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THE RESEARCH MATHEMATICIAN AS STORYTELLER

William Yslas Vélez and Joseph C. Watkins
Southwest Regional Institute in the Mathematical Sciences
Department of Mathematics
University of Arizona
Tucson, Arizona 85721

The Southwest Regional Institute in the Mathematical Sciences (SWRIMS) was funded by the National Science Foundation (NSF) as an effort to integrate research and education. One of the visions that NSF has formulated is that the researcher should be simultaneously involved in both research activities and in educational initiatives. One of the purposes of SWRIMS was to bring into focus what these educational initiatives might be. SWRIMS wanted to answer the following questions. How can a researcher use research achievements, especially their own personal research achievements, to further the educational goals of the mathematics community? How can this research be used to motivate our children to pursue higher mathematical studies? One of us addressed these questions in [Ve1ez]. This article served to catalogue SWRIMS activities for the academic year 1995-96 at its three funded sites (University of Arizona, Northern Arizona University and Utah State University). The Vélez article also served to formulate some ideas that could be used to reach the goal of integrating research and education. The present article aims at highlighting one of the products of these activities and at the same time to remind the mathematics research community of one of its responsibilities, namely our role as the teachers in our society. We are the ones who are responsible for transmitting our mathematical knowledge to the community.

It is the role of teacher that we have come to re-think as we carried our SWRIMS activities. This role is very nicely articulated in the words of Robert A. Williams, Jr., in his Forward to The Rodrigo Chronicles [Delgado]. Williams, drawing from his background as a member of the Lumbee tribe states:

In the Native American tradition, to assume the role of Storyteller is to take on a very weighty vocation. The shared life of a people as a community is defined by an intricate web of connections: kinship and blood, marriage and friendship, alliance and solidarity. In the Indian way, the Storyteller is the one who bears the heavy responsibility for maintaining all of these connections.

To be a Storyteller, then, is to assume the awesome burden of remembrance for a people, and to perform this paramount role with laughter and tears, joy and sadness, melancholy and passion, as the occasion demands.

There is an art to being a Storyteller, but there is great skill as well. The good Storytellers, the ones who are most listened to and trusted in the tribe, will always use their imagination to make the story fit the occasion.

The stories that SWRIMS would like to tell are stories of relevance, of applicability, of excitement, of importance. They are the stories of our adventures as we forged new tools to address old problems or found new uses for old tools. These adventures have all of the high drama that we would expect from journeys into unexplored territory, and yet we have kept these adventures hidden. We have not been effective Storytellers in our tribes, and our children have been kept ignorant of this important aspect of our culture. So much so that many in our community state with pride how ignorant they are of mathematical reasoning. The research community is one who should develop these stories and begin to tell them. If we do not, then who will?

In this article we want to describe two stories that SWRIMS developed for the high schools. These stories were developed in a particular setting, a grouping of a very diverse set of individuals. One of the mandates of NSF in funding SWRIMS was that activities should include undergraduate and graduate students, high school and university faculty. Groups of these individuals were formed at the various universities. One of the tasks of these groups was to re-package mathematical research in such a way that the results of the research could be understood by a much larger community. It was hoped that these activities would provide a valuable learning experience for all of the individuals in the group. The researchers would better learn how to communicate mathematics while the high school faculty would come to understand more fully the utility of mathematics. When the groups felt that they had developed ideas that would work in a high school classroom setting, these ideas were tried out in the classroom by all of the participants in the group.

Why such a broadly based team?

For the students on the team, high school was not so long ago. They understand student capabilities and student anxieties. Their closeness in age to the students adds to their credibility. As Storytellers, the student's mainstay is the joke placed to give levity when the circumstances become unduly serious or tense.

High school teachers understand the class management issues that university faculty rarely, if ever, consider. For the most part, teachers operate under conditions that researchers would find intolerable and do so with grace, imparting little of the frustrations created by these conditions. As a Storyteller, the teacher's genre may be the anecdote or the riddle, a short piece well placed in the classroom hour that ties stray facts together and starts heads pondering.

