Introduction

Ubiratan has asked me to share my thoughts about the state of ethnomathematics in Brazil. From July to December 1998, I had the privilege of working with colleagues at the Instituto de Ciências Exatas at the Pontifícia Universidade Católica de Campinas, in Brazil. During this time I was able to work with teachers enrolled in the Mathematics Specialization Program (PUCC - SME) and also met frequently with the ethnomathematics research group at the Universidade Estadual De Campinas (UNICAMP) and the Universidade of São Paulo (USP). PUCC-SME trains cohorts of mathematics teachers to use an ethnomathematical approach (using mathematical modeling see appendix I) for research and development of mathematical activities for students. Students in the program learn to examine and then develop various aspects of mathematics found outside of school and to analyze it for content and usefulness for use in teaching and learning environments. The primary reason for my request to come to Brazil as a Fulbright Scholar was to work closely with Geraldo Pompeu Júnior and Ubiratan D’Ambrosio. I wanted to learn how to develop an ethnomathematics-based course for my students and community at California State University, Sacramento. I also wanted to take advantage of the resources and data archived in São Paulo / Campinas, Brazil related to ethnomathematics.
 
 

PUCC

The Instituto de Ciências Exatas, Pontifícia Universidade Católica de Campinas (PUCC) offers a year-long program entitled "Specialization in Mathematics Education" (SME) for teachers working in schools corresponding to the primary, middle, and high school levels in the United States. The main objective of the course is for students to learn to analyze the implications that come from the utilization of methodologies that teach mathematics. Students learn how ethnomathematics / mathematics modeling can be used in the development of new teaching and learning methods and resources. SME participants were interviewed and enrolled in early February of 1998, with the program continuing until February of 1999. Classes were held Friday afternoons from 1 to 6pm and Saturday from 8:00 AM to 1:00 PM, with small groups of students often working on research and projects into the late afternoon on Saturday.
 
 

Mathematical Modeling As A Methodology

Mathematical modeling seeks to develop a learning environment that allows for the learning and exploration of mathematics in an interdisciplinary context. The development of this form of understanding and problem solving gives mathematics a larger area to explore for learners, researchers and teachers. Mathematical modeling creates an environment that senses the applicability and practicality in concrete situations, and that will resolve day to day problems that are diagnosed and elaborated during the process of doing research.
 
 

Research in Public Schools

As an outgrowth from conversations with students in the PUCC-SME, I was invited to visit a number of public school classrooms. One of the teachers, Milton Rosa in the PUCC-SME told me that he taught over 55 hrs a week. In questioning him about this, he extended an invitation for me to see how this was done. Every Friday evening and Monday I have spent in Amparo working with the teachers and students. I was also invited to participate in a panel for monographs" the graduate students presented.

In observing Professor Rosa and his students’ interactions, I used ethnographic description (approx. 100 hours of observation), and interviews of students and teachers. Every five minutes I randomly selected a student and asked them about the lesson in progress. In Prof. Rosa’s classroom, there are no books, no homework, no overhead projectors or any of the so called "conveniences" that many of us in the United States have come to believe that we cannot live without. Instead the technology that Prof. Rosa had access to was a blackboard and colored chalk. His students have notebooks that they keep notes in and may refer to at anytime.

His students clocked, despite the late hours of the school (most of them worked during the day) an astoundingly accurate rate of success, nearly 99.9% on task, and with correct answers. Students in the United States average about 50%of the time. My data here conflicts greatly with recent state and national test results as published here in Brazil. After a conversation with Ubiratan D’Ambrosio, I have come to believe that it is the kinds of questions we ask that influence the performance of our students. Together Milton and I translated a series of four questions, one from the soon to be published book for middle and high school from the Family Math / Equals project at the Lawrence Hall of Science (LHS) (see appendix II). We used these questions to gather quantitative data related to performance = tests we graded using a 6-point scale adapted from the LHS (Appendix III).

What did this tell me? The ethnographic data suggests that the kinds of questions I ask learners are as important as the task we give. In my questioning with the students in Amparo, I always asked them to explain to me what the instructor was talking about, this allowed them to explain in their own words, their understanding. It also says, much to my chagrin, that too much technology can be just as detrimental to learning as not enough, certainly, how we use the technology we have access to, is an important question as well when constructing learning environments. It also reinforced my growing concern about how or even should we be using textbooks.

Prof. Rosa and his colleagues use a form of instruction that is in-tune with their students. Rosa takes great pride in teaching one idea per class period - quality vs. quantity. In Amparo, as in many smaller cities in the world, the school is a meeting place a "ponto de encontro" for many kids. In some middle-sized cities, the school takes on a social importance for the community that has been lost in large urban areas with many, many distractions and alternatives.
 
