration, the habitat’s attractiveness to individuals, and whether they are net exporters or importers of individuals (Table 1). Source Patch High-quality habitat Attractive Stable or growing Net exporter

Sink Patch Low-quality habitat Avoidable Declines to extinction Net importer

Figure 13 – Contour plot of (1-p)N* where 500 > (1-p)N*. Table 1: Defining characteristics of source and sink populations.

From inspection of Figure 13, we see that (1-p)N* < 500 for all values of p and f1. Therefore, it is always true that G > 1, and thus possible for species 2 to invade species 1. CONCLUSION

Many factors determine whether a population will persist or go extinct in its environment. Extensive studies into source-sink populations have shown that some of these factors are related exclusively to the species’ habitat. The following list describes some of the properties correlated with population persistence: 1. Connected habitats are better than disjointed habitats. Over time habitats may act as either sources or sinks, depending on environmental and demographic variability. As such, linkages between sources and sinks significantly affect the persistence of the population. For example, productive populations (sources) can contribute individuals to less productive populations (sinks) through immigration, thus rescuing them from extinction. Linkages between populations often help to minimize extinction and genetic isolation of

Keywords: source-sink population dynamics, fundamental niche, realized niche, population persistence

1

Environmentalists and ecologists are becoming increasingly interested in preserving and restoring habitats of rare, threatened, and endangered species. According to Pulliam (1988), such species preferentially dwell in high quality habitats (i.e. sources); thus, it is important to understand how to identify high quality habitats and how populations respond to habitat loss or change. Our first model showed that the loss of source habitat had a significant effect on the total bird population because excess individuals from the source habitat helped to populate the sink habitat. A large proportion of a species’ population can exist in sink habitats; thus, ecologists may misinterpret the species’ habitat requirements by focusing the majority of conservation efforts where the larger population exists. In the same way, source habitat may be ignored or even destroyed if only a small proportion of the population resides there. The degradation or destruction of the source habitat may, in turn, heavily impact associated sink populations, potentially over large distances (Tittler et al. 2006). For an already

16

18CHAPTER 3. CLIMB: COLLABORATIVE LEARNING AT THE INTERFACE OF MATHEMATICS AND BIOLOGY

3.3

Using Technology: Online Course Management Tool

SmartSite is the new UCD online course management tool and it has provided many useful tools for the CLIMB program accessible from any computer with a working browser and internet connection. Although Smartsite still has its set of bugs it is nice because it is not just for classes. The site offer several tools that can be useful to research labs. That’s why we set up the CLIMB SmartSite, it allows us to arrange the CLIMB research project as we continue throughout the year, introducing the students to the collaborative tools before they use them this summer.

3.3.1

CLIMB Smartsite Tools

• Announcements: We can post (and email) the latest CLIMB news, assignments and other updates. • Chat Room: This is where students can live chat with one another. This works well if students are all trying to work collaboratively from different locations. It is especially useful for quick questions that can be sent out by one student. They can see if any of the other students are logged in and want to discuss their questions, often leading to a quicker response then email. See Section 3.3.3. • Mailtool and Email Archive: The site provides email tools and archives the previous emails to the site members. This helps students and faculty stay aware of group progress and important news and the archive provides a useful resource for students to recall group work when completing group reports. • Forum: A message board to discuss current projects/ readings/ problem sets. See Section 3.3.2. • Resources: online space for sharing and archiving project-related files. Students are able to upload files, have access to the readings/ problem set and share their group work progress. • Schedule: a calendar of CLIMB events and group meetings • Wiki : This is a webpage that you can edit- useful for personal information and group edits. CLIMB students have used the wiki to: – Organize group schedules: each student can edit their availability schedule and check to see if others are available to work. – Personal contacts: Students and Faculty collaborators post personal contact information that is only accessible to other CLIMB Smartsite members. This information allows for better and quicker collaboration. – Research Tools: As research issues and questions arise, we have created an archive of useful research tools and tutorials like the LATEXtutorial that I created for the students. For examples of well developed wiki’s check out: http://wiki.sfwiki.org/, http://daviswiki.org/, http:// www.wikipedia.org/ • Site Info, Support & Training and Help : We want this site to be the students so we allow them site management permissions to that they can access and change the site info, learn more about Smartsite and continue to develop the site to meet their needs. The use of collaborative technology and group facilitation has helped the group process and made it easier for students to work together. Although Smartsite provides this set of tools together in one location, the same approach for using a set of web-based course management technology can be done independent of this particular site management tool, I look forward to learning what tools that you use in your department and incorporating them into my lectures and group work.

3.3. USING TECHNOLOGY: ONLINE COURSE MANAGEMENT TOOL

3.3.2

19

Smartsite Forum

I used the Smartsite Forum to facilitate group by starting discussion outside of the classroom about the research papers and problem set. Students were requested to post one question and one comment before every discussion session. This started student group discussion that was continued on the forum and in the Smartsite Chatroom.

Send To Printer | Close Window Messages & Forums / Fall 2007: Research Units Discussion Forum / Eisen: Phylogenomics and the study of microbial diversity

Bits and pieces for the paper - Gonzalez, Tania (Oct 26, 2007 6:19 PM) Last Revised By Gonzalez, Tania on Fri Oct 26 18:24:21 PDT 2007 Everyone please post notes. ---------------------------------------------------------------------Citation: Gross L (2006) Bacterial Symbionts May Prove a Double-Edged Sword for the Sharpshooter. PLoS Biol 4(6): e218 doi:10.1371/journal.pbio.0040218 http://biology.plosjournals.org/perlserv/?request=get-document&doi=10.1371/journal.pbio.0040218&ct=1 endosymbionts - live inside cells; bacteria help hosts synthesize essential nutrients The agricultural pests, Sharpshooters (Homalodisca coagulata), house two endosymbionts: Baumannia cicadellionicola and Sulcia muelleri. They're unrelated and they work together, fulfilling different needs for their host and each other. The host can't make its own vitamins, for example, so the bacteria in its gut take care of that. The first endosymbiont matches AY676896 on GreenGenes. It's "specific host" is Paromenia isabellina. EU156142 wasn't found on GreenGenes. Endosymbiont genomes have similarities: (1) smaller genomes size, (2) less G-C pairs than A-T pairs, (3) fast evolution of proteins. Other endosymbonts share these characteristics, inevitably leading to the question... WHY? Possible explanations presented in article: - Effects of genetic drift due to small populations. - High rate of mutation due to loss of DNA repair genes.