Researchers often spend years focused on a tiny piece of a speciality inside one of hundreds of possible scholarly subjects. We become researchers only after many years of training, understanding that our achievements can have relevance, have importance, beyond our own tiny piece of a specialty. The researchers story is an epic.

We will now work telling our epics with jokes, anecdotes, and riddles "as the occasion demands", using our imagination to "make the story fit the occasion."

Looking back over the two years, we realize how uncertain our beginnings were. We had no fixed ideas on how to proceed and no models from the mathematical community to guide us. As we met with mathematics and biology teachers from several of the Tucson high schools and became acquainted with them and with their programs, our confidence in our ability to make an impact waxed and waned. Part of our early activities had us retreating to a familiar place, the mathematics lecture. The Core Group settled on attending a series of introductory lectures by Jim Cushing, a mathematician whose research focus is mathematical ecology and population dynamics.

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 Pena - began holding planning sessions in conjuction 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 contexts - simpler at first, more subtle as we continued. These experiences are summarized in the Southwest RIMS Class Notes: The Exponential Function and the Dynamics of Populations.

Sunnyside High School is on Tucson's south side. This barrio school has a predominantly Chicano population. The University of Arizona is six miles to its north. Very few of these students have managed to bridge this six mile journey northward.

Day 0. Our First Story Begins. Doug, Paul, and Jeff invited the Core Group to Sunnyside to see how high school life proceeds. Our plan was to be anonymous observers trying to get a feel for the place. As we would later learn, personal daily greetings are a tradition in the mathematics building at Sunnyside. The students found us quickly and immediately began decribing their activity for the day. After much design and construction, they had completed two large paraboloids and were ready to place them at extreme ends of a hallway crowded with students. Could they communicate over the din using the paraboloids to direct sound? They had had a successful test earlier in the day, and they were eager to repeat the test in the Core Group's presence.

The second test was an overwhelming success.

This is how mathematics instruction proceeds at Sunnyside. Mathematical concepts are turned into physical reality by the students themselves.

The typical classroom at Sunnyside has multiple levels - students enrolled in Pre-algebra, Algebra I, Algebra II, and Pre-calculus can regularly be seen working side-by-side. The topics are common for every student. The mathematical techniques and goals are specialized to be appropriate for each student.

Day 1. Powers of Two. Our first aim was to experience the exponential function -- both growth and decay -- in a large variety of ways. To make the case as elegantly as possible, we first suggested activities that generate powers of 2.

Our first exercise was the simple successive folding of a sheet of paper in half. How many sheets thick is the paper after n folds? Some pre-algebra students, not looking for a formula or pattern, counted the sheets of each fold. The advanced students wrote a simple statement: "After n folds, the number of layers is 2 to the power n." The pre-algebra students ask, quite correctly, "What does it mean to have 1024 after 10 folds when I can't even fold the paper in half eight successive times?" The ensuing discussion give us our first glimpse into the applicability and the drawbacks of a mathematical model.

One student hustled through the exercise, showed us his result, and then asked us, "What mathematics do I need to study if I want to be an economist?" He had been holding this question for a long time, waiting for just the right chance to ask it. We had brought a university catalog for just this occasion, and went over the program in mathematics and economics at The University of Arizona, where he is now a sophomore.

The second activity uses a distance probe and a computer to give a graph of the distance between a moving person and a probe. The graphs were displayed as they were generated using a projection panel and an overhead projector. Because the use of a distance probe is likely to be a new experience for many students, we began with a series of warm-up questions and kinesthetic answers to help them become familiar with this technology. The students were slow to start participating, but soon wanted to create their own graphs. Very soon, every student could describe and create a graph having properties usually introduced in a calculus class.

