Southwest RIMS

Mathematics Department

University of Arizona

• Summary Report for 1994-1995

• Outreach Activities for 1994-1995

• Integration of Research and Education
  paper from the Notices of the American Mathematical Society, October 1996

Summary Report

Formation:

The Southwest RIMS (SWRIMS) began operations in September of 1994 under a two year grant from the National Science Foundation. SWRIMS is administratively housed in the Department of Mathematics at the University of Arizona. The Director of SWRIMS is Dr. William Yslas Velez. SWRIMS has chosen as its theme the notion of mathematical modeling, that is, the motto for SWRIMS will be that mathematics can be made to be meaningful and germane by seeing how mathematics is used to elucidate the workings of nature, in its many facets.

Working Structure:

The model that has been taken for SWRIMS is not a centralized one where individuals come to one place to carry out mathematical activities. Rather, we have chosen to carry out our activities in the local communities, where we can engage the broad spectrum of mathematicians in carrying out the goals of the institute, which are to increase the ability of all of us to better communicate to our students the wonder and utility of mathematical thinking.

Beginnings:

Towards this end, the enlisting of the great intellectual abilities of our research mathematicians in this endeavor, William Velez spent the Fall semester, 1994, visiting mathematics departments throughout the Southwest (Arizona State University, Northern Arizona University, New Mexico State University, University of New Mexico, University of Texas at El Paso, and Utah State University). At these universities, Dr. Velez presented what the National Science Foundation expected from a Regional Institute and how it was that the SWRIMS had chosen to address the guidelines that NSF had enunciated. The way in which SWRIMS is to operate is as follows: A scientific theme for the institute is to be chosen. This theme is to be the intellectual guide for at least an academic year.

Three themes had been proposed: population biology, cryptography, and optics. By the end of the Fall 94 semester one of those themes was to be chosen and SWRIMS activities were to begin with the Spring 95 semester. Three sites were to be selected by Spring 95. One of these sites would be The University of Arizona and the other two sites would be selected from those universities expressing interest in participating in these activities. A site would be funded to support a Core Group, consisting of two university faculty, one or two graduate students, one or two undergraduate students, perhaps an adjunct faculty, and two or three high school teachers.

Progress:

These Core Groups would spend the academic year working together to achieve the goals of SWRIMS. In December 94, the other two sites were chosen, namely, Northern Arizona University and Utah State University, and the Core Groups began their operations in January 1995.

The goals of SWRIMS were briefly mentioned above, but how are we to achieve them? This was addressed very eloquently by the Core Group at Utah State University, headed by Dr. James Powell and Dr. James Cangelosi. In describing the goals of their Core Group project, they state:

Project Goals:

1. Team building, idea sharing goal

Establish a vehicle by which secondary and middle school mathematics and science teachers and university faculty members contribute to the professional development of one another so that:

A. The secondary and middle school mathematics and science teachers refine and extend their understanding of how:

i. scientists and mathematicians collaborate on research projects

ii. mathematical discoveries and inventions can be stimulated by the pursuit of problems in science

iii. scientists apply mathematics so that they are better able to develop more meaningful academic experiences for their own students.

B. The university faculty members refine and extend their understanding of

i. the problems and complexities faced by secondary and middle school mathematics teachers

ii. how their research projects can serve as an impetus to building more meaningful mathematics and science curricula

iii. how learners construct mathematical and scientific knowledge so that they broaden the audience benefiting from their research efforts.

Conference on the Research Mathematician as an Educator:

In order to encourage greater participation among the university faculty, a conference was held in February, 1995. The purpose of the conference was to bring to the Southwest individuals who had participated in the Regional Geometry Institutes (RGI). The goals of the NSF sponsored RGIs were very similar to the goals for the Regional Institutes. The conference was entitled, "The Research Mathematician as an Educator". The invited speakers were Dr. Anthony Phillips (SUNY, Stony Brook), Dr. Herb Clemens (University of Utah) and Dr. Marjorie Senechal (Smith College). Due to health problems, Dr. Senechal had to cancel at the last minute. A panel of high school teachers, Doug Cardell, Jeff Uecker, and Paul Dye, gave a presentation on the program of study at Sunnyside High School. One day was devoted to presentations by these speakers and discussions. The second day was devoted to examples of their work. On this second day, Dr. Phillips and Dr. Clemens gave workshops for pre-college teachers. The participants at the meeting attended these workshops in order to get first hand experience as to how research mathematicians can contribute to educational issues.

