Ethnomathematics - Bishops (1979) and Barton (2009)
In the book, Critical Issues in Mathematics Education, there is a section pertaining to Cultural and Social Aspects. This section begins with a paper written by Bishop (1979). This paper seems to be the cinch pin of getting the enthnomathematics way of thinking to begin. Bishop, along with many other researchers, states the fact there seems to be one dominant culture influencing most of mathematical thinking – the Western culture (Bishop, 1979; Barton, 2012; Orey and Rosa, 2012). Within his paper, he studies the culture within Papua New Guinea. While interviewing a few subjects, he discovered that, without context, based on students’ background/culture, a simple question may have different meanings. Although this may seem obvious now, it was a huge discovery at the time, which propelled him to investigate more. At the end of his paper, he mentioned, “… perhaps if we consider[ed] mathematics education as a form of cultural induction we would realize both the enormity of the task and the range of influences that can be brought to bear” (Bishop, 1979). The take away message from his discovery is that mathematical meaning is socially constructed and more research must be done.
Barton (2008), inspired by Bishop’s proposal to research mathematics as a form of cultural induction, made some significant discoveries and proposed some difficult questions. In his paper, Barton writes, “meaning is culturally-based and therefore an individual’s constructed meanings will be at odds with those of others from different cultures” (2008). This implies that if mathematics is to have meaning to an individual, it must be in the context of their culture. As such, if different cultures constructed different meanings, there could be a conflict in the interpretation of mathematics. Thus, as Barton mentioned, this would be a challenge when the culture of the teacher differs from that of the students (2008).
Barton asked four questions in his paper, to which he answered throughout. The questions he presented were:
If mathematics education is a form of cultural induction, then:
1. What is it an induction to?
2. How does it interact with other cultural models?
3. Who decides the subject of the induction?
4. How could the induction proceed?
Barton (2008)
To address the first question, Barton points out that there have been at least three new forms of mathematics that has arisen due to ethnomathematics. For example, there are now “the visual computer language embedded in Indian Kolam drawings, the curved ‘linear’ functions of the double-origin geometry inspired by ways of reference in Polynesian languages, and, most recently, the cyclic matrices inspired by Angolan sand-drawings” (Barton, 2008). Thus, if mathematics is a form of cultural induction, it is an induction to new methods of mathematics that lies in the context of different cultures.
Barton responded to the second question by relating the relationship between mathematics as a cultural induction to other cultural models to the relationship between pure and applied mathematics. That is, between the two forms of mathematics, there lies applications of math, such as economics – which in itself provide original mathematics. However, this is not to say that economics is math, nor is mathematics economics. Rather, we can look at economics in a mathematical view, just as we can look at mathematics in an economic lens.
The answers to the third and fourth question remain open, according to Barton (2008). He mentions some studies that involve ethnomathematics programs with some success, however, also warrants caution from over generalizations. The only condition is that an enthnnomathematical curriculum is mainly effective “primarily in situations where the class is predominantly from one culture that is not close to the culture of mathematics” (Barton, 2008). Having an enthnomathematical curriculum requires that culture of the class be different than the culture of mathematics in order for the teacher to bridge the understanding of mathematics to the understanding and mathematical aspects of their own culture (Barton, 2008). Nonetheless, if the ethnomathematics curriculum is the only curriculum in place, it negates the national standards that have been in place. This is another issue mentioned by Barton. The mathematical induction may not proceed as long as students are held to the National Standards as each culture would be learning different methods. In addition, the context of the standardized tests would be too general for an ethnomathematical curriculum. Thus, there is much that needs to change in order to reap benefits of an enthmathematical program. Overall, Barton ends his paper with, “considerations of language, hermeneutics, anthropology, and other socio-cultural areas enhance mathematics, they do not destroy it” (2008). Approaching mathematics with an open mind allows it to grow and enhances the learning for students.
Barton (2008), inspired by Bishop’s proposal to research mathematics as a form of cultural induction, made some significant discoveries and proposed some difficult questions. In his paper, Barton writes, “meaning is culturally-based and therefore an individual’s constructed meanings will be at odds with those of others from different cultures” (2008). This implies that if mathematics is to have meaning to an individual, it must be in the context of their culture. As such, if different cultures constructed different meanings, there could be a conflict in the interpretation of mathematics. Thus, as Barton mentioned, this would be a challenge when the culture of the teacher differs from that of the students (2008).
Barton asked four questions in his paper, to which he answered throughout. The questions he presented were:
If mathematics education is a form of cultural induction, then:
1. What is it an induction to?
2. How does it interact with other cultural models?
3. Who decides the subject of the induction?
4. How could the induction proceed?
Barton (2008)
To address the first question, Barton points out that there have been at least three new forms of mathematics that has arisen due to ethnomathematics. For example, there are now “the visual computer language embedded in Indian Kolam drawings, the curved ‘linear’ functions of the double-origin geometry inspired by ways of reference in Polynesian languages, and, most recently, the cyclic matrices inspired by Angolan sand-drawings” (Barton, 2008). Thus, if mathematics is a form of cultural induction, it is an induction to new methods of mathematics that lies in the context of different cultures.
Barton responded to the second question by relating the relationship between mathematics as a cultural induction to other cultural models to the relationship between pure and applied mathematics. That is, between the two forms of mathematics, there lies applications of math, such as economics – which in itself provide original mathematics. However, this is not to say that economics is math, nor is mathematics economics. Rather, we can look at economics in a mathematical view, just as we can look at mathematics in an economic lens.
The answers to the third and fourth question remain open, according to Barton (2008). He mentions some studies that involve ethnomathematics programs with some success, however, also warrants caution from over generalizations. The only condition is that an enthnnomathematical curriculum is mainly effective “primarily in situations where the class is predominantly from one culture that is not close to the culture of mathematics” (Barton, 2008). Having an enthnomathematical curriculum requires that culture of the class be different than the culture of mathematics in order for the teacher to bridge the understanding of mathematics to the understanding and mathematical aspects of their own culture (Barton, 2008). Nonetheless, if the ethnomathematics curriculum is the only curriculum in place, it negates the national standards that have been in place. This is another issue mentioned by Barton. The mathematical induction may not proceed as long as students are held to the National Standards as each culture would be learning different methods. In addition, the context of the standardized tests would be too general for an ethnomathematical curriculum. Thus, there is much that needs to change in order to reap benefits of an enthmathematical program. Overall, Barton ends his paper with, “considerations of language, hermeneutics, anthropology, and other socio-cultural areas enhance mathematics, they do not destroy it” (2008). Approaching mathematics with an open mind allows it to grow and enhances the learning for students.