Re: Bits and pieces for the paper - Jacklin, Mary (Oct 29, 2007 6:51 PM) FUTURE AVENUES OF RESEARCH There are many future implications for this type of genomic analysis for microbial populations from various environments. With a seemingly never-ending index of microbial species, there is much work to be done obtaining and sequencing rRNA and entire genomes. By using databases like Genbank, phylogenetic trees can be made to estimate organism relatedness and make inferences about ecological properties of species which are difficult or impossible to culture in a lab. One of the problems apparent in the Genbank database is the lack of naming conventions yet has a policy of listing all published data providing much room for error and repetition. To help address this issue, databases such as GreenGenes have been created to make an attempt at condensing and verifying species uniqueness and identity.

20CHAPTER 3. CLIMB: COLLABORATIVE LEARNING AT THE INTERFACE OF MATHEMATICS AND BIOLOGY

3.3.3

Smartsite Chatroom

Students used the Smartsite Chatroom throughout the course for group meeting arrangements, completing problem assignments, sharing references and editing the final report. This was particularly useful for students with children and other outside obligations who weren’t able to make it to meetings, allowing them to work from home.

Currently viewing messages for 'Main Chat Room' All chat messages are archived and can be read by any site participant. Tushar Rawat (Oct 11, 2007 7:43 PM PDT) For some reason I can't access the forums when I log in. I click the link and the page loads, but no forums show up. Brandy Wiegers (Oct 11, 2007 8:25 PM PDT) You might try the arrow at the top, next to the Forum, Brandy Wiegers (Oct 11, 2007 8:25 PM PDT) or ckick on the forum link at the top of the "forum" section. Tushar Rawat (Oct 11, 2007 10:13 PM PDT) I'm not sure what you mean by clicking the forum link at the top of the forum section. But when I click Forums (at the left side navigation) all that loads is the bar with the arrow, Forums, and the help icon. Nothing underneath it appears. I'm not sure why. Michelle Jensen (Oct 11, 2007 10:37 PM PDT) That's odd. I'm not having any troubles... Matt did post some information about the problem you are working on. Are you still having trouble? Tushar Rawat (Oct 11, 2007 10:54 PM PDT) Andy posted some stuff as well, around 6pm earlier today. But I'm at home now and I can't access it. I'm going to try logging out and logging back in. Tushar Rawat (Oct 11, 2007 11:34 PM PDT) OK, Ying helped me out. Apparently my ad-block plug-in for Firefox was blocking the iFrame which loads the forums. Annoying, because it was working fine yesterday. Michelle Jensen (Oct 15, 2007 7:50 PM PDT) Hey Matt, Are you there? There's no phone number listed for you under "wiki". We're going to try to have a chat session with everyone at 9pm. Michelle Jensen (Oct 15, 2007 7:57 PM PDT) Mary - I talked to Dan and he's in a meeting until 9:30 or so, but he'll give me a call and I will fill him in on the news. Ying and Matt do not have phone numbers listed. How did it go on your end? Mary Jacklin (Oct 15, 2007 8:33 PM PDT) I talked to Tania and Chris and they said they'd be on around 9. I had to leave a message for Andy and Tushar saying to come online or at least check their email and tell us what they think about the authorship ideas. Mary Jacklin (Oct 15, 2007 8:35 PM PDT) Let's just do the best we can to come to a consensus at 9 and then email out the results and hopefully if any one has a problem with it they'll get back to us before we need to print tomorrow morning. Michelle Jensen (Oct 15, 2007 8:50 PM PDT) That sounds good. I just realized that I needed to refresh this page for new postings to come up... weird. Matthew Reed (Oct 15, 2007 8:59 PM PDT) I should be free to chat on and off from now on... Mary Jacklin (Oct 15, 2007 9:00 PM PDT) i'm here :) Tushar Rawat (Oct 15, 2007 9:00 PM PDT) I'm here Michelle Jensen (Oct 15, 2007 9:01 PM PDT) Cool. So, to fill everyone in, the layout form we received in class mentions that we should come up with a consensus for the authorship listing. Christopher Mosser (Oct 15, 2007 9:01 PM PDT) yo Michelle Jensen (Oct 15, 2007 9:01 PM PDT) Everyone so far has been expressing that they really don't care where their name goes. Christopher Mosser (Oct 15, 2007 9:01 PM PDT) ditto Michelle Jensen (Oct 15, 2007 9:02 PM PDT) However, we all need to agree on this... So do you have any suggestions on how we come to a consensus? Mary Jacklin (Oct 15, 2007 9:02 PM PDT) so should we just draw straws? lol Christopher Mosser (Oct 15, 2007 9:02 PM PDT) I think that we all worked hard. I don't honestly think that I could vote for someone to take last position Tushar Rawat (Oct 15, 2007 9:02 PM PDT) How do you draw straws online :) Mary Jacklin (Oct 15, 2007 9:03 PM PDT) ya that's the hardest part....

Users in Chat Michelle Jensen Brandy Wiegers

Chapter 4

Numerical Analysis As the assistant for Math 128 I held with office hours, graded technical reports & exams, and did substitute lectures for Numerical Analysis. The course instructor recognized my interest in teaching and allowed me extra responsibilities and teaching opportunities while I was assisting with the course. As part of this learning I also took the opportunity to create my own course syllabus and other course materials in preparation to teach the course in the future. Technical Reports As the assistant to the course, my main personal contribution came in creating the personal course learning objective that “by the end of the quarter [the] student should be able to list the components of a technical report and to create such a report”. I was able to achieve this goal by ensuring that office hour sessions, assignments and grading addressed how to write a technical report. By providing the report format and much individual feedback, by the end of the quarter students knew how to write a technical report and had gained a more thorough understanding of the course material because they had to move beyond implementing and testing a numerical method to comprehensively appreciate the numerical method in order to complete the assignments. In grading these assignments in the past I have seen that not all student gain the same understanding of the technical report because they miss the one lecture where it is covered. When I am teaching this course I would also spend time discussing technical reports in class and providing examples of technical reports throughout the quarter. Course Syllabus As I discussed in my teaching philosophy, I design my syllabus with the course learning objectives. After my experience with this course, I have defined a future goal for teaching a numerical analysis course that deals with the diversity of student experience which is the primary barrier to student success in the course. There are thirty students in the course that range in academic experience from undergraduate sophomores (where this is their first upper division course) to first and second year graduate students. It is difficult, especially when assigning the computing assignments, to reach out to this grouping and expect the same level of achievement. Although this has had little effect on the homework and the exams the programming assignments have been a big indicator of how little experience the younger group of students has. More specifically they don’t know how to write a technical report and don’t understand the distinction between this being a mathematical class with a little bit of programming and a programming class with a little bit of mathematics. With this said, I have focused my organization of my future syllabus around the computer assignments and created the learning objective focusing on the technical report aspect of the assignment (Refer to Section 4.1 to see my Numerical Analysis Syllabus and Technical Report format). Other Responsibilities Beyond the personal goal of assisting with technical reports I also had the responsibility to create comprehensive exam solutions. Visit Section 4.2 to see the midterm problem key. In addition, I was responsible for the website management where I used similar tools to those described in the CLIMB program.