Educational Information Services:

SWRIMS also recognizes that it is important to catalogue the outreach activities of mathematics departments and SWRIMS has committed itself to collecting this information and putting it together. A SWRIMS produced Tour of Outreach Activities document (for the Department of Mathematics, University of Arizona) is now available on the Web Server ``http://math.arizona.edu.'' We hope soon to have the data together for the other two sites that are presently carrying out RIMS projects.

This document not only describes the activities that are being carried out, but also details how the programs are funded and whom to contact if an individual is interested in starting such a project at their home institution.

Spring 1995:

During the Spring 95 semester, all of the three sites began their SWRIMS activities. Each of the sites had the autonomy to determine their course of action and it is interesting to see the diversity of programs that were developed.

Northern Arizona University:

The Core Group consisted of Dr. Terrence Blows and Dr. Joseph Lingerfeldt (Northern Arizona University Faculty), Deborah Wolf (High School Faculty), Valerie Young (Graduate Student), Kiona Ogle (Undergraduate Student). This group met once a week for several hours during Spring 95. In the beginning, research was conducted to assimilate current, high interest population data. Next the scope and sequence of the high school curriculum was explored. Finally, the team spent time bridging the "knowledge gap" by developing several creative and instructive lessons. These lessons involved "hands-on" activities and used technology to demonstrate visualization of the mathematical concepts. The results of their visits to high school classrooms are contained in a document that they developed entitled, "RIMS PROJECT IDEAS - Concerning the study of population dynamics'". The Core Groups visited several high school classrooms in order to try out the ideas that had been developed.

The Core Group also developed the idea of offering a university course on population biology for high school teachers in Fall 95. Northern Arizona has already had this course approved for graduate credit and has recruited teachers. A syllabus for this course includes applications to Spruce Budworm models, Harvesting, and Fishery Management. The mathematics that is to be studied is very much tied to physical applications.

Utah State University:

The Core Group consisted of Dr. James Powell and Dr. James Cangelosi (Utah State University Faculty), Mindy Bell and Nancy Puhlmann (High School Faculty), Anne Marie Harris (Graduate Student). The Core group began by meeting together to read through the text, Mathematical Models in Biology by Leah Edelstein-Keshet. This text, like many mathematics texts, is couched in its own terminology, and the reading was difficult for the high school teachers, as would be expected. In order to make the reading more germane, the Core Group developed a strategy. All of the participants in the seminar bought a tomato, took out the seeds and planted them. The Core Group members then kept track of the number of seeds that were produced by the plants. This physical model was then used as a basis to understand the mathematics that was involved. All of the participants found that this activity was very helpful in making a connection between their own experiences and the mathematics that was being discussed.

A seminar was also developed for the Core Group. Several scientists gave talks on their research which dealt with the mathematics of population biology. The Core Group was to take some of the ideas that had been discussed in the seminar and develop material to be used in the classroom. This experience led this Core Group to develop a model for future work by emulating the idea of the joint work that was often presented in the aforementioned seminar, the idea of a mathematician and a biologist jointly working together to address a problem. With this model in mind, they formed teams of teachers, one a mathematics teacher and the other a biology teacher. These teams consisted of teachers at various levels of the educational spectrum, ranging from university, junior college, high school and junior high school. Though elementary school teachers do not specialize in this fashion, they have identified a pair of elementary teachers.

These teacher pairs will work together to address biological problems from a mathematical viewpoint and they will try to integrate these activities in their classrooms. The topics that they will address have come from the seminar. The teacher pairs will also attempt to carry out this experiment in their classrooms by forming groups of students, one group from their biology class and the other group from their mathematics class. As the teacher pairs carry out their activities, they will keep a detailed account of what actually transpires in their classroom. This account, which the Core Group has called a Case Study, will be used to disseminate the material and the experiences that are to be developed.

The University of Arizona:

The Core Group consisted of Dr. J. M. Cushing, Dr. Joseph Watkins, Dr. Larry Grove, Dr. William Yslas Velez (University of Arizona Faculty), Doug Cardell, Jeff Uecker, Paul Dye (High School Faculty), Martin Garcia (Graduate Student), Dianna Pena, Tyler Bayles (Undergraduate Biology Majors). The Core Group met twice a week at the University of Arizona, though occasionally the Core Group would hold its meeting at Sunnyside High School. Dr. Cushing is the scientific coordinator for SWRIMS and he began the Core Group activities by giving some lectures on the mathematics of population biology. The Core Group also thought it was important to hear from the high school community. Several high school teachers gave presentations on the kind of innovations that were being attempted in the high schools.

