Mathematics Learning

In simple terms, numeracy can be defined as the ability to understand basic mathematical concepts and operations. Numeracy thus encompasses a wide range of topics, including formal symbolic mathematics, cultural practices, children's intuitions about mathematics, and everyday behaviors mediated by mathematics. There are different forms of numeracy, and their realization in various cultural contexts has commonly been called ethnomathematics. Researchers have explored how pedagogy can be changed to incorporate cultural practices related to numeracy.

Numeracy in Cultural Context

All cultures have developed various representational systems that provide ways of thinking about quantitative information. Different systems highlight different aspects of knowing. For example, the Oksapmin of Papua New Guinea have a counting system that uses body parts to express numbers from one to twenty-seven. Though no base is used in this system, it is adequate when trading goods using a one-to-one correspondence. It is inadequate, however, for computing or counting objects beyond twenty-seven. A contrasting example is the enumeration system of many Asian languages that is congruent to the structure of base ten. Asian children using this system tend to recite more number names in correct sequence and show earlier mastery of place-value concepts (i.e., relations among number words, multi-digit numerals, and quantities) than children using less regular base-ten systems.

In addition to enumeration systems, cultures have developed representational systems for locating (geometry, navigation), measuring, designing (form, shape, pattern), playing (rules, strategies), and explaining (abstraction). These representational systems entail beliefs and values associated with numeracy, and they support numeracy activities using tools such as abaci, clocks, and digital computers. A broad array of human activities to which mathematical thinking is applied is thus interwoven with cultural artifacts, social conventions, and social interactions.

Cultural variations in mathematical behavior are also seen in the ways people use mathematical representations in their everyday activities. For example, children working as street vendors in Brazil were found to use different computational strategies when selling than when doing school-like problems. While selling they used such strategies as oral computation, decomposition, and repeated groupings, whereas when given school-like problems they used standard algorithms. These children were found to be much more accurate in the context of selling than in the school setting. Research in other domains, such as measurement and proportional reasoning, further confirms that informal mathematics can be effective and does not depend upon schooling for its development.

Mathematics is used in everyday life in pursuit of goals that differ from the goals of academic mathematics found in schools and universities. This type of mathematics is often referred to as informal, or naive, mathematics. Representational systems and practices deriving from everyday activities are modified as new goals emerge. Thus, everyday mathematics is an adaptive system that can be used to creatively meet new challenges. For example, as the Oksapmin became more involved with the currency system, their body counting system described earlier began to change toward a base system. Although knowledge acquired through informal experiences are often distinguished from school mathematics, skills developed in the informal domain can be used to address goals and practices in the school setting. For example, the most successful elementary students in Liberian schools combine the strategies from their indigenous mathematics with school algorithms. Use of informal mathematics in school settings, therefore, may be an effective way to help children learn school mathematics. Several authors have argued for building bridges between informal and formal mathematics.

Although most research on numeracy and culture has been done outside of the United States, understanding informal and everyday mathematics is important for educators in the United States for a number of reasons. According to the census conducted in October 1999, about 2.5 million foreign-born children came to U.S. schools, bringing with them different mathematics representational systems and associated computational skills. In addition, everyday mathematical activities and language repertoires for American children of different ethnic groups have been shown to differ both across groups and when compared to the school curriculum. In the case of some groups, such as Native Americans, mathematical reasoning derived from cultural traditions is distinct from that of the schools, posing major conceptual problems for these children in the regular school curriculum.

Research on Curricular Change

A number of projects have attempted to make mathematics instruction more culturally relevant for groups of children who have traditionally underachieved in the U.S. school system. Mary Brenner has worked with teachers to improve mathematics teaching for Native Hawaiian children. She interviewed parents and children and observed children in everyday settings to determine what kinds of numerical skills children brought with them to school. At the kindergarten level, adapting the existing curriculum consisted of reordering topics to begin with counting and computation (areas of student strength), more use of the students' nonstandard dialect in mathematics lessons, and more emphasis upon hands-on and game-like activities. At the higher grade levels, adaptations focused more on including activities, such as a school store, that enabled students to move from informal mathematical activities to more standard mathematical practices.

Ethnographic research has revealed many mathematically rich activities in everyday adult life. Luis Moll and James Greenberg have developed a culturally relevant pedagogy for Latino students by building classroom activities from the "funds of knowledge" that are present in their family networks. Teachers and researchers worked together to plan lessons based upon ethnographic data, and they also invited parents to teach. New mathematics units, such as one involving candy making, were developed as the contexts for teaching specific mathematical ideas.

In a different approach, Jerry Lipka and Ester Ilutsik, who work with the Yup'ik people in Alaska, advocate giving the community control over the process of curriculum development. The goal is to make the schools a local institution, rather than having schools act as representatives of the dominant society. Researchers, Yup'ik teachers, and tribal elders have worked together to translate Yup'ik mathematical knowledge into a form that can be utilized in classrooms. Like the Funds of Knowledge project, this group has analyzed everyday adult activities, such as fish camps, to understand the culturally relevant mathematics. In addition, they have worked to better understand the Yup'ik number system and how it can be used in the classroom. The goal is to create an entire mathematics curriculum based upon Yup'ik culture, rather than adapting existing curricula.

Gloria Ladson-Billings has conducted research on culturally relevant mathematics instruction for African-American children. This research has highlighted a variety of attitudinal changes that teachers must make in their teaching, including expecting higher academic standards for students, emphasizing cultural competence, and instilling critical consciousness in students.

Teaching and Cultural Context

Teaching is an inherently cultural activity; it is situated in a bed of routines, traditions, beliefs, expectations, and values of students, teachers, administrators, parents, and the public. Thus, the inclusion of cultural and everyday mathematical knowledge in school mathematics must take into account the school-based assumptions about the appropriate way to teach mathematics.

For example, cultural assumptions about effective ways to improve teaching in Japan include lifelong professional development activities carried out by ordinary teachers. Typically, a few teachers with similar goals and interests form a study group. They select a few lessons that need improvement and analyze what is and is not working in the current practice in terms of learning goals for students, students' misunderstandings, and use of activity. They gather information on the topics by reading about other teachers' ideas, as well as other sources of recommended practices. A revised lesson is then planned, and one of the teachers from the group implements it while the others observe and evaluate what is and is not working. This process of evaluation, planning, and implementation is repeated until a satisfactory lesson is crafted, and may consume an entire school year. This is in sharp contrast to the type of model lesson developed by expert teachers and handed down to ordinary teachers in the United States. This is also different from the "one-day, make-it-and-take-it" type professional development workshops often implemented in the United States–a practice whose long-term effectiveness is questionable.

Ideas for Teachers

The importance of raising awareness of cultural diversity among teachers has been extended to the teaching of mathematics in the United States. For example, in the mid-1990s a task force for the National Council of Teachers of Mathematics recommended the publication of a series, Changing the Faces of Mathematics, in order to help make the slogans of Mathematics for All and Everybody Counts real. Particular efforts were made to focus on education of ethnic and cultural minority students. Included in this series are volumes on African-American perspectives, Latino perspectives, and Asian-American and Pacific Islander perspectives. Each of these volumes includes articles that discuss successful pedagogical strategies for culturally diverse groups of students. The volumes also feature articles that may help educators develop a deeper understanding of the cultural differences that influence classroom dynamics, behavior, and environment.