21

22

4.1

CHAPTER 4. NUMERICAL ANALYSIS

Syllabus- Numerical Analysis

Math 128A: Numerical Analysis Fall, 2006

Contact Information Instructor:

Reader:

Brandy Wiegers, PhD candidate Email: Office: Undergrad Email:

[email protected] Mathematical Sciences Building 2139 [email protected]

Lecture Monday, Wednesday, Friday

1:10-2:00 PM

Cruess 107

Office Hours Tuesday Thursday

2 PM - 4 PM noon - 2PM

MSB 2139 MSB 2139

Course Website: http://my.ucdavis.edu All course information including assignments, due dates, and grades will be available on the http://my.ucdavis.edu website. If you have any questions about logging into ”my ucdavis” please ask the instructor as soon as possible.

Text: Numerical Analysis (8th Edition). RL. Burden & JD. Faires. Published by Brooks/Cole (Pacifica,CA). Dec, 2004. ISBN 0-534-39200-8. The 7th and 6th editions contain the same discussion of the course material as the 8th edition, but the homework problems are different in the new edition. A copy of the 8th edition will be available on reserve at the library to copy the homework problems if you have one of the earlier editions. An errata for the 8th edition is available online at: http://www.as.ysu.edu/∼faires/Numerical-Analysis/

4.1. SYLLABUS- NUMERICAL ANALYSIS

23

Course Summary This course is an introduction to many basic methods used in numerical analysis. The main topics include: Interpolation and approximation of functions, numerical integration and differentiation, solution of non-linear equations, acceleration and extrapolation, solution of systems of linear equations, eigenvalue problems, initial and boundary value problems for ordinary differential equations, and computer programs applying these numerical methods. Learning Objectives By the end of this course you will be able to... • Define numerical analysis. • Write a technical report that presents your research findings and conclusions. • Analyze the error propagation that results from mathematical algorithms. • Have several working algorithms that can be applied to your research.

Course Prerequisites: This is a mathematics course with a substantial programming component. The focus of the assignments is on the mathematical analysis of these programs. Any programming language can be used to complete the assignments and it is assumed that you are already proficient in a programming language. The programming component will only count for 30% of the grade of the assignment so plan to spend your time accordingly. Course Outline Chapter 1 (1.1-1.3) Chapter 3 (3.1-3.6) Chapter 4 (4.1-4.8) Chapter 8 (8.1-8.4) Additional:

Review of calculus, roundoff errors and computer arithmetic. Interpolation and polynomial approximations. Numerical differentiation and integration. Approximation Theory We may cover additional special topics including sections 8.5-8.6

Grading Midterm Exam Final Exam Homework Computer Assignments

2-3 lectures 8-10 lectures 7-8 lectures 8-9 lectures time permitting

18% 35% 12% 35%

Exams Exams will be comprehensive and can not be made up without a doctor’s note.: Midterm Exam: Monday, October 31, 2005. During class, 1 hour. Final Exam: Thursday, December 15, 2005, 4:00-6:00 PM, 2 hour. Homework Homework will be due everyday and will be turned in at the beginning of class. No late homework will be accepted without a university accepted excuse note. Collaboration You are encouraged to talk to classmates about your computer assignments and other problems from classwork but you must complete all assignments by yourself. This means that you can discuss your algorithms as a group but you need to create individual codes and individual results. If you do talk with others please indicate who your group members were on your assignment.

24

CHAPTER 4. NUMERICAL ANALYSIS

Computer Assignments There will be 6 computer assignments, but only 5 will be counted toward your grade. The lowest score will be dropped. Students will be assigned to six computer assignments throughout the course that will highlight the particular mathematical methods. You are welcome to write your code in any computer language that you feel comfortable with however, it must be a general purpose programming language that does not give any special assistance in implementing the algorithms we’re studying. More information will be available on http://my.ucdavis.edu. Assignment Descriptions Assignment Due Dates Program 1 October 10 Program 2 October 26 Program 3 November 9 Program 4 November 23 Program 5 December 7 Program 6 December 15

Description Error Propagation Interpolating Polynomials Cubic Splines Derivation Approximation Integral Approximation Integral Approximation

Assignment Due Dates Incomplete problem solutions will NOT be accepted for credit. In order to receive full credit for an assignment, it MUST be completed and turned in by class time on the specified due date. Any assignment turned in late, but on or before the absolute due date (typically the following class period) will receive a maximum of one-half credit. Any assignment turned in after the absolute due date will not be graded and no credit will be given for it. Assignment Format To meet the learning objective of technical report writing, for each of the programming assignments you will write a brief technical report which answers the given questions and illustrates the fundamental ideas in clear, concise, descriptive English prose. The report should separate the required tasks of the given project and document each in the appropriate section, i.e. Analysis, Computer Program, or Results. Below is a brief description of each section of the report. Refer to the Assignment Format Directions for more details. The focus of the assignments is on the mathematical analysis of these programs, not the program. Please plan your time accordingly. Analysis, 30% This section should begin by stating all the problems posed in the handout, derivations and mathematical proofs necessary, brief description of the algorithms and discussion of numerical considerations for the algorithms you have just described. The focus of this section is on the theory discussed in class and predictions of how your algorithm will perform based on that theory. Computer Program, 30% Internal comments should describe algorithms and variables, relating them to those described in your Analysis section. And this section should describe the input and output to and from your code. Ten percent of the points for your program will be for your programming style. If you have bugs in your program, do not expect them to be found during the grading process, rather come to the office hours for assistance. Results and Discussion, 30% This section contains the output of your program and an explanation of the results. Explaining your output should include comparing the results to the predictions in Analysis along with appropriate comparisons between the performance of different methods used and/or cases solved. Style, 10% Any English course has a minimum standard for quality of written expression of ideas, and you would not consider handing in a rough draft as a final copy. The same holds true here in Math 128. Not only are you expected to understand the given project and program it correctly, but you are also required to express this through your report in clear, concise, and readable English or math notation. With the exception of mathematical equations please type your report.

4.2. EXAMPLE: MIDTERM SOLUTION

4.2

25

Example: Midterm Solution

This is the midterm solution for Problem 3 from Math 128 A - Fall 2005. The original write-up was 2 pages long and incorporated all of these elements. I have expanded on my original writeup so that it makes sense to the reader by including the definitions that are referenced from the book and extra steps. Some notes about this solution: • Discussion of 2+ methods of solving the problem. Although only method 1 was discussed in the course lectures, some students memorized the formulas and used methods 2 and methods 3 on the midterm. • Methods 2 and 3 weren’t covered in lecture because they are applications of method 1 but now that students understand method 1 they can be reviewed in terms of the exam, allowing review of this exam key to serve as learning process, expanding for those students who already aced method 1. • In order to facilitate the learning, all three methods would be discussed in the post-exam review.