A decision was made early on that the Core Group would develop material that would be used in the classroom. After several meetings a series of activities was developed that could be used in the high school classroom. These activities dealt with the fact that exponential growth occurs very naturally in nature. To ensure that these ideas would work in the classroom, it was decided to experiment by actually carrying out these activities in a local high school classroom. The Core Group spent four days, from 7 a.m. until 3 p.m. at Sunnyside High School, carrying out these activities and modifying them as they went along. Dr. Watkins took up the duty of actually writing up the results of these activities in a set of notes called the "SWRIMS Class Notes: The exponential function and the dynamics of populations". The notes are quite extensive and detail the four days that the Core Group spent in the high school classroom at Sunnyside High School. These notes have been widely distributed and will be used at other sites in this country.

Another project that is currently being implemented at The University of Arizona is a modification, by Dr. David Lovelock, of some software to aid in the analysis of the mathematics dealing with the discrete modeling of populations, as opposed to models involving calculus.

Conference on Mathematical Modeling in Population Biology:

On August 17-19, 1995, a research conference, entitled, "Mathematical Modeling in Population Biology", organized by Dr. Cushing and Dr. Powell, and sponsored by SWRIMS, was held in Logan, Utah. Members from each of the three Core groups attended the conference where invited mathematicians and biologists from around the world presented their research. Two special sessions were held for the Core Groups, one before and another after the meeting. The first session of the Core Groups was used to share the results of their activities with each other and the second session was used to formulate plans for the next year's activities.

Researchers were asked to give general talks, since there would be a few high school teachers and undergraduates in the audience. Most of the speakers made a good attempt to give their talks with a minimum of mathematical notation. This attempt at communication was appreciated by all attendees. This conference was not aimed at high school teachers and only the high school teachers who belonged to a Core Group attended the conference. Besides bringing together investigators to share the results of their investigations, there was another agenda to this meeting. The research conference was supposed to provide the Core Groups with ideas that were motivated by biology and used mathematics to elucidate the biology. The Core Groups were presented with some very good topics during the conference that they will be able to use during the following academic year to help achieve the goals of SWRIMS.

The Core Groups will begin working on some of the ideas presented and they will invite some of the presenters at the conference to the Core Group sites to aid in the development and educational implementation of these ideas.

Outreach Activities

Beginnings: In January, 1995, the University of Arizona organized its inaugural core group of university professors, high school teachers, graduate and undergraduate university students, and high school students. Here is our original team:

University Faculty

J. M. Cushing

Larry Grove

Shandelle Henson

William Vélez

Joe Watkins

Students

Tyler Bayles

Martin Garcia

Dianna Peña

High School Faculty

Doug Cardell

Paul Dye

Jeff Uecker

We also invited several prominent research mathematicians who had experience presenting research ideas to audiences that included high school teachers and students. This two day long conference, "The Research Mathematician as an Educator", held on February 17 and 18th, is described elsewhere in this report.

Interactions: Our major goal for the semester was to go to the high school classroom and teach. We had the good fortune to meet the mathematics teaching staff at Sunnyside High School. Their interest, flexibility, and enthusiasm convinced us that Sunnyside was a good place to start. The classroom team - Professors Larry Grove, Bill Vélez, and Joe Watkins, high school teachers Doug Cardell, Paul Dye, and Jeff Uecker, and university students Tyler Bayles, Martin Garcia and Dianna Peña - began holding planning sessions in conjunction with Jim Cushing's lectures. We settled on four full day visits to Sunnyside, each visit separated by about a week. Our plan was to bring exponential growth and decay in a wide variety of situations - simpler at first, more subtle as we continued - in the context of the dynamics of populations. This variety gives each student several invitations to engage in the subject and to see that mathematical knowledge, once they make it for themselves, can provide insights in contexts that otherwise seem unrelated. The two hundred students from Sunnyside High School that we worked with on each of these days were an essential part of the SWRIMS team.

We met classes at all levels, from pre-algebra to pre-calculus, with some of the classes having multiple levels. Our classroom methods were simple. We presented the ideas of the day as succinctly and clearly as we could. The students then chose their own methods of investigation and analysis. The classroom was provided with all the materials that the students required - paper, pencil, rulers, calculators, computers, and, most importantly, their fellow classmates. Students often found groups of four comfortable and would begin their activities. The teachers would circulate around the room, listening to the deliberations, asking questions, and providing quick tutorials. Near the end of the class hour, the students would summarize their findings, and place them prominently on the board. We were all then able to compare points of view and methods of inquiry.