When the time came to perform the exponential walks, 2 to the power t and 2 to the power -t , the students quickly realized that such walks are impossible to sustain for any length of time.

One student, whose friend was having a baby, had heard that a fetus doubles in size every month. If we use the fact that a healthy baby born at full term is about 8 pounds, then we can fill in a table of monthly birth weights. Using this table as a guide, the students embarked on a sophisticated discussion on pregnancy. Everyone had been close to at least one classmate who is now a parent, and so the exponential function had meaning to them.

Day 2. Random Models of Growth and Decay. Our second visit to the high school classroom was devoted to more experiences with the exponential function. Today, we will use random models and so we will not have exact powers. Throughout the classroom hour we conducted two activities simultaneously -- the growth of bacterial colonies and coin experiments.

Tyler was the "biologist" in charge of the experiment. During each of the six class hours, he chose a group of laboratory assistants. We grew two colonies of a locally gathered soil bacteria, one at room temperature, and another at the optimal temperature for growth of this strain of bacteria. We ran the experiment throughout the day, generating a long table of spectrophotometer readings on the blackboard. For some students, this connection between mathematics and biology was the best part of the whole experience, and many would drop by between classes to examine the data.

At the same time that the bacteria growth experiment was proceeding, the students were also performing two coin experiments -- one on growth and one on decay. These coin experiments are the simplest examples of simple branching processes having growth rates are 2 and 1/2 with the variance being as small as possible while still being positive.

Graphs of the experimental results convinced the students that the growth or decay was exponential. Eventually, several students could make a confident conjecture on the growth rate and give an explanation based on an intuitive understanding of the probabilistic mechanism underlying the experiment.

By this time, the Core Group knew quite well, through many personal interactions, that Sunnyside was filled with talented and creative students. Many remarkable students did not know how remarkable they indeed were. The Sunnyside teachers wasted no opportunities in arranging some unstructured time for the Core Group members to meet with these students.

One Chicana had just returned from a year abroad in Germany and we conversed for awhile in German. She was a top mathematics student but her love was the theater. Her parents had objected strongly to her year in Germany and were not in favor of her going to New York to attend a renowned acting school even though she had been offered a scholarship covering tuition and living expenses.

This is a story of "kinship and blood". At this time, many families feel that a child who leaves the community physically also leaves the community spiritually. This young woman was truly remarkable. When she returns to the community after her time away, she can undertake, through her acting, a parmamount role in this barrio. Can we as teachers use our skills to listen to this young woman's family and to earn their trust?

Day 3. Population, Plenty, and Poverty. The goal for the third day of classroom activities was to develop ideas on the purposes for mathematical modeling, and, at the same time, gain some understanding of the limitations that all models have. We placed our discussion in the context of the dynamics of human populations. The students were given copies of "Population, Plenty, and Poverty" by Paul and Anne Ehrlich. This article, which appeared in the December, 1988 issue of National Geographic, gave us short introductions to the lives of six families, one each from Kenya, China, Hungary, India, Brazil, and the United States. To provide context for each of the stories, we provided a summary on each of their countries obtained from the CIA World Factbook .

Many students live in large families, and their families are important aspects of their day-to-day lives. So, the prospect of a society, as China is now building, with no aunts, no uncles, no nieces or nephews, and no cousins was very unappealling to them. The concensus was clear. "If the government is going to institute such a strong policy, then they had better get it right." How do the Chinese know for sure that the population will not collapse someday? How do they know that a country can function effectively with such a large number of elderly? The policy makers are obliged to develop the best model possible and to study that model with utmost care.

This discussion brought the students logically to the role of any mathematical model. What can these models do for us? First of all, in creating a model, we decide what concepts are important and we state our beliefs on the interactions of these concepts. We can use mathematics to study and computers to simulate our model. We can test our hunches and, after a period of study and deliberation, make well informed conjectures. Models can sometimes reveal to us how important things are, and suggest to us what more we need to investigate. If we have competing strategies, then we can use the model to assess each of their impacts.