Problem 3 (10 pts) A natural cubic spline S on [0,2] is defined by  1 + 2x − x3 S(x) = 2 + B(x − 1) + C(x − 1)2 + D(x − 1)3

0≤x≤1 1≤x≤2

Find B, C, and D.

4.2.1

Solution 1: Use Definition of Natural Cubic Spline

Definition 3.10. Given a function f defined on [a, b] and a set of nodes a = x0 < x1 < ... < xn = b, a cubic spline interpolant S for f is a function that satisfied the following conditions: 1. S(x) is a cubic polynomial, denoted Sj (x), on the subinterval [xj , xj+1 ] for each j = 0, 1, ....n − 1; 2. S(xj ) = f (xj ) for each j = 0, 1, ..., n; 3. Sj+1 (xj+1 ) = Sj (xj+1 ) for each j = 0, 1, ..., n − 2; 0 4. Sj+1 (xj+1 ) = Sj0 (xj+1 ) for each j = 0, 1, ..., n − 2; 00 5. Sj+1 (xj+1 ) = Sj00 (xj+1 ) for each j = 0, 1, ..., n − 2;

6. One of the following sets of boundary conditions is satisfied: (a) S 00 (x0 ) = S 00 (xn ) = 0 (free or natural boundary); (b) S 0 (x0 ) = f 0 (x0 ) and S 0 (xn ) = f 0 (xn ) (clamped boundary). - (Burden and Faires, pg 143)

For our given S(x) with n = 2 and x0 = 0, x1 = 1, x2 = 2 we need to find the B, C, and D values that fulfill the requirements of the definition of a cubic spline:

26

CHAPTER 4. NUMERICAL ANALYSIS 1. S(x) is a cubic polynomial, denoted Sj (x), on the subinterval [xj , xj+1 ] for each j = 0, 1;  S0 (x) = 1 + 2x − x3 0≤x≤1 S(x) = S1 (x) = 2 + B(x − 1) + C(x − 1)2 + D(x − 1)3 1 ≤ x ≤ 2 Thus: S0 (x) = 1 + 2x − x3 S00 (x) = x − 3x2 S000 (x) = −6x

(4.1) (4.2) (4.3)

S1 (x) = 2 + B(x − 1) + C(x − 1)2 + D(x − 1)3 S10 (x) = B + 2C(x − 1) + 3D(x − 1)2 S100 (x) = 2C + 6D(x − 1)

(4.4) (4.5) (4.6)

2. S(xj ) = f (xj ) for each j = 0, 1, 2; This doesn’t help us for this problem because we don’t know our f(x) function. 3. S1 (x1 ) = S0 (x1 ), where x1 = 1 S0 (x) = 1 + 2x − x3 = (1)2 + B(x − 1) + C(x − 1)2 + D(x − 1)3 = S1 (x) S0 (1) = 1 + 2 − 1 = 2 + 0 + 0 + 0 = S1 (1) S0 (1) = 2 = 2 = S1 (1) 4. S10 (x1 ) = S00 (x1 ) , where x1 = 1 S00 (x) = x − 3x2 = B + 2C(x − 1) + 3D(x − 1)2 = S10 (x) S00 (1) = 1 − 3 = B + 0 + 0 = S10 (1) S00 (1) = −2 = B = S10 (1) −2 = B 5. S100 (x1 ) = S000 (x1 ) ; S000 (x) = −6x = 2C + 6D(x − 1) = S100 (x) S000 (1) = −6 = 2C = S100 (1) −3 = C Using the definition we have defined B and C. Now we must pick a boundary condition in order to find D. Natural Cubic Spline: Using a natural cubic spline boundary condition requires S 00 (x0 ) = S 00 (x2 ) = 0 S 00 (x0 ) S 00 (x0 ) = S000 (x0 ) = −6(x0 ) S 00 (0) = S000 (0) = 0

=0 =0 =0

S 00 (x2 ) = 0 S 00 (x2 ) = S100 (x2 ) = 2C + 6D(x2 − 1) = 0 S 00 (2) = 2C + 6D(2 − 1) = 0 S 00 (2) = 2C + 6D = 0 D = C3 Thus B = −2 , C = −3 and D =

C 3

= −1

4.2. EXAMPLE: MIDTERM SOLUTION

4.2.2

27

Method 2: Formulas in the Book

We want to find the interpolating cubic polynomial Sj (x) = aj + bj (x − xj ) + cj (x − xj )2 + d(x − xj )3 for each j = 0, 1, ...n − 1. Using the definition we know that Sj+1 (xj+1 ) = Sj (xj+1 ) for j = 0, 1, ..., n − 2 then aj+1 = Sj+1 (xj+1 ) = Sj (xj+1 ) = aj + bj (xj+1 − xj ) + cj (xj+1 − xj )2 + d(xj+1 − xj )3 for each j = 0, 1, ... n-2. We then define hj = xj+1 − xj for each j = 0, 1, ... n-1. Thus: aj+1

= Sj+1 (xj+1 ) = Sj (xj+1 ) = aj + bj hj + cj h2j + dj h3j

(4.7)

holds for each j = 0, 1, ...n − 2. 0 In a similar manner, examine Sj0 (x) = bj + 2cj (x − xj ) + 3dj (x − xj )2 . Using the definition, Sj+1 (xj+1 ) = Sj0 (xj+1 ) for each j = 0, 1, ..., n − 2 we see that:

bj+1 bj+1

0 = Sj+1 (xj+1 ) = Sj0 (xj+1 ) = bj + 2cj (xj+1 − xj ) + 3dj (xj+1 − xj )2

= bj + 2cj hj +

3dj h2j

(4.8) (4.9)

for each j = 0, 1, ...n − 1. 00 Looking at the second derivative Sj00 (x) = 2cj +6dj (x−xj ) and using Sj+1 (xj+1 ) = Sj00 (xj+1 ) for each j = 0, 1, ..., n−2 then

2cj+1 cj+1 dj

00 = Sj+1 (xj+1 ) = Sj00 (xj+1 ) = 2cj + 6dj (xj+1 − xj ) = cj + 3dj hj 1 = (cj+1 − cj ) 3hj

(4.10) (4.11) (4.12)

holds for each j = 0, 1, ...n − 1. Plug (??) into (4.17) and (4.9) to reveal:

aj+1 bj+1

h2j (2cj + cj+1 ) 3 = bj + (cj + cj+1 )hj

= aj + bj hj +

(4.13) (4.14)

for each j = 0, 1, ...n − 1. (4.13) can be solved for bj : bj

=

bj−1

=

1 hj (aj+1 − aj ) − (2cj + cj+1 ) hj 3 1 hj−1 (aj − aj−1 ) − (2cj−1 + cj ) hj−1 3