The Sunnyside High School teachers report that the impact of the visits on the high school students is marked. The nature of the impact is difficult and certainly varies from student to student. Students saw mathematics being relevant to their own lives. This relevance provided impetus to study mathematics for its own sake. Regular interaction with high school and university faculty gave many students more confidence in reporting what they learned. Their active participation in the development of curriculum gave them a more realistic impression of how mathematical ideas are developed by professionals and how these ideas are turned into activities.

SWRIMS Class Notes: After having completed the first of a series of visits to Sunnyside High School, the activities were recorded by Joe Watkins as a case study in "Southwest RIMS Class notes: Exponential Growth and the Dynamics of Population". These notes contain supplementary documents so that a classroom teacher can adapt the ideas contained in the Class Notes for ready use in the classroom. In addition, David Lovelock developed software program "maps" - now a part of the University of Arizona Mathematics Software to complement our teaching efforts.

The contents of these class notes have been presented to over a hundred teachers. Joe Watkins gave a presentation in a PRIME workshop on April 26, 1995. PRIME (Promoting Reform in Mathematics Education) is a University of Arizona based, NSF sponsored project designed for teachers in the later elementary school or in middle school. Elias Toubassi directs activities of PRIME. William Vélez gave a series of workshops in June to middle school teachers as a part of a second NSF sponsored program - Making Everyone Count. David Gay directs the activities of Making Everyone Count. On June 15, Joe Watkins delivered a workshop sponsored by the Tucson Unified School District (TUSD) to area high school faculty. This workshop, one of a series, was coordinated by Becky Montaño, director of faculty development at TUSD. Also in June, Shandelle Henson gave a workshop to area high school mathematics faculty in Chattanooga, Tennessee.

The activities of PRIME and Making Everyone Count are more completely described in A Tour of Outreach Activities in the Department of Mathematics at the University of Arizona, compiled by Catherine Yslas.

Feedback: The reaction of teachers and students has been tremendously enthusiastic - from the elementary school through the middle school to the high school. Activities involving coins or the distance probe are finding their way in the elementary schools. Teachers are making good use of the notes and are adapting them to a wide variety of situations. At risk classes, honors classes, and traditional classes are planning to try these activities. They are being used in mathematics classes, in mathematics and biology combined classes, and in mathematics and social science combined classes.

The philosophy of SWRIMS has received an even more favorable response. Teachers are visibly impressed that the National Science Foundation and the University of Arizona would spend its resources in a way that benefits every type of classroom in every type of school.

To date, we have received requests for the Class Notes from schools in Arizona, Utah, Tennessee, North Carolina, Iowa, and Hawaii. Dianna Peña is disseminating information on SWRIMS activities at an NSF Diversity Conference in Washington, D. C. As a consequence, we expect that several additional widely dispersed schools will be participating in our activities.

The core group has now reconvened for the fall semester with some new core members - Catherine Yslas, a graduate student in mathematics, Cristina Enriquez, an undergraduate student in mathematics, and Brenda Ugalde, a mathematics teacher from Pueblo High School who is spending this year at The University of Arizona as an adjunct faculty member. In addition, Sue Adams and Becky Montaño from the Faculty Development Department of TUSD will be attending the biweekly SWRIMS meetings.

Fall 1995: This fall we will be supporting the teaching efforts related to our class note at several Tucson area high schools - Catalina, Cholla, Mountain View, Pueblo, Rincon, Sunnyside - as well as sites outside the Tucson area. We plan to accomplish this with an introductory and a follow-up workshop. During these workshops, we will refine the Class Notes and prepare ourselves and the participating high school faculty for the second series of school visits. In addition, all participating schools are invited to subscribe to the SWRIMS listserv. This internet connection will provide a forum so that all participants in the SWRIMS activities can share their experiences and contribute their perspectives. During the SWRIMS sponsored research conference, several of the presenters expressed interest in working directly with high school students. Consequently, we will also be using the internet to establish direct lines of communication between high school students and researchers in mathematical biology. The results of these student-researcher interactions will be a central part of a SWRIMS capstone conference in population biology. Plans are currently being developed to have this event in the fall of 1996.

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