 

Trips

Part of the extraordinary opportunity Fulbright gives to its scholars is the opportunity and support to travel and meet with people within the country of their stay. Brazil being larger than the continental United States has many varied regions and cultures. So it was important that I travel as much as possible to learn as much as I could in the short time I had there. So I invite you to take-out your atlas, and follow me around Brazil, as I briefly describe where I went.

I began my time in Curitiba, the state capital of the state of Paraná, where I took an intensive Portuguese class and met with students in the mathematics specialization program at the Federal University of Paraná. Curitiba is well known for its innovative social programs, urban planning and over-all quality of life. While in Curitiba, I traveled to attend the Encontro Nacional de Educação Matemática (ENEM), where I was able to meet with many mathematics educators and learn about current concerns and ideas related to mathematics education here in Brazil. After three weeks, I moved on to my home base in Campinas, in the state of São Paulo, where the first order of business was a regional conference for mathematics educators, where I gave a closing keynote address, "A Reforma Curricular na Califórnia para o Próximo Século". I also met for the first time with the SME class where I enjoyed sitting in on a series of classes that Ubiratan D’Ambrosio gave to the SME students at PUCC.

With an invitation from colleagues in Vitória, I had an opportunity to visit and speak with the mathematics educators at the Universidade Federal do Espírito Santo in Vitória. Here, I gave a workshop and a couple of talks. Vitória is a great city, with superb beaches and fine food, it was my first real dose of the Brazilian jeito — that work and play can co-exist.

After Vitória I was able to spend some quality time in at home in Campinas, and with my host and colleague Geraldo. I also began my research project in Amparo. It was through the SME students that I began to receive invitations to visit schools and work with teachers in various locals. I traveled to the University of Sorocaba and the Faculdade de Ciências e Letras "Plínio Augusto do Amaral" in Amparo were I participated in a series of "cultural weeks" as a visiting scholar and speaker.

Then it was off again to the state of Rio Grande do Sul, where I participated by giving a "mini-curso" called "Jeito: Fazendo Etnomatemática" for the II Seminario International de Educação no MERCOSUL in Cruz Alta, RS. The next week took me to Parati and Barra Mansa in the state of Rio de Janeiro, where I took my research colleague (Milton Rosa) from Amparo with me to present our preliminary results and his perspective re: mathematical modeling.

My final trip culminated in participation as the opening keynote speaker for the "Semana Cultural" at the Universidade Federal de Mato Grosso in Barra do Garças (the geographic center of South America). It was the perfect way to end my stay here in Brazil, as I found that the end of the world, contrary to popular belief, is indeed paradise. It is full of the kindest most wonderful people one could imagine. The math group picked me up each morning and we went to a hot water resort, we soaked in the sun and water as we discussed mathematics education (really!). So it was that my final trip confirmed my thesis that Brazilians know far better than anyone else how to work.
 
 

Future Plans

Once you have been to Brazil, it is impossible to forget about it. I have been visiting regularly here for over 4 years, and life for me has become divided in half — time when you are in Brazil, and time when you are planning for your return to Brazil. Thus it is with me. At this point in time, I hope to return to Brazil next year to assist in putting together a long-distance education proposal for the University of Cruz Alta (UNICRUZ). As well, I hope to give a couple of a minicourses at the University Center of Barra Mansa and the University of Mato Grosso in Barra do Garças.

The State of Ethnomathematics in Brazil

Ubiratan has asked me to write about my thoughts about the state of ethnomathematics in Brazil. In so doing, I gave a quick tour and report of my time in Brazil. Without a doubt, Brazilians know the names of Freire, D’Ambrosio and ethnomathematics. Because of the unique social conditions that exist in Brazil, there is probably more freedom and access to information from a wide variety of sources than in most places in the world. There is more experimentation going on than probably anywhere in the world as well. The political and economic changes in Brasil have made for a dynamic, creative, active and an optimistically oriented culture of inquiry. I have been deeply moved by the desire of teachers and students to learn, and apply new methods of instruction — all of this despite some very rugged teaching and learning realities.

The quality of work being produced by the SME students is superlative. The projects and research being produced by Geraldo Pompeu Junior and his students at PUCC and those of Maria do Carmen Mendonça at USP is of the highest possible diversity and quality. The discoveries and data being produced are very exciting, I dearly hope that the students will begin to publish their findings. This all being done with some very disheartening prejudice against some the ideas and philosophy of ethnomathematics.

I wish to thank a number of people, though the list should go into the thousands… Ubi, Geraldo, Maria do Carmen, Pedro Paulo and Milton —you all gave so much of your hearts, souls, time and energy to make my stay in Brazil so successful. Your efforts and friendship have changed my life forever. Your patience and humor with this ignorant gringo will always be appreciated…saudades!