We ended by building a mathematical model. Our population would have two age groups - children and adults. The dynamics would be given through a birth rate, a maturation rate, and death rates. We then proceeded to perform the model dynamics.

Some of us were designated children; some of us became adults. A few of us were consigned to limbo. The class chose three members of the census bureau - a "stork" to count the adults and determine the number of new babies by the time of the next census, an "escort" to count the number of children and compute the number of those who had matured and a "grim reaper" who would cull the adult population. The grim reaper was never hard to recruit.

Day 4. Models of Populations. Our fourth day of activities was devoted to investigating the model we had developed at the conclusion of Day 3. The rates were carefully chosen so that the eigenvalues for this system were 1 and 1/2. This gave us a constant population total and displayed clearly the role of the eigenvalue 1/2 as the rate of convergence to the eigenvector having eigenvalue 1.

We imagined a large spaceship with 40,000 adults and no children headed for Alpha Centauri and looked at the dynamics of the population through 24 censuses. These population numbers were obtained either by using a programable calculator or by using Stella, a computer software tool.

After careful working with the data, the students could sort out that powers of 1 and powers of 1/2 had something to do with this model.

The next question turned out to be quite challenging: If we know the initial population, can we predict the stable population?

To address this question, each group picked an initial population of 10,000, variously distributed between children and adults. They reported their stable population - 4000 children and 6000 adults in every case.

Next we varied the initial population sizes and recorded the stable populations. The students still saw no pattern. At this point, the Core Group had a caucus. Eigenvectors, we believed, were a natural concept. How could we get this point across?

In a near panic, we suggested that they run the model a third time, and record the stable population in a pie chart.

Aha, all the pie charts looked the same. So, at Sunnyside High School, we do not have eigenvectors, but rather stable pie charts. This experience and the ideas are described in "The Teaching of Eigenvalues and Eigenvectors: A Different Approach".

The students then changed a parameter and were able to predict the stable growth rate and the stable pie chart for their new model.

In this hubbub of activity, we end our first story. We began by folding paper and ended by simulating age structured populations. The Sunnyside students were ready for more. An account of the activities they saw as possible continuations could very well be a list of projects drawn by a researcher in population biology.

An Interlude. Exchanging Stories. The midpoint of SWRIMS two year plan of outreach activities on the dynamics of populations was a mathematics conference organized by Jim Cushing of the University of Arizona and Jim Powell of Utah State University. One purpose of this conference was entirely typical - bring together mathematicians who share research interests and schedule lectures for them to report their most recent findings. However, this gathering had an additional purpose. The presenters were informed that high school biology and mathematics faculty would be attending. Jim Cushing and Jim Powell asked the presenters to prepare their remarks with this larger audience in mind.

On the whole, the lecturers should be praised for their efforts and their successes in communicating their ideas. This success was summarized by one participant researcher who remarked: "I am inviting high school teachers to all of my conferences - the extra effort made in preparing the talks made them more accessible and enjoyable for all of us."

The conferees heard many talks that could be adapted to provide an exciting experience for high school students. The talk particularly noteworthy for this story was presented by James Matis of Texas A and M's Statistics Department. Jim had developed a probabilistic model for the migration of the Africanized honey bee and used it to make a highly successful prediction of the Africanized honey bee's arrival to the United States through south Texas.

Sunnyside teachers Doug Cardell and Jeff Uecker labled this talk a "must do" and we arranged to talk to Jim and begin the work in preparing the ideas in this research for a high school audience.

Our second story began in 1956 in Rio de Janeiro. A bee researcher there hoped that he could combine the aggressive characteristics of the African honey bee with the nectar collecting predilections of their European cousins to produce a bee that aggressively collects nectar.

The result realized was the escape of 26 colonies of African honey bees.

