THE ALGORITHM COLLECTION PROJECT (ACP):
AN EXPLORATION OF THE
ETHNOMATHEMATICS OF BASIC NUMBER SENSE ACQUISITION ACROSS CULTURES
CERME 3: Third Conference of the
European Society for Research in Mathematics Education
28 February - 3 March
2003 in Bellaria, Italy
Clark Orey, Ph.D.
California State University, Sacramento
This discussion focuses on the interactions this
writer studies that occur between culture, the language(s) spoken, and the
particular algorithms used by recently arrived high school immigrant students
attending an inner-city high school in Northern California. This activity
allows students to develop an understanding between the relationships of an
individual’s language and their algorithms. This work is based on a Brazilian
research paradigm employing strategies of ethnomathematics and mathematical
Esta discussão focaliza as
interações, que o escritor estuda, e que ocorrem entre cultura, língua falada, e
a peculiaridade dos algoritmos que são utilizados por alunos imigrantes,
recém-chegados aos Estados Unidos, e que freqüentam uma escola secundária no
norte da Califórnia. Esta atividade permite aos alunos desenvolverem a
capacidade de compreender o relacionamento entre a língua falada e os algorítmos
por eles utilizados. Este trabalho tem como base um paradigma brasileiro de
pesquisa que utiliza etnomatemática e a modelagem matemática como estratégias de
A large number of students experience mathematics
negatively. Many of these same students have difficulty in performing basic
arithmetic operations that may ultimately exclude them from participation in
society. Comparable populations across the world have fewer problems with
mathematics than do students in the United States (TIMSS, 1997). Facility with
the algorithm one uses, including its unique history and cultural significance,
certainly contributes to one’s success or failure in mathematics.
A fundamental aspect of this project respects the
importance of the interaction between languages spoken and algorithms used that
combine to form individual abilities or disabilities in mathematics. Success
during early attainment of arithmetic operations forms the basis for how
successfully one learns advanced mathematics. Work with mathematics learners in
other countries (Orey, 2000, 1999, 1998) suggests that true reform should not
exclude an ethnomathematical perspective. An in-depth study of the type of
algorithms used by many immigrants to California has important international
significance as well, and represents a clear case for studying “what works”
across cultures in relation to number sense acquisition.
This writer studies these interactions from an
ethnomathematics perspective by using techniques as learned as a Fulbright
Scholar in Ethnomathematics at the Pontifícia Universidade
Católica de Campinas in Brazil. The design of this project encompasses a
Collection: The ACP uses a Brazilian interview protocol (Orey &
Rosa, 2002) that is used to conduct interviews with recently arrived immigrants
in relation to their mathematics. Cohorts of graduate and student teachers were
trained to interview recently arrived immigrant students in relation to their
attainment and use of the four basic arithmetic operations. As part of course
assignments, students make interviews in relation to how immigrants learned and
use addition, subtraction, multiplication, and division algorithms in their
country of origin. The data taken from these interviews have assisted in the
development of a vocabulary collection of over 21 languages currently spoken in
the researcher's immediate community.
Analysis: Each semester, the principle investigator trains groups of preservice
teachers to contact and interview immigrants to our region. The samples are
analyzed for diverse protocols, and error patterns distinctive to each
algorithm. During the pilot study, 16 groups were represented; it has grown to
over 35 in the Sacramento region, some of which are shared on the ACP Website.
The initial pilot study uncovered four patterns for long division being used by
the high school students (see appendix).
Dissemination: ACP study data has been used to construct a new
course objective for mathematics methods students at California State
University, Sacramento. Students are asked
to interview a newly arrived immigrant along the same lines as outlined above,
and descriptions written by the students are posted on the ACP website. Data
gleaned from this work has been used to develop curriculum strategies for
teacher educators in both California and Brazil through numerous seminars and
workshops given by this writer. As well, the findings have been shared in the
United States, Costa Rica and in Brazil, as well as Capital Public Radio, the
California Mathematics Council Asilomar Meeting, NCTM, and CERME3 and though The
International Study Group on Ethnomathematics network. The ACP website has
generated a great deal of interest from researchers in Europe, North and South
America. Recently the Algorithm Collection Project was awarded a Pedagogy
Enhancement Award from the Center for Teaching and Learning at California State
University, Sacramento to further develop the data and upgrade the website.
