WORKING TITLE:
ETHNOMATHEMATICS AS A PEDAGOGICAL ACTION: THE TEACHING OF MATHEMATICS IN A DIVERSE AND GLOBALIZED SOCIETY

Submitted to: J. Wiley & Sons, September 2002

Daniel C. Orey, PhD.
California Sate University, Sacramento

Introduction
Not unlike most other texts in the field the NCTM Standards will form the basis for the content and the instructional work that will be presented by this book, but there is will be an overall difference; the emphasis will be on the learning to teach mathematics in highly diverse contexts.  In so doing, three overall themes related to diversity will be used throughout: ethnomathematics, access and equity and the mathematics to technology.

Ethnomathematics seeks to identify the techniques, abilities, and practices as used by members of distinct cultural groups as they have come to explain, know and understand their environment.  Ubiratan D’Ambrosio, in referring to the cultural context in which ethnomathematics emerges, defined cultural groups as something quite broad, including such considerations as language, jargon, codes of behavior, myths, symbols, work, political and religious values.  Alternative forms of mathematics have come about as people work to explain and resolve practical problems in their daily lives.  The diverse ways people have developed to quantify, compare, classify, measure, and explain day-to-day phenomena are of equal and important value in ethnomathematics.

Ethnomathematics views modern academic mathematics we teach and learn in school as an integral part of the ongoing developmental human legacy of explaining, knowing and understanding life in an interdependent and globalized world community. 
Many researchers agree that an individual’s facility and ease with basic mathematical algorithms contributes to success or failure in mathematics.  However, highly individualized and unique interactions between one’s culture and the culture of others; language; and the algorithms we use, combine to construct individual abilities or even disabilities in mathematics.  Math is a language.  Language is a function of both thought and culture.  The ability to use and communicate using this language (mathematics) is important.  Mathematics has its own culture as well – it possesses a language and set of norms.  It is the ability to use and communicate mathematics well that has become paramount to success in our increasingly technological and global society. 

Accomplishment in mathematics allows further access to a greater variety of life choices related to careers and educational opportunities; issues equally important to education, gender equity, and diversity.  This is directly connected to the disempowerment of underrepresented minorities and women in our culture by lack of access to good mathematics teaching and learning experiences.  Ethnomathematics collects and disseminates important aspects of mathematics used by distinct cultural groups that may not have had a voice or access to mainstream mathematics.  Brazilian colleagues (representing more than 80% of the research and applications) have done work with:

•    The mathematical modeling of distinct themes: beverages, coffee, esoterics, transportation, and tourism to name a few;

•    Work with the documentation of the mathematics used by distinct and divers indigenous groups - most notably the work being done with the native peoples of the Amazon and in the state of São Paulo;

•    The mathematics used to empower landless peasants, homeless people, and street children is also equally impressive; and finally

•    Applications of ethnomathematics as directly implemented in many schools as outlined in the Brazilian national mathematics curriculum standards.

We want to train all learners in the primary grades to be able to do good mathematics, similar to that outlined above when they reach secondary school.  What questions that this text will discuss are following:

•    What is the role culture plays in coming to understand what and how we learn (mathematics)?

•    What is the role culture plays in how we teach mathematics when expectations understandings by parents and different communities vary by background, social class, and culture;

•    What is the roll that culture plays in how we communicate mathematically with others which is strangely connected to;

•    What is the role of monolingualism, bilingualism and multilingualism in the mathematics classroom?

Pre-service teachers will participate in assignments that are related to interviewing newly arrived immigrants about the mathematics they learned, and the development of their own professional development plan for future filed placements. This will become an important outcome of interacting with this text – the reader will be assisted in developing their own plan for further professional development. 