European honey bees were brought to the Americas in the sixteenth century by the Spaniards and have spread throughout the hemisphere. Soon after the 1956 escape in Brazil, the European and African honey bees in Brazil began to interbreed. Their progeny of Africanized honey bees took on the behavior of their African ancestors. Over the succeeding decades, the Africanized bee habitat has been spreading into the regions that did not sustain mean temperatures below 50F. Their migration front has been moving northward at an average speed of 400 miles per year. By 1990, the time and place of their arrival to the United States of America was the subject of scientific research. James Matis gave us the results of his investigations.

Joe Watkins, a mathematician in the University of Arizona Core Group, took the task of recruiting a team to implement a strategy for presenting these research ideas. Tyler Bayles made a return visit to the SWRIMS project. Robert Lanza, a geography graduate student, was brought on the team to work with the bee biologists to design remote sensing maps for use in predicting migration. Rhonda Fleming represented a fine group of environmental biology teachers at Tucson High School.

The tremendous stroke of fortune was the discovery that Tucson hosts one of the five United States Department of Agriculture's bee research centers. Gloria DeGrandi-Hoffman, an entomologist at the center, works on models for the dynamics of hive populations and on mechanisms for the Africanization of European honey bees. This "bee team" met weekly at the center from October, 1995 to January, 1996, to design the activities.

Our goal was to connect many of the broad themes found in a high school environmental biology course to a common topic - the honey bee. For high schools, the novelty of the activities is that mathematical ideas be used to provide insights into biologically meaningful questions. We hoped to intrique students by bringing them to the current research questions in the biology of bees. The results of the six days of activities are documented in the Southwest RIMS Class Notes: BEEPOP: The dynamics of Honey Bee Populations in the Hive and in the Wild. We wanted to assess our activities with a broad range of students. Thus each day's activities were tested in six different classes at Tucson High School - one regular class, two with at-risk students, two with predominantly Spanish speaking students, and one with a class having many high achieving students - with "bee team" members presenting the activities. We made our visits on consecutive Fridays.

Each day's activities began with a reading. These readings were composed by the members of the team by gathering information from beekeepers' handbooks and then circulating the readings around the bee lab for updates and clarifications.

During the class, we would discuss the biology of bees extensively. We would try to present the mathematical activity in clear and concise language. This allowed the students to form their own working groups to formulate the pertinent questions and to seek out methods of investigation. This method freed the teachers to move among each group of students and to work with them as they addressed the issues and prepared their presentations for the end of the classroom hour. These presentations revealed that many imaginative methods are possible.

These Fridays could be quite intense and the teachers - Oscar Romero, Richard Govern, Kay Wild, and Rhonda Fleming - made careful preparations before and extensive follow-ups after so the day could go well.

Day 1. Sizing up the Population. Part of a scientist's work is to evaluate how the language of mathematics can convey the concepts of the science. For more than a century, education in mathematics and the evaluation of its role in physics has been considered an essential part of the training to become a physicist. For some number of years, a physics teacher could state a reasoned opinion on the mathematics necessary to do physics effectively.

The fact that questions in biology could yield to mathematical techniques is a much more recent phenomenon. Consequently, only the most extraordinary biology or mathematics teacher from a previous generation would have integrated the two subjects - mathematics and biology.

At this time, many mathematical biologists could agree on a syllabus for a biomathematics course whose emphases were ecology and population biology. A consensus for a course content that included genetics, molecular biology, and cellular biology would be difficult to build. The high school students of today will form the next generation of mathematically well trained biologists. To them will fall the question of the role of mathematics in the development and understanding of issues in biology. In this classroom activity, we wanted to bring the idea that mathematics and biology are two subjects that can work side by side. In so doing, we decided to keep the day's mathematics gentle.

Working methods for obtaining a reliable census differ dramatically depending on the animal or plant whose population is being tracked. For bees, this can be accomplished by photographing the frames in a hive and estimating the number of bees or brood in the photograph. This task in estimation was today's activity.