Research Question and Significance
Do algorithms we use have cognitive, as well as
pedagogical significance for their users? What is the consequence for language
attainment (multi-bilingualism) and a specific algorithm? Recent research
suggests that one’s mother tongue affects one’s personal form of cognitive
processing (Holtz, 2001; Devlin, 2000). It appears that some language groups
have significantly more incidences of dyslexia, most notably native monolingual
speakers of American English. This interaction of language and written
language, suggests that monolingual children in California possess equal
difficulty with the standard algorithm as taught here, and may be influenced by
factors related to the unique linguistic interactions between American Standard
English and the algorithms taught in the United States. Educators in many
countries are surprised to find that common day-to-day algorithms differ by
culture and by national origin.
For example, there are at least five major
patterns used for long division, four of which are used by immigrants to this
part of the world. For purposes of this study, they are called: North American;
Franco-Brazilian; Indo-Pakistani; and Russo-Soviet. A fifth, represented by a
colleagues Norway, has not been found in the samples gathered from this part of
the world as of this writing.
Context and Background to the Study
Early colonization enabled, for better or worse,
the exchange of alternative ways of thinking and learning between diverse groups
in North and South America. The ongoing processes of immigration and
urbanization have brought together members of diverse cultures; Northern
California is part of this phenomenon, where people of all cultures and
languages interact with some frequency. Sacramento, the state capital (metro
area population of 2 million) has over 80 languages and cultures represented.
The Sacramento region is one of the fastest growing metropolitan regions the
United States, suggesting as the population and urban area expands, schools here
will have to adapt to the increased migration and diversity.
The pilot study uncovered relationships between
mono, bi and multilingualism; the algorithms used; and student ability and
confidence. For example, some language groups use the comma (,) for the
decimal, which can cause for some confusion for scientists, business people,
students and educators. Mistakes, often deadly, in translation between standard
and metric measurements are legendary in this country. Thus, a strong case can
be made for studying "what works" across cultures in relation to number sense
acquisition and language. A surprise to many educators is the successful
methods for learning, memorizing, calculating and communicating answers often
differ across cultures.
Much of what has been studied in both
ethnomathematical and multicultural contexts has often been related to the
ancient ways of doing algorithms. For example, it is not uncommon for textbooks
in many countries to introduce Roman and Babylonian number systems,
medieval-Russian peasant addition, Napier’s bones and other such activities as
historical curiosities. It is also quite common to study certain aspects of
A Brazilian graduate student, Milton Rosa, serving
as a visiting foreign exchange teacher in mathematics, was having difficulty in
using and explaining the standard North American algorithm to his students as
prescribed by the curriculum, and realized that his method also differed from
that of a number of his students (see examples from Brazil and Kazakhstan). He
asked his students to demonstrate how they learned to do long division in their
What is Ethnomathematics?
Ethnomathematics forms the intersection between
mathematics and cultural anthropology. It was introduced by Ubiratan D’Ambrosio
(2001) who explained that ethnomathematics is the "art or technique of
explaining reality within a proper cultural context", and describes all the
ingredients that form the cultural identity of a group: language, codes, values,
jargon, beliefs, food and dress, habits, and physical traits. Ethnomathematics
defines a broad view of mathematics, and includes ciphering, arithmetic,
classifying, ordering, inferring, and modeling. In this context, "ethno" and
"mathematics" are understood in the broadest possible sense. Ethno
refers to a broad concept of cultural groups, and not an anachronistic concept
related to race or exotic groups of people; mathematics is to be seen as
a set of activities such as calculating, measuring, classifying, ordering,
inferring, and modeling. The traditional curriculum in the United States
has encouraged “only one way to solve a problem,” the ethnomathematical
perspective as adopted by the Algorithm Collection Project enables this
researcher to use the diversity in California classrooms as a resource.