There are numerous textbooks on the market that can be used to teach mathematics methods for pre-service elementary teachers.  The ones used and consulted by this author are: Fuys & Tischler, Kennedy, Van De Walle, and M. Burns.  However they seem to be primarily aimed at a middle-class teacher reality that does not always match a complex, multicultural, urban, multilingual, international and highly diverse contexts found in many urban regions of the world.  These urban regions are not necessarily the large urban centers of São Paulo, London, or the San Francisco Bay Area.  The diversity is found in Denver, Portland, Albuquerque or Sacramento as well as even smaller communities nationwide.  For example Sacramento was recently named one of the best integrated cities in the nation by Time Magazine.  As the capital region of California, it is an urban region of over 2 million people, with as many as 180 languages spoken here.  It is not uncharacteristic for a public school in this area to have 35 or so languages, and classrooms with individual kids who can use between one and three languages or children not really literate in their home language and now needing to learn English.  This same classroom has numerous races and social classes and abilities as well.

The emphasis of this book is on the use of this classroom diversity as a resource, thus reflecting one overriding bias to this book which is that this diversity is a resource and not a problem.  This book will include numerous vignettes - personal stories - from students and teachers that overcame difficulties, and made their own connections between culture and mathematics. These vignettes will range in scope from solving problems to demonstrating aspects such as the alternative algorithm.

Mathematical challenges that "at-risk" populations experience begin with the early, formative experiences in the elementary school, which can be either intentionally or unintentionally racist, sexist or disempowering to many learners.  The locus of control is often found to be outside of both the child and the teacher, with the fault often unjustly, falling upon the teacher.  Yet when a child fails a computer game they do not blame the computer, they normally go back and relearn and then return to the game, why should this be any different in school?  What is it about learning mathematics that can’t be like learning to beat a computer game?  Another essential aspect of this book is to support teachers in constructing learning environments that allow all kids (see NCTM’s Every Child Statement) to take control of and own their own mathematical knowledge.  This is the central point to true empowerment as demonstrated by Burns and Parker.  We are faced with many negative images related to mathematics.  When both teachers and kids are not given the tools to defend themselves from it they can be lost to the community, and nation at large.  It is a central aspect of this book to assist the teacher to be able to build a learning environment where all learners will find success and learn to use mathematics as a tool for self improvement and growth.

In the writer’s community, their are numerous native-born 15 year olds who, despite the best efforts of both elementary and middle school cannot calculate, read or write at grade level, who have low self-esteem. They are placed in the same classroom as newly arrived immigrants who know a great deal about the world and might speak and read in 2 or 3 languages.  Both groups of students have experienced extremely difficult, indeed life threatening experiences in getting to that particular classroom, yet their academic abilities are widely differentiated. 

The author’s student teachers want to know how these students (demonstrating what I refer to as “spectacular forms of ignorance”) arrived at this place, and with their own particular mathematical context.  They want to reduce this form of ignorance and disability to an absolute minimum, and want to know how to be able to do so.  Often, current texts do not offer this perspective, putting all learners of a particular age into a particular ability or expectation level.  Traditional texts fail to answer this need.  Along with numerous realistic examples and useful ideas, student teachers need equally realistic and honest examples of what works in difficult contexts.  They crave concrete examples of what works in the context of their students and by using resources and materials that exist in their community.  They do not want to see perfect kids, behaving well, in schools that are fully appointed.  They want to know what to do when they get their first job in a tough school.  This book seeks to provide the resources for them to do this successfully.  The curriculum we currently teach, the methods teachers currently learn, the textbooks used in teacher training, do not adequately prepare teachers for teaching mathematics to kids that may not fit the “norm”.
Other questions asked by this project, are related to individual teacher understanding of the mathematics.  As well, how will the teacher know if they come from vastly different social backgrounds and experiences than their own?  My students are trained to work in classrooms that have members of various interests and abilities, disabilities, social classes, sexual orientations, religious beliefs, and political persuasions.  This book will be honest, open and inclusive.