(4.15) (4.16)

28

CHAPTER 4. NUMERICAL ANALYSIS

which can be substituted into ( 4.14): bj = bj−1 + (cj−1 + cj )hj−1 1 hj 1 hj−1 (aj+1 − aj ) − (2cj + cj+1 ) = (aj − aj−1 ) − (2cj−1 + cj ) + (cj−1 + cj )hj−1 hj 3 hj−1 3 hj−1 hj 1 1 (2cj−1 + cj ) − (2cj + cj+1 ) − (cj−1 + cj )hj−1 = (aj − aj−1 ) − (aj+1 − aj ) 3 3 hj−1 hj 3 3 (aj − aj−1 ) − (aj+1 − aj ) hj−1 (2cj−1 + cj ) − hj (2cj + cj+1 ) − 3(cj−1 + cj )hj−1 = hj−1 hj 3 3 −hj−1 cj−1 + (−2hj − 2hj−1 )(cj ) − hj cj+1 = (aj − aj−1 ) − (aj+1 − aj ) hj−1 hj So: hj−1 cj−1 + 2(hj + hj−1 )(cj ) + hj cj+1

=

3 3 (aj+1 − aj ) − (aj − aj−1 ) hj−1 hj

(4.17)

for each j = 1, ...n − 1. n−1 n−1 Thus, using a linear combination of {hj }n−1 j=0 and {aj }j=0 , the {cj }j=0 can be determined and the remained of the n−1 constants, {bj }j=0 and {dj }n−1 j=0 can be calculated via (4.15) and (4.12) respectively.

So, given:  S(x) =

1 + 2x − x3 2 + B(x − 1) + C(x − 1)2 + D(x − 1)3

0≤x≤1 1≤x≤2

We have

S0 (x) = 1 + 2x − x3 = a0 + b0 (x − x0 ) + c0 (x − x0 )2 + d0 (x − x0 )3 S0 (x) = 1 + 2x − x3 = a0 + b0 (x) + c0 (x)2 + d0 (x)3

, x0 = 0

thus a0 = 1, b0 = 2, c0 = 0, d0 = −1

S1 (x) = 2 + B(x − 1) + C(x − 1)2 + D(x − 1)3 = a1 + b1 (x − x1 ) + c1 (x − x1 )2 + d1 (x − x1 )3 S1 (x) = 2 + B(x − 1) + C(x − 1)2 + D(x − 1)3 = a1 + b1 (x − 1) + c1 (x − 1)2 + d1 (x − 1)3

, x1 = 1

thus a1 = 2, b1 = B, c1 = C, d1 = D We want to solve for a natural boundary condition so we know that c2 = 0 = c0 . Thus, using (4.16), j = 1, hj = hj−1 = h = 1: 1 1 (a1 − a0 ) − (2c0 + c1 ) 1 3 1 2 = (2 − 1) − (2(0) + C) 3 C 1 = − 3 −3 = C b0

=

4.2. EXAMPLE: MIDTERM SOLUTION

29

-orUsing (4.11): j = 0 c1 C C

= c0 + 3d0 = 0 + 3(−1) = −3

(4.18) (4.19) (4.20) (4.21)

Now, use (4.14), j = 0, h =1 and C = -3 = c1 : b1 B B

= b0 + (c0 + c1 ) = 2 + (0 + C) = −1

Now, use ( 4.12), j = 1, h =1 and C = -3 = c1 :

4.2.3

d1

=

D

=

D

=

1 (c2 − c1 ) 3 1 (0 − C) 3 1

Method 3: Linear System of Equations - Checking the Previous Work

Burden and Faires, pg 143-146 outlines a method of creating a linear system of equations that extend from the (4.17). More specifically, Theorem 3.11 says: Theorem 3.11. If f is defined at a = x0 < x1 < ... < xn = b then f has a unique natural spline interpolant S on the nodes x0 , x1 , ...xn ; that is, a spline interpolant that satisfies the boundary conditions S 00 (a) = 0 and S 00 (b) = 0. The proof of Theorem 3.11 provides the following vector equation, Ax = b, where A is the (n+1) x (n+1) matrix:   1 0 0 ... 0 h0 2(h0 + h1 ) h1 ... 0    0 h 2(h + h ) h ... 0  1 1 2 2   A= ... .... ... ... ...   ...  0 ... ... hn−2 2(hn−2 + hn−1 ) hn−1  0 ... ... 0 0 1

and b and x are the vectors: 

 0 3 3   h1 (a2 − a1 ) − h0 (a1 − a0 )    and ... b=  3  3  (a − a ) − (a − a ) n n−1 n−1 n−2  hn−1 hn−2 0

  c0  c1     x=  c2   ...  cn

Pn |aij |) so Ax = b is a linear system with a unique solution (j=1 j6=i ) for c0 , c1 , ...cn (for more information review Burden & Faires, pg 398).

Matrix A is strictly diagonally dominant (ie: |aii | >

30

CHAPTER 4. NUMERICAL ANALYSIS

Starting with:  S0 (x) = 1 + 2x − x3 = a0 + b0 (x − x0 ) + c0 (x − x0 )2 + d0 (x − x0 )3 S(x) = S1 (x) = 2 + B(x − 1) + C(x − 1)2 + D(x − 1)3 = a1 + b1 (x − x1 ) + c1 (x − x1 )2 + d1 (x − x1 )3

0≤x≤1 1≤x≤2

we’re already found a0 = 1, b0 = 2, c0 = 0, d0 = −1, a1 = 2, b1 = B, c1 = C, d1 = D, c2 = 0 (refer to previous method). Also, h0 = h1 = 1. Thus: 

1 A = h0 0

  0 0 1 2(h0 + h1 ) h1  = 1 h1 1 0

0 4 1

 0 1 1

(4.22)



     0 0 0 b = 3(a2 − a1 ) − 3(a1 − a0 ) = 3(a2 − 2) − 3(2 − 1) = 3(a2 ) − 9 0 0 0     c0 0 x = c1  = C  c2 0 Using (4.17), j =1 a2 a2

= a1 + b1 + c1 + d1 = 2+B+C +D

Thus 

 0 b = 3B + 3C + 3D − 3 0 Combining into matrix form:  1 1 0

0 4 1

    0 0 0 1 C  = 3B + 3C + 3D − 3 1 0 0

We can solve this quickly, as only the middle row is significant: 4C = 3(B + C + D − 1) We previously found B = −1, C = −3, D = 1, does this solution work for (4.23)? 4C = 3(B + C + D − 1) 4(−3) = 3(−1 + −3 + 1 − 1) −12 = 3(−5 + 1) −12 = −12

YEAH

It is correct!