Day 2. Practical Knowledge of European and Africanized Honey Bees. On any warm and sunny day, those of us spending time out of doors are likely to encounter a honey bee stopping by a flower to collect pollen and nectar. Our familiarity with the honey bee gives us a practical curiosity for them. For this day's activities, we will address the practical side of human co-existence with the honey bee from the viewpoint of the gardener, the pet owner, the outdoor adventurer, the consumer of food, the homeowner, and the beekeeper. Now that a significant fraction of the southern United States is populated by the Africanized honey bee, we must add some scientific knowledge to our practical information.

Using materials prepared by the Carl Hayden Bee Research Center, we presented this practical training to several hundred students at Tucson High School. In this way, we could advertise our presence and give a context for discussion of bees outside of class. In the college classroom, the instructor can stick to a syllabus. In high school, the students' demand to respond to all of their concerns is much more pronounced. In particular, some students just could not study mathematical models for honey bee population and migration dynamics until their basic questions were addressed. They wanted to clear all doubts about Africanized honey bees and to know precisely how to deal with possible stinging incidents.

This training had more relevance than anticipated. One student realized during the presentation that she had encountered a swarm of Africanized bees just to the west of campus. They were safely removed later that day.

Day 3. Hands on Models of Population Dynamics. The exponential and the logistic curves have seen ubiquitous application in describing the trends in a population. Because their appearance is so widespread, the mechanisms that result in these two types of curves must have an elegant description.

On this day, we engaged in two activities using pencil, paper, styrofoam cups with lids, and black and white beads to generate exponential and logistic curves.

On this day, we also had to deal with the tragedy of a drive-by shooting. In this case, the victim had nothing to do with the altercation. Students quite naturally had priorities other than playing with beads, recording data in tables, and making graphs. Today a part of our story had tears of sadness and the connections of friendships. The storyteller's skill is needed in respecting this saga of violence and tragedy. We have seen the high school teacher use this skill far too frequently.

Day 4. A Month in the Hive. We asked a lot from the students for this day. Before our arrival, they took their knowledge of bee biology and consolidated it into a flow diagram. Most of the day was dedicated to navigating through the technological apparatus necessary to see how our model predicts the change in a hive population over a month's time.

The effort had its reward. Each pair of students chose a month in the year, a city in the United States, an initial population of brood, house bees, and foragers, and a reasonable guess on the abundance of nectar and pollen. By magic, the three equations used for the model produced a biologically meaningful result over and over again.

This was an entirely new experience for them - mathematical biology had something to it. Concepts from biology can be turned into equations and equations can be used to make predictions. These equations were our best shot at choosing the important concepts. Constructing the model forced us to clarify the relationship between concepts and choose the most important relationships. On this day, the mathematics itself was telling the story.

What was the backdrop for this day?

Presidential candidate Patrick Buchanan traveled to Tucson to participate in La Fiesta de los Vaqueros, and to articulate his position on immigration across the border just 60 miles to our south - a border that was to our north 150 years ago. Tucson High has a racially diverse population - about 40% white, 50% Latino, and 10% African American. More than 5% of the students have recently arrived from Mexico. For many students, Patrick Buchanan's remarks were quite personal and a repudiation of their culture. They wanted to walk out of school in protest. The principal noted that they were reasonably upset. However, their quarrel was not with Tucson High, but with Mr. Buchanan. If they stayed in school, then she would give them the time to discuss their grievances.

Thus, at 5 o'clock on the day before the presentation, the bee team was informed that we would be working on a revised class schedule. We had to adapt our story at a moment's notice. University professors never have to do this. High school teachers see this as a skill they have to acquire.

The National Science Foundation site visit took place on this day. They, like us, had no choice but to accomodate the revisions in schedule.

Day 5. A Year in the Hive. Mathematical models appear in our day-to-day lives to predict population changes, weather patterns including global warming and ozone depletion, the economy of a nation, or the spread of a disease.