Using primarily a Constructivist–Freireian-Gardener based perspective, this text will model techniques found and modeled by AIMS, The Exploratorium, EQUALS/Family Math, Kay Toliver, Marilyn Burns, Arthur Benjamin, Robert Moses, the Algorithm Collection Project, and the Mathematical Modeling Project.  The activities demonstrated will be of the type that emphasizes sheltered English techniques, cooperative learning, hands-on manipulatives, and a diversity of forms of communication and assessment.   All these will be used to develop a starting point for mathematics instruction; this is the essence of ethnomathematics as pedagogical action.

The other reality of the world in which pre-service teachers are entering is one that needs a definition of what it means to be literate.  Traditional literacy models define and focus on arithmetic, reading and writing.  Yet in actuality, 90% or more of the population no longer uses, or limits their own access to print media to gain information.  These same individuals rely on numerous other techniques to solve problems – calculators, rounding, waiting, asking others, and questioning.  What are the consequences of a population deeply influenced by visual forms of communication - computer images, billboards, film, video etc and machine / computerized data but cannot understand how to make sense of this visual feast?  Where do people in our community find and use mathematics? What kind of mathematics is needed to be successful, or to obtain certain kinds of jobs in this technology-based reality? The NCTM Standards speak specifically to this as well as will this text.

As mentioned earlier, traditional literacy models and definitions (The 3 R's) are no longer enough in this new world that is technology-based and diverse.  A case in point:  what are steps that we need to teach kids to make sense of money in a cashless economy – where credit cards, ATM’s and the internet have entered the marketplace?  How we teach such abstract ideas when concrete symbols no longer exist or are less and less valid? 

A methods course and text must assist new teachers in dealing with multiple assessments, a variety of learning styles, needs, abilities and perspectives as supported by Fuys & Tischler and Howard Gardner.  Over the past 16 years I have had far fewer new teachers who say that they do not like math, or bring with them issues related to math avoidance / phobia.  Now new teachers want to know how to make these connections. They want to know how to assess kids with instruments that do not seem to fit the reality or understanding of the children and community in which they want to work.  The population at large is bombarded by visual images and reads less (especially for fun), yet the traditional curriculum ignores film, video, computers, commercials, advertising, and other important aspects of modern global culture. All these represent aspects that can and should be brought into the mathematics classroom, and the methods courses, not for marketing purposes but for purposes of making critical sense of this media context, mathematically speaking. 

It is goal of this book to discuss ideas towards making mathematics, and to some extent literacy in general, more realistic, honest and useful (powerful) to the new generation of teachers and learners.  Empowered users of mathematics are better able to both ask and answer questions that are connected to the reality found around us. 

Paulo Freire taught a method of literacy that enabled millions of illiterate adults to read and communicate by using day to day objects and common ideas.  Teachers and students can actively and critically examine the mathematics related to issues of diversity, poverty, access and equity, AIDS, war, the environment.  It is an active model for literacy, and one that I seek to develop and share within the context of state and national mathematics standards.  The best way to do this is to train teachers to use mathematics that is effective both for their lives, and for the lives of their students.

An ethnomathematics perspective encourages teachers to find the mathematics around them, mathematics that finds expression in their daily lives, a mathematics that is both real for kids and community.  Models used in Brazil link standard classroom mathematics goals to those as outlined by the community.  Thus linking reality, usefulness, service and responsibility to what we teach and learn.  American teachers need to learn about, adapt and use tools and ideas that have proven effective not only in the United States but abroad.  Because of this diverse reality, and in order to understand our students better, research in mathematics education must look further than what is being done in North America, the textbooks that are used for in-servicing and training future teachers must model this as well.  This is a lesson from TIMMS and Liping Ma’s work.  We live in a world where most people in the world know much more about the United States and how we think than we do about the rest of the world.  Events related to 9/11 have shown us exactly how dangerous this may be for us.