(4.23)

Chapter 5

MME: Math Modeling Experience I co-organized a series of workshops for high school and undergraduate students, preparing them for the COMAP (Consortium of Mathematics and its Applications) MCM (Mathematical Contest in Modeling) and HiMCM (High School Mathematical Contest in Modeling) competitions. Most importantly, I prepared a series of 2 hour lesson plans to introduce students to math modeling topics including disease and population growth models while providing them with the tools to develop their own models for competition modeling problems. In preparing for this program I needed to create promotional materials, program schedules and lesson plans. See the following sections for more information: Summary of the MME Program Section 5.1 Learning Objectives Section 5.2 Topics: Research, Mathematical and Competition Topics Covered throughout the year Section 5.3 Schedule: High School and Undergraduate Course Topic Schedule Section 5.4 Saturday lesson: Modeling in Nature Section 5.5 I particularly proud of my Modeling in Nature lesson plan because it shows a simplified application of my research in the MME program. Students were introduced to my research and gained an important competition skill through spreadsheet mathematics.

31

32

5.1

CHAPTER 5. MME: MATH MODELING EXPERIENCE

Summary of MME

The MME, Mathematical Modeling Experience, is a multi-level experience for high school students and undergraduates. The focus of the experience is to use the COMAP sponsored HiMCM and MCM to introduce both sets of students to mathematical modeling. The goal of the program is to use the university support to provide the high school with motivation and confidence to start their own team. What is Math Modeling? Math modeling is an extension of already existing math skills to real world problems. Thus math modeling means different things to different people. The overall idea is something like this: start with an interesting problem from the real world; make decisions about how to simplify the complex problem and approach the simplified version with mathematics; do the mathematics; interpret the result so that (ideally) something interesting can be said about the original real-world problem. ALL kinds of math are of interest to us. HiMCM: High School Mathematical Contest in Modeling Designed to provide students with the opportunity to work as team members in a contest that will stimulate and improve their problem solving and writing skills. This competition takes place online with teams of up to four students-working on a real-world problem for a consecutive thirty-six hour period. Teams are allowed to work on the contest problem at any available facility and then submit their solution papers via mail to COMAP for centralized judging. The teams solutions are then judged compared to other team solutions, receiving designations in descending order: National Outstanding, Regional Outstanding, Meritorious, Honorable Mention, and Successful Participant. MCM: Mathematical Contest in Modeling for Undergraduates This setup allows for the undergraduates to participate in 2 programs. First they get to work on their skills in teaching as teaching assistants for the high school teams, 1 TA for each high school team. Second they work on their own mathematical modeling skills by preparing for the MCM. The MCM takes place online with teams of up to three students-working on a real-world problem for a consecutive seventy-two hour period. The teams receive the problems on Thursday night and submit the solution on Monday night. Teams are allowed to work on the contest problem at any available facility and then submit their solution papers via mail to COMAP for centralized judging. The teams solutions are then judged compared to other team solutions, receiving designations in descending order: Outstanding, Meritorious, Honorable Mention, and Successful Participant.

5.2. SMART LEARNING OBJECTIVES FOR MME

5.2

33

SMART Learning Objectives for MME

(S - specific M - measurable A - achievable R - relevant T - timely)

By the end of the MME, we should be able to...

Instructors: • advise other instructors about the MME program. • promote the MME program to different audiences (academic, administrative, corporate). • Speak intelligently about teaching and mathematics outreach philosophy, including gender issues, effective classroom management, student mentorship, undergraduate research and applied mathematics education.

Students: • feel confident entering the (Hi)MCM by: – understanding the format, rules, and expectations of the contest. – having a good relationship with teammates. – being aware of resources available during the contest including departmental computers, library services and MME library. – formulating a contest strategy. • present ideas in a concise professional manner including a demonstration of written ability (referencing) and an ability to present ideas. • speak persuasively about the purpose and value of math modeling. • be aware of continuing research opportunities. • have an awareness of the following topics: – differential equations and difference equations. – stochastic-deterministic models. – the difference of continuous and discrete models. – graph theory. – linear regression. – optimization and linear programming. • have a facility with the computers which can be measured by the ability to share files, print, save, and manage directories. • use the following programs: MATLAB, Spreadsheets, Mathematica, • do purposeful internet searching, create weblinks, and understand what makes reliable internet sources. • have undergraduate teaching assistants learn about the teaching experience.

34

CHAPTER 5. MME: MATH MODELING EXPERIENCE

5.3

MME Topics

• Defining the Modeling Process: – Start with a real world problem – Define the problem: state assumptions – Build a model: ∗ List constraints ∗ Brainstorm different ideas for models ∗ Research – Test the model: how to put the numbers in and get results, using real-world data. – Evaluate the model – Refine/improve the model and restart the testing and evaluation process. – Write the Report (see below) – Relating this process back to the real world problem. • Computer Resources: – Internet Resources: judging the value of an internet source and do purposeful internet search. – Spreadsheets: functions, charts, linear programming – Open Source Office quirks (specific to the Linux based UCD Computer system) – Programming: Matlab, Mathematica – Typesetting ( Latex ) • Mathematical Topics: – Combinatorics – Differential Equations (Require High School students to know calculus) – Discrete Math: including difference equations, and graph theory. – Optimization and linear programming. – Stochastic-deterministic models. – Statistical Overview • Mathematical Approaches: – How to simplify math problems. – Discuss how equations are related to the data they model (regression, etc.) – How to transfer a conceptual model into some that can generate numbers and make predictions based on model parameters. – Data Analysis: discussing all possible explanations for relationships among data. – Real World Data. • Other Topics: – Team Work Strategy: Job assignments, how to get out of a rut, how to decide on a problem. – Using the Library: How to get to the information that you need. – Writing a technical report: basic structure, how it compares to an essay, does and don’ts, COMAP guidelines. – References: How to find references, what makes a good reference?

5.4. MME 2005-2006 SCHEDULE

5.4 5.4.1

35

MME 2005-2006 Schedule Schedule for High School Meetings

Basic Plan for High School Meetings: Have a weekly theme to discuss, be careful to provide several real world examples and to provide a combination of theoretical and computer solutions. At the end of the lessons the students should feel comfortable with their own skills and should feel like they can discover the solutions themselves, without requiring an authority figure to tell them what to do. Note: Every week have an activity for them to do when they get in the door. Ideas: find modeling in the newspaper, solving a simple word problem, do a thought process of this week’s problem. This allows students who will be late to not miss too much. 2005 MME Schedule • Meeting 1: Introduction, Showing Math Modeling is Fun! – Waiting Time, waiting for people to arrive- do brain teasers (15 mins) – Welcome, Introductions, Overview Program (10 mins) – Ice Breakers for the group (7 mins- 3 mins) – Many Models (Using TA demo models- use TA’s to bring out more) (30mins) – Activity Wrap up (15 mins- 2 mins on each model) – Team Activity- Voting (35 mins) – Wrap Up- Handout technical writing information, assignment, and discuss next week (5 mins) – Individual Assignment: Type up your voting model, bring for next week. – Paperwork: Send home a description of the program, timeline (emphasize parent’s meeting), our contact information for Saturday mornings, call for volunteers • Meeting 2: Population Models – Waiting Time- brain teaser (15 mins) – Sign up for their computer accounts (10 mins) – Split the groups: Need spreadsheet refresher group & the good with spreadsheets group ∗ 1 group gets the basics ∗ Other group does the plant growth kinematics spreadsheet – Heifer Intl. Population Model (30 mins) – Deer Population Model – Problem Solving Session • Meeting 3: Spreadsheets and Volume – Computer Basics: Using a spreadsheet and Open Office quirks – Continue Developing the Model- either population or volume in a room. – Assignment: 1 member of group write up further developed model. • Meeting 4: Evaluation and Validation – Individual Plant Growth Model: Develop a model for plant growth