The students emerged from this exercise with a sophisticated appreciation of the effort involved by ecologists, meteorologists, economists, or epidemiologists in designing and evaluating a model. In addition, we hoped that these students would now be able to evaluate the quality of the predictions they encounter in the popular media. Many models are not based on good science or do not use appropriate mathematical methodology. Many journalists do not have the necessary expertise to make a reliable report of scientific research.

Finding the appropriate balance between mathematics and biology is the continuing challenge in constructing a mathematical model. If the students set out with a model that includes all of the concepts in their flow charts, they will be quickly overwhelmed by computation. If their reaction has them removing too many concepts from the biology, the mathematics will be easy. However, the information will not say much about bees (populations, the weather, the economy, or an epidemic).

Because the honey bee has such a large economic importance, they have been studied extensively. Thus, when the scientists at the Carl Hayden Bee Research Laboratory set out to build a mathematical model for the dynamics of the hive, they were fortunate to have an abundance of information that allowed them to design the model and to evaluate the model.

The research team's model, BEEPOP, is comprehensive. However, such a model is prohibitively difficult for almost any high school student. Thus, the bee team faced the task of choosing a model that incorporates less biology than the researchers' model. On the other hand, the model must convey something meaningful. The resulting model, for whose name we use the Spanish diminutive, BEEPOPITA, gave useful information for a large variety of climates.

Day 6. Birth, Death, and Migration. BEEPOP is a model that works well when resources are available, and the hive does not face any circumstances that are overly stressful. These conditions cannot continue forever. At some point, the hive will become overcrowded or will be disturbed. Environmental conditions may change - the hive may be under attack by a bear, by ants, by a pesticide, by disease, or by fire. The queen may no longer be productive. In these instances, the hive must make a critical decision.

The colony of bees can divide or abscond. It can supercede an unproductive queen. If it does nothing, the hive will die.

Jim Matis turned the rates in which these critical events happen for Africanized honey bees into a model based on local environmental conditions. Robert Lanza, the geography graduate student on the "bee team", incorporated these model parameters to design a false color composite map of Arizona.

Because the colors are not what the eyes see, some experience with false color composite maps is necessary before they reveal their relevant information. We began our classroom activity by projecting a slide of a false color composite map of the central part of Tucson. After a while, students could obtain information from the map and find significant landmarks in their lives - parks, malls, schools, and their own homes.

Bees and humans differ in their views on which landmarks are important, and so the remote sensing map of Arizona requires a bit more examination before it is useful. However, both bees and humans agree that the rivers, the mountains, the deserts, and the Grand Canyon are important landmarks in Arizona. We also presented a false color composite map of the Mexican state of Sonora. Many Tucsonans have connections to Sonora and know its geography. Also, the Africanized honey bee made its arrival to Arizona via Sonora, and so we can learn about the possibilities of migration in Arizona from the information collected in Sonora.

The birth-death-migration model developed by James Matis and Thomas Kiffe used the types of models that we had previously seen using beads and styrofoam cups. We added to this model the process of migration - modeled by a random walk. For Matis and Kiffe, this is a two dimensional process. In Arizona, because the movement of the Africanized honey is along riparian areas, the process is one dimensional, and hence easier to simulate.

What are the parameters in our simulation? We can learn this from migration data along the rivers in Sonora. Similar regions in Sonora and Arizona will be revealed by the remote sensing maps. Could we obtain these data? To date, the answer is no. We know all the ingredients to make the model, but we can not predict when the Africanized honey bee will first reach Hoover Dam until we put our hands on these data.

The Rest. After the Sixth Day. As the days continued several students were building a portfolio of their work. One of them would become the SWRIMS Fellow and be given an internship at the bee lab. After the presentations, Rhonda recruited applicants for the fellowship. Tyler and Joe made a return visit to Tuscon High to meet with the applicants. We planned to help them assemble their portfolios and lend tips to prepare for their interview. Gloria would do the interviewing.