About the Market

Primary Course

The primary course for which this book is intended is EDTE 304: Introduction to Mathematics in the Elementary School, it is a 2 credit course taken for 1 semester with approximately 9 sections of 35 pre-service teachers each.  It is a required course for teacher certification in California.  The Purpose of the Course  is:

1.    To introduce and to build the student’s confidence in developing a mathematical teaching/learning environment using a variety of teaching strategies;

2.    To introduce the student to the urgency of issues related to access, equity and multicultural issues in mathematics education in California;

3.    To help the student to develop a personal philosophy of mathematics education based on current research and findings;

4.    To provide the student with time to work to develop, share and collect ideas and resources; and

5.    To assist the student in acquiring a set of activities and resources to work with and practice in future field and professional placements.

Students

This course is a credential requirement and is part of the fifth year component to teacher credentialing.  The success rate for this course is higher than other courses at CSUS, due to the nature of our program (B+ or above average is needed, the CBEST, a writing proficiency exam, and passage of basic skills exams in English and Mathematics). 

Instructors

The course is usually taught by 3 full-time faculty members at CSUS.  Who take a number of approaches to this course depending upon the program and phase (we have a 2 semester / and a 3 semester (phase) option.  As well sections are cohorted and taught in 7 different school districts / locations. Depending on a particular school district’s expectations, the course and pre-service teachers may have certain kinds of goals.  Overall, due to the limited length of time we have with our students, and the enormous amount of material to be covered, the course is in reality can only serve as an introduction to mathematics education. 
The course provides an introduction to mathematics education, and is designed to give pre-service educators access to various forms of instruction, problems in the community, and access to hands-on experiences and activities that will be useful to them as they make their way into the profession.  The main criteria other instructors use to select a text for this course are based upon numerous factors, most notably the ability to provide an introduction, mastery is impossible due to the short length of the course. 

Competition

I have most recently used both Van de Walle and Fuys & Tischler texts.  Though in many aspects it is dated, I prefer Fuys as do most of my pre-service teachers because they keep this particular book, and use it as a reference, almost a “cook book” as it were.  I envision the text being proposed here as being very similar to that of Fuys – in kind and style.  I want pre-service teachers to keep the book, to use it when they are stuck or need review, go to it as a reference.  Fuys succeeds in that, as it has a great deal of valuable ideas and activities that can be tried and “played with” in field placements and classrooms.  It fails in that technology is absent; it doesn’t talk to NCTM Standards and has very little if any talk about culture, though “multiple embodiments” speaks to different learning styles.  It is also expensive, going for about $90.00. 

Secondary Markets

I hope that this book would be the type of book that people purchase for professional development in cross-cultural multilingual mathematics education environment.  I in vision this book as imbued with the flavor of Fuys,  EQUALS/Family Math, Marilyn Burns’ and Zaslavsky’s work for professionals.  As well it will interest those in ethnomathematics, multicultural education, and mathematics education who want to gain continued knowledge related to mathematics in cross cultural environments.  Significant international, professional, direct mail, or trade markets exist in multicultural education, bilingual education and ethnomathematics.

Coverage

This book would be designed to empower pre-service teachers to continue this process by providing a number of activities and resources (modeled after the Burn’s book and Fuys & Tischler) that can be turned to when they plan future lessons and may need a resource to go to for more information.  This aspect could easily become an on-line resource / CD-ROM.  Included in this outline are the following topics.  Their functions as a future Table of Contents:

• Mathematical Power
• Ethnomathematics / Multiculturalism
• Number & Its Uses
• Informal Geometry
• Developing Operations
• Whole Numbers
• Basic Facts / Alternative Algorithms
• Measurement
• Probability and Statistics
• Algebra

• Objects to Think With (Manipulatives)

Attribute Materials
Lids and other manipulatives found in the environment of the learner)
Cuisinare Rods
Geoboards
Color Tiles

• The Vocabulary Wall

And other Sheltered English Techniques in the math classroom
The Algorithm Interview

• Unit Construction   

Evaluation & Assessment

• Further professional development

    Developing your own plan

Approach

Marcia Ascher recently wrote,

For all of us, on whatever level of learning, knowledge of the ideas of others can enlarge our view of what is mathematical and, in particular, add a more humanistic and global perspective to the history of mathematics.  This enlarged view, in which mathematical ideas are seen to play a vital role in diverse human endeavors, provides us with a richer and fuller picture of mathematics and its past…  Twenty-first century mainstream mathematics is reaching people of more and more diverse cultures as the teaching of it continues to spread across national and continental boundaries as people move from one country or region to another, and as several cultures are represented in the backgrounds of more individuals. An enlarged view of the past can help in furthering the realization that people of different cultural traditions will enrich mathematics itself by bringing to it different perspectives and different ways of perceiving and categorizing the world.