36

CHAPTER 5. MME: MATH MODELING EXPERIENCE ∗ What is a plant? Just the seed or the whole plant? – Steps to Evaluate and Validate – SIR Model Lecturette – Group SIR Model Evaluation and Validation – Assignment: Group Paper of SIR model – TA Meeting: Discussed the topics that we still need to cover, gave them the chance to get out of the program. • Meeting 5: Optimization – Worked on Graph Theory: http://www.utm.edu/departments/math/graph – Reminder: Long Meeting – Problem Solving Session • Meeting 6: Practice Competition (longer meeting) – Competition Practice (3 hours): ∗ ∗ ∗ ∗ ∗

Provide Problem Statement They create team strategy 1 hour check in: Review model (assumptions, is it the simplest?) and team strategy 3 hour check in: Review Teach Report status final: Evaluate team performance and strategy

– Parent’s Meeting (1/2 hour, during last 1/2 hour of practice): ∗ Competition Timeline (schedule for Sat and Sun). ∗ Volunteers: Food, drivers ∗ paperwork – Student Presentation (1/2 hour): Students present their competition results to their parents • Meeting 7: – Return reviewed competition reports – Discuss results – Discuss Team Strategies – Do Team Building Exercises – Review previous lessons – Resource Scavenger Hunt: practice saving files, sending files, building web links, mme library, using OOwriter, printing, and looking at the HiMCM instructions. • Meeting 8: HiMCM • Meeting 9: Evaluation/HiMCM Wrap-up Party – Discuss their results- handout the certificates and t-shirts – Discuss other Explore Math Programs (ARML & Math Circle), plans for the next year, and possible summer math programs. – Have a Party! – Arrange for students who didn’t come to get their stuff.

5.4. MME 2005-2006 SCHEDULE

5.4.2

37

Schedule for Undergrad TA Meetings

Basic Plan: Have a weekly theme to discuss, focusing on teaching them how to teach and work with the high school students. It’s important that the TA’s don’t do the work for the high school students, instead they should help guide the students to find their own solutions. These weekly meetings should focus on this but also should provide time to discuss their weekly computer assignments (so that they can also learn how to use the programs). • Pre-Meeting Preparation: Each TA should spend the summer preparing a demonstration model. • Meeting 1 (on first day of classes): Introduction to MME – Introductions (5 mins) – Present individual demonstration models (quick- 20 mins) – Overview Details: What is this contest(s), Payment, Program Time-lines, Website (10 mins) – Model Problems (10 mins) – How to work with high school students- being constructive (5 mins) – Computer Assignment: Spreadsheets (spreadsheet growthexample.xls) • Meeting 2: Canceled, students were asked to work on their spreadsheet assignment, emailing the results to the instructors by Friday evening. • Meeting 3: ?? – Discussed spreadsheet assignment • Meeting 4: Volume Model – Reviewed Saturday’s Lesson of Spreadsheets and Volume. – Continue to develop volume models – Discuss Evaluation and Validation of volume models – Assignment: Read the HiMCM past problems, what topics do we still need to cover? • Assignment: Write a problem for the long competition based on the general format of the previous HiMCM problems. • Meeting 5: – Decide problem to bring to long competition – SIR model discussion, plotting and studying equations without solving them. – Discuss leaving team and finding replacements – Assignment: • Meeting 6: Finalize Practice Competition – Look at both of the chosen problems and make them more high school friendly. • Meeting 7: Review the Practice Competition Reports – Discuss changes that need to be made. – Discuss team selections – Discuss the mathmodels.org problem • Meeting 8: HiMCM Preparation This is the last meeting prior to the HiMCM – Set schedule for TAs during the competition

38

CHAPTER 5. MME: MATH MODELING EXPERIENCE – Finalize team selections – Work on mathmodels.org problem: Have each group present their approach and take parts to start the solution. • December Meetings: Completing the mathmodels.org problem (The undergrad practice problem) took the place of meetings for the rest of Fall quarter. The undergraduates had a really hard time finishing this problem and they just kept dragging out this process. We tried to help them separate the tasks and set deadlines but they didn’t turn anything in. • January Meetings: Note: The following meetings took place in January, prior to the HiMCM. Initially they were not going to be required but after students were having problems bringing together their teams we changed them to be required to decrease team frustration about inability to meet each other and develop team strategy. The format of these meetings were: 1. As long as teams are motivated to keep working on problems, we will assign them every 1-2 weeks, and then arrange meetings to provide feedback about their work. 2. For the other topics, we make group appointments with everyone who is interested. For example, if 3 people want to work on Matlab, we’ll find a time to do that for a couple hours together in the lab. • Meeting 9: Introduction to Library Resources Presented by UCD Librarians Ruth Gustafson ([email protected]) and Bob Heyer-Gray ([email protected]). The librarian walked through the lib.ucdavis.edu website and gave example searches to help students understand the library website. – Information provided to the speakers: information about our computer lab (PC on Linux based system running mozilla with projector available), COMAP website, highlighting the previous competition problems, sample topics to help them put together their presentation. – This session was in the MSB Undergraduate computer lab, we reserved a projector so that the presenters could work with the students. – This session was open to all graduate students, invitation including a request that they not interfere with the high school students. • Meeting 10: Matlab and Team planning – An online MATLAB tutorial was available for those who are interested to work through it and ask the instructors questions. – The teams will meet to make plans for: ∗ finishing their component of the room capacity problem ∗ agreeing on strategy for the competition. Those who were not interested in the MATLAB tutorial can use the rest of the hour to make progress on these two issues. • Meeting 11: MCM Preparation & Typesetting – LaTeX Tutorial: Teams completed an online LaTeX tutorial. – Discuss possible competition conflicts (projects, assignments, midterms) and possible teamwork solutions to these conflicts. – Work on the room capacity problem, and/or ask any other questions that have come up (questions about MATLAB? about how to do the writeup?) • MCM • Evaluation & Dinner – Discuss their MCM process – Discuss plans for the next year

5.5. MME SATURDAY LESSON

5.5

MME Saturday Lesson

Schedule: Saturday October 8, 2005 Todays Theme: Modeling in Nature 10:00 am

Opening Activities: Population Brain Teaser, Introductions (15 mins)

10:15am

Computer Time (25 mins): Start by getting everyones computer accounts set up. Then do the spreadsheet activity rootgrowth.xls. Split the groups into spreadsheet knowledge- those that need a refresher and those that don’t. Goal: Prepare students to be able to use spreadsheet computations in the competition. • Those that need a refresher- go over the basics. Step them through the spreadsheet root growth example using the undergraduate assistant. • Those that have enough experience and don’t need a refresher let them try it for themselves.