The construction of mathematical concepts
incorporates the reality of each individual. It begins by placing new
situations and problems in front of a child for them to master within their own
context and experiential reality. It is on this basis that math concepts are
learned. It is with a focus on the understanding and resolution of problems,
that we "do" mathematics. Students must be given experiences that enable them
to learn how to mathematically: break a problem situation into manageable parts,
create a hypothesis; test the hypothesis, correct the hypothesis; and make
transference and generalizations to their own reality. Activities involving
mathematics should enable opportunities for open-ended exploration, appropriate
project work, group and individual assignments, discussion and practice using a
variety of mathematical methods, tools, and techniques. It is no longer
acceptable that the intellectual activity of a child is exclusively based on
memorization and testing, or for that matter only with the application of
archaic knowledge, which serves only to increase math avoidance. Using an
ethnomathematics based pedagogy, a teacher can introduce their students to new
tools and techniques directly connected to the real life of the learner, help
them to practice becoming proficient in their use, and guide them towards
sophisticated mathematical applications. It is through “context” that the
teacher creates additional explanations and ways to work within a
"mathematically based reality". This work within a mathematics reality is
related to the "transforming action” (Freire, 1997) that looks to reduce its
degree of complexity through the choice of a model where representations of this
reality are derived by enabling the exploration, explanation, and increased
comprehension of the concept.
A cross-cultural study of basic algorithms allows
us to reflect on the inherent possibilities of each context and for these same
possibilities to become the object of critical analysis by the students. The
process in which we consider, analyze and make ongoing reflections on different
algorithms is called modeling and is an important element in uncovering
ethnomathematics in general. The way to introduce students to mathematical
modeling is to expose them to a diversity of problems and models that include
mathematical interpretations of problems, which in turn, are representations of
models under study. When we analyze a given situation for its mathematical
perspective, the teaching and learning process becomes more than an over
emphasis on rote memorization of basic facts. An excellent place to begin this
study is in the way that people have learned to personally compute or calculate
thought a study of the algorithms used on a day-to-day basis. Mathematical
questions are used to explain and to forecast phenomena in the real world. What
is interesting from an ethnomathematical point of view is that many of these
explanations are unique from one culture to another, and of course this works
for algorithms that people use to make basic day-to-day calculations. Many of
these perspectives are used in representing situations for the study of
alternative techniques used to make calculations.
By bringing the alternative strategies and
algorithms found in our community together, teachers and students can learn to
flexibly solve problems, use alternatives when one strategy does not work, and
look at "best practices" from a global view.
A number of interesting things happened as an
outgrowth from this activity. A few preservice students were extremely nervous
about contacting someone to interview, "I do not know any one from another
country" was heard a number of times. Together as a class, we brainstormed a
number of possibilities which included, going to the Student Union, sitting and
listening, and politely introducing yourself and asking if the person speaking
another language was a foreign student, and if they would be interested in being
interviewed; contacting a church or other religious community, contacting the
University Multicultural Center, and contacting the Department of Foreign
Interestingly enough, students who expressed the
most reluctance stated that they enjoyed the activity the most. All
interviewees in the sample mentioned how happy they were that Americans were
interested in knowing something about their culture and ways of doing something
as basic as arithmetic. The classes developed lists of vocabulary words,
mathematical terms, numbers from 0-10 etc. , and wrote the words on index cards
that were placed in the classroom to compare each language.
Additions to this activity often include:
conversion formulas — standard to metric, time zones, etc.; vocabulary lists in
languages not represented by this study; the four basic operations, and square
roots; tricks for resolving problems; and a discussion of how the participants
memorized the basic facts. Almost all students who have participated in this
activity have come away with a greater appreciation of “more than one way to
solve a problem” a greater appreciation for cultural diversity, a new
understanding for how hard it is for newly arrived students to adapt to life in
a new country.
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