This book is to be professional in tone but not heavy, it will be accessible and useable and real in both tone and function.  After reading a section or passage, I want people to say, “this makes perfect sense, I want to try this".  The treatment here will be broad yet comprehensive.  The reader will be introduced to the universe of ethnomathematics, but will be given a number of specific areas by which to make it work in the multicultural classroom.  The emphasis will be, as a Persian mystic once stated, "to walk the mystical way with practical feet".  That means the book will introduce some theory, to enable the reader to develop their own philosophy, and a usable plan that enables the pre-service teacher to practice practical aspects, connected to a sound and consistent pedagogy and philosophy.  M. Burns has recently said,
We've learned that professional development for mathematics teaching must address three essential issues -- mathematics content, how children learn, and effective instructional strategies.

Outlined in this text is a design that seeks to assist the reader in developing their own ongoing professional plan along the guidelines of M. Burns’ three areas:

•    Mathematics Content:      What content do I need to master? 
•    How Children Learn:      What developmental strategies do I need to master?
•    Effective Instructional Strategies:  What pedagogies do I need to master?

Specifications

Hardcover approximately 600 pages including references, bibliography and attachments / resource pages (these might be put on a CD-ROM).  Illustrations will range from concrete examples of the materials being discussed, to photos of people being interviewed and classrooms.


Schedule

The manuscript (completed in first draft form) could be available in about one and on half years from contract signing.


The Author

Daniel Clark Orey is professor of Multicultural and Mathematics Education at California State University, Sacramento (CSUS), where he has resided since 1987. Dr. Orey is the former Director of Professional Development and the Center for Teaching and Learning at CSUS. He earned his doctorate in Curriculum and Instruction in Multicultural Education from the University of New Mexico in 1988. His Mellon-Tinker funded field research took him to Highland Maya Guatemala and to Puebla, Mexico. He is a founding board member, and served as Vice President for North America (1996 - 1999) and General Secretary (1995) of the Sociedade Internacional para Estudas da Criança. In 1998, he was a J. William Fulbright Scholar to the Pontifícia Universidade Católica de Campinas, Brazil where he worked with the mathematical modeling / ethnomathematics program with Ubiratan D’Ambrosio and Geraldo Pompeu, Jr. Prof. Orey continues to travel to Brazil as a lecturer and teacher every year.

Vita available at:    http://www.csus.edu/indiv/o/oreyd/res.html
 
Bibliography

Ascher, M. (2002). Mathematics Elsewhere: An exploration of Ideas Across Cultures. Princeton University Press.

_________.  (1998). Ethnomathematics: A multicultural view of mathematic ideas. Chapman & Hall: New York.

D’Ambrosio, U. (2001). Etnomatemática: Entre as tradições e a modernidade. Autentica: Belo Horizonte, Brasil.

_____________.  (2001, February).  “What is ethnomathematics, and how can it help children in schools?”  In: Teaching Children Mathematics, Reston, VA: National Council of Teachers of Mathematics.

_____________.  (1998). Ethnomathematics: the art or technique of explaining and knowing.  International Study Group on Ethnomathematics: Las Cruces, NM.

Fuys, D. & Tischler, R. (1979). Teaching Mathematics in the Elementary School (2nd Edition). Boston: Little, Brown & Co.

Langbort, C. (1988, November). “Jar Lids - An unusual math manipulative”. The Arithmetic Teacher. (Vol36) no 3 pp.22-25.