Twenty students stepped forward. Most were intimidated by the procedure and even though they were interested in the work, seemed certain that they had no chance. The United States Department of Agriculture rules stated that an intern had to be a United States citizen, at least 16 years old, with a B or better average.

What happened?

Some students were 15 years old, some were not citizens. The rest were from the at-risk class. These students had made the move beyond their time as a gang member or worked through the initial stages of parenting. Everyone had at least one year in their academic record that they would rather forget. Their average was not B, and once again, they were not able to put their past behind them.

We had not anticipated these difficulties and did not give ourselves the time to appeal to the USDA. The experience was difficult - we had an opportunity and circumstances conspired so that we could not even give this opportunity away.

BEEPOP began as a research project to be read by entomologists. We have shown that it can be delivered to a broad audience of high school students. Mark Templin, a USDA webmaster, is working to deliver it to another audience - beekeepers. Soon, we will have an interactive web site that will allow a beekeeper to use the model to simulate beekeeping strategies. Our friends from Tucson High will be making trips to the bee lab to help improve the web site. Some of these students are now 16, others are now citizens. Perhaps a few more have improved their grade point average. They will know that the bee center is not a intimidating place and we will find our SWRIMS Fellow.

A Final Note. Gloria has been working on queen development time as an explanation for the Africanization of the honey bee population. By July, 1996, the data were in, but the statistical analysis was not revealing the secrets that the researchers knew were true. In a one hour meeting, Joe and Gloria were able to work out the model that established this fact - any place that the Africanized honey bee can live, it will take over. Invasion of species and the inheritance of complex characteristics are basic questions in ecology. This question can now be informed by the example of the Africanization of the honey bee population and the mathematical model that comes with this example.

The bee team met with a purpose - to create a valuable educational experience for high schoolers. As a consequence, the team will also have a significant research achievement.

All of us have faced the embarassing moment after new acquaintances learn that they are conversing with a mathematician. They feel no connection to us - mathematicians are a distant and foreign breed. Now, when we meet someone, we have stories to tell of friendship, alliance, and solidarity. The reponse to these stories is always warm. We plan to have more stories to tell and we encourage you to find yours.

Nearly two years have passed since we began this activity. Students, beginning with the very first group at Sunnyside, still run to meet us and to tell us how much they enjoyed our visits. By telling stories of mathematics, we set out to change the way people view the world. People's views did change, beginning, most pointedly, with our own.


REFERENCES

DeGrandi-Hoffman, G., Collins, A. M., Loper, G. M., Watkins, J. C., Martin, J. H., Arias, M. C., and Sheppard, W. S., Queen Development Time as a Factor in the Africanizaion of Honey Bee Populations.

DeGrandi-Hoffman, G., Roth, S. A., Loper, G. and Erickson, E., BEEPOP: A Honeybee Population Dynamics Simulation Model, Ecological Modeling, Volume 45, 1989, pgs. 133-150.

Delgado, Richard, The Rodrigo Chronicles, Conversations about America and Race, New York University Press, New York and London, 1995.

Matis, J. H., T. R. Kiffe and G. W. Otis. Use of Birth-Death-Migration Processes for Describing the Spread of Insect Populations. Environ. Entomol. 23.

Vélez, William Yslas, The integration of research and education, Notices of the American Mathematical Society, Volume 43, Number 10, October 1996, pgs. 1142-1146.

Vélez, William Yslas, and Watkins, Joseph C. The teaching of eigenvalues and eigenvectors: A differenct approach. (submitted)

Watkins, Joseph C., SWRIMS Class Notes: The Exponential Function and the Dynamics of Populations.

Watkins, Joseph C. et. al.,SWRIMS Class Notes: BEEPOP: The Dynamics of the Honey Bee Populations in the Hive and in the Wild.


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