Notes about this for the future: None of the students had enough spreadsheet knowledge, even if they thought they did. Keep them together as a big group and plan to spend more time on this because this will be a useful resource that they can use during the actual competition. 10:40am 11:05 am 11:15 am

Rabbit growth (25 mins) - see attached lesson Bring the groups back together and recap. Have snack (10 mins) Deer Population modeling. - work in HiMCM teams. (40 mins) • Play “Oh Deer” (10 mins) • Students analyze the results of the game. (25 mins) – Students should use spreadsheet to graph the data collected while playing “Oh Deer”. – They should review their data and the information provided about deer populations. If they were a wildlife manager what sort of deer management would they use? • Wrap up the activity (5 mins): What did they learn? They should use the TA modeling results to write up their final reports.

Note: The ‘Oh Deer’ game really helped drive the point home about environmental control of population growth. This was made apparent in the writeups where the students often referenced it. They also referenced the model throughout the quarter so it must have really made a mental impact. 11:55 am

Wrap up the day (5 mins) 1. We find modeling in nature, where else can we find it? 2. Review/Suggestions from the last report.

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40

CHAPTER 5. MME: MATH MODELING EXPERIENCE

Carrying Capacity Brain Teasers Q: A pair of mallard ducks begins breeding at one year of age. They average eight offspring each year. If they original pair lives ten years (quite possible) and all the offspring survive, how many ducks will there be at the end of ten years? A: 1,953,125 Q: In the above example, if there is no increase in the population (zero population growth), how many ducks will there be at the end of ten years?] A: 2 Q: If there is a 100% increase in the population, how many ducks will there be? A: 4 Q: What happened to all the other offspring? A: Group discussion/ introduction to the topic Other Data ? Field mice produce an average of 17 litters of young a year with 6 young per litter. ? Snowshoe hares and jackrabbits have the potential to produce 13 million offspring per couple every 3 years. ? It took thousands of years for the human population to reach 1 billion. The second billion was reached in 75 years. In 1960, after only 35 years there was 3 billion of us. By 1976, there were 4 billion. By the year 2000, it is estimated that the population will exceed 6 billion. Did this in fact occur? What is the estimate for 2050?

5.5. MME SATURDAY LESSON

5.5.1

Plant Growth Spreadsheet Lesson

Problem Set on Growth Kinematics- Learning to Use

The data on the following page are taken from a biology lab, representing the movement of a plant root tip growing away from the surface. The distance from the root tip, x , and the rate of displacement from the soil surface, u (x ), are shown. We are going to use this data to do a thought excercise to understand that there is different ways to think about root growth: (1) Watching from the soil surface as the plant root tip grows away from you. -or- (2) Watching from e root tip backwards as the tip moves away from the surface. If you are actually sitting on the ip then you can think of the soil surface moving away from you at some constant rate equal to the growth rate of the root.

The best way to think about this growth process is to think about putting a mark on the side of a plant root at 2mm and watching how the mark moves away from the root tip as the root grows. You expect that the mark and the root tip will move away from the soil surface but you will find that the mark will change the rate as it moves through the plant growth zone ( which is only 10 mm long). When the mark is out of the growth zone then it will stop moving/growing and the tip will continue to grow away from the surface. Look at the marking experiment on the right to Variables: x: distance from the root tip where x = 0 is at the root tip (mm), 0.2 mm increments between x=0 and x=12.4 mm u(x) : rate of displacement from the soil surface (mm h-1) v(x) : rate of displacement from the root tip (mm h-1) REG: Relative Elemental Growth Rate (h-1) t: time in hours m: location of mark on plant through time that was originally at x0 = 2.0mm. (mm) v(m) : rate of displacement from the root tip of the mark that was originally at xo= 2.0mm (mm h-1) n: location of mark on plant through time that was originally at x0 = 3.0mm. (mm) v(n) : rate of displacement from the root tip of the mark that was originally at xo= 3.0mm (mm h-1) s: length of segement that was originally 1 mm, between x0= 3.0 and x0=2.0 mm (mm)

Assignment Goals: * Learn how to use spreadsheet formulas that reference cells in the spreadsheet * Learn how to plot and manipulate those plots. * Show your understanding of spreadsheets. * Do a basic plant physiology model of plant root growth.

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CHAPTER 5. MME: MATH MODELING EXPERIENCE

Problem Set on Growth Kinematics- Learning to Use Spreadsheets a) Convert u(x), the displacement rate from soil surface, to v(x), the displacement rate from root tip. i ) Calculate values of v(x) into column C in worksheet 'Initial Data' using the equation: v(x) = u(x) - u (tip), where the tip is located at x = 0. ii ) Create a new worksheet called 'plots'. All plots made for this assignment will be kept in 'plots' iii ) Plot v(x) against x and put the plot in your "plots" worksheet (note: you do not need to copy your data into this worksheet, instead when you reference your data and the plot function can put the plot in a different worksheet). Make sure to label the plot "Velocity of Displacement from root tip" and label the xaxis "Distance from the tip (mm)" with the y-axis labeled "Velocity of displacement from the root tip (mm/hr)". b) Estimate relative elemental growth rate (REG): i ) Use a two-point backward difference formula to estimate REG rates in Column D, ƒv / ƒx ie: ƒv / ƒx = (v3 - v1) / (x3-x1). Note: this formula won't work for x=0, at x = 0 assume REG = 0 or use a two-point forward difference formula. ii ) Plot the relative elemental growth rates against the distances from root tip. c) Growth Trajectories: Use your data in column C to find the position of a mark (m) on the plant root as a function of time (t) I ) In column F, count from 0 to 10 hours, in 0.25 hr increments. ii ) In column G, multiply the velocity for the point two mm from the tip, v(x=2.0), by a time increment of 0.25 h. Add the resulting distance increment to the former mark position (m0 =2.0 mm) to find the new mark location. mnew = 2.0 + v(m=2.0) * (.25) iii ) The new mark location (mnew) will be between two x values (xold