Ma, L. (1999).  Knowing and teaching elementary mathematics. Mahwah, NJ: Erlbaum.

Moses, R. P. (2001).  Radical Equations: Math literacy and civil rights. Boston: Beacon Press.

Orey, D.  (2000). Chapter.  "Geometry of the Tipi and Cone: Using Mathematical Modeling as Applied Ethnomathematics” in Mathematics Across Culture: The History of Non - Western Mathematics.  (Selin, H. Ed.).  Dordrecht, Netherlands: Kulwer Academic Publishers.

_____________.  (2000, December).  "An Ethnomathematics Postcard: July - August 2000.”  International Study Group on Ethnomathematics Newsletter.  New York: ISGEm, 15 (2).

_____________.  (1998). “Mathematics for the 21st Century.” In: Teaching Children Mathematics, Reston, VA: National Council of Teachers of Mathematics.

(2002). Fasciculo Didático (text book) with Milton Rosa.  Modelação Algébrica. Modelação Algébrica. São Paulo: Escolas Associadas Pueri Domus.

(2001). Book with Milton Rosa. Ethnomathematics as Pedagogical Action: Introducing Mathematical Modeling to Adolescent and Adult Learners.  (Unpublished manuscript available from Authors).

Parker, R. E. (1993). Mathematical power: Lessons from a Classroom. Portsmouth, NH: Heinemann

Powell, A. B. and Frankenstein, M. Eds.  (1997). Ethnomathematics: challenging eurocentrism in mathematics education. Albany, NY: State University of New York.

Rosa, M. (1998).  Matemática: Seqüências e Progressões (Mathematics Sequences and Progressions). São Paulo: Editora Érica.

Shoenfeld, A. H.  (2002). “Making mathematics work for all children: Issues of standards, testing, and equity”.  Educational Researcher, Vol. 31. No. 1, pp.13-25.

Van de Walle, J. (1997). Elementary and Middle School Mathematics (3rd Edition). New York: Longman.

Zaslavsky, C.  (2001, February). “Developing number sense: What can other cultures tell us?”  In: Teaching Children Mathematics, Reston, VA: National Council of Teachers of Mathematics.  7 (6).



A.    Making Subject Matter Comprehensible To Students

STANDARD 8A (a) Multiple Subject Mathematics

1.    During interrelated activities in program coursework and fieldwork, MS candidates learn about the interrelated components of a balanced program of mathematics instruction: computational and procedural skills; conceptual understanding of the logic and structure of mathematics; and problem-solving skills in mathematics.

2. They learn to:

(1) recognize and teach logical connections across major concepts and principles of the state-adopted academic content standards for students in mathematics (K - 8),

(2) enable K - 8 students to apply learned skills to novel and increasingly complex problems;

(3) model and teach students to solve problems using multiple strategies;

(4) anticipate, recognize and clarify mathematical misunderstandings that are common among K - 8 students;

(5) design appropriate assignments to develop student understanding, including appropriate problems and practice; and

(6) Interrelate ideas and information within and across mathematics and other subject areas

TPE 1A:  Subject-Specific Pedagogical Skills for MS Teaching Assignments:  Mathematics

1.    Candidates for a Multiple Subject Teaching Credential demonstrate the ability to teach the state-adopted academic content standards for students in mathematics (K-8).

2.    They enable students to understand basic mathematical computations, concepts, and symbols, to use these tools and processes to solve common problems, and apply them to novel problems.

3.    They help students understand different mathematical topics and make connections among them.

4.    Candidates help students solve real-world problems using mathematical reasoning and concrete, verbal, symbolic, and graphic representations. They provide a secure environment for taking intellectual risks and approaching problems in multiple ways.

5.    Candidates model and encourage students to use multiple ways of approaching mathematical problems, and they encourage discussion of different solution strategies.

6.    They foster positive attitudes toward mathematics, and encourage student curiosity, flexibility, and persistence in solving mathematical problems.