In this assignment, you need to engage with cultural aspects of mathematics and share your reflections on this engagement with the wider education community.

1. Choose a topic of interest that relates to either contemporary applications of mathematics, the history of mathematics, or ethnomathematics. Independently or collaboratively, source and read appropriate material on your chosen topic.

2. Create and maintain a reflective blog, showing how your thinking about mathematics is impacted by your reading. In particular, reflect on how this reading enables you to contextualise learning to meet the learning needs of a particular social context.

3. Share your knowledge and critical interpretation into a culturally responsive pedagogical approach with others through online discussions, via presentation at a conference or in a contribution to a collaborative publication. Your culturally responsive pedagogical approach should aim to create an inclusive classroom learning environment for children in diverse or specifies social contexts

In the absence of any concrete and unanimously accepted definition of mathematics, it can broadly be understood as a science of quantity, space, and numbers, either as abstract concepts, i.e. pure mathematics, or as applied to other areas of studies such as engineering or physics, i.e. applied mathematics (Schoenfeld, 2009). D’Ambrosio (2001) described mathematics as a medium of communicating ideas of number, space and time between people (D’Ambrosio, 2001). Mathematics as a discipline of study has evolved over centuries and across civilizations and cultures. Eliot (2010) describes culture as the collective manifestation of arts and other human intellectual achievements, transmitting from one generation to the next through arts, language, institutions, material objects, and rituals. The word culture is also used intertwined with ethnicity. Cultural groups can be defined as the group of people which share the same form of art, music, literature, customs, festivals, rituals, language or other characteristics (Eliot, 2010).

Mathematics, although, is a complete and independent subject in itself, can be seen as cultural constructs. The wide range of mathematical models has evolved in different cultures and civilisations in different times in history. Each such cultural groups devised their own methods of numbering, counting, or calculating based on their cultural contexts and requirement. This resulted in a variety of mathematical models, some of which could be related to each other due to some similarities or analogies, and others are yet discreet and undeciphered (Ascher, 2002).Different cultural groups used different methods of mathematics and made considerable human progress and scientific advancements, reflecting the merits of such different methods. This compels one to study and explore the interplay of different cultures and their practice of mathematics.

In the broader parlance, the study of culture and mathematics together, along with its numerous aspects, is coined as Ethnomathematics.This paper discusses a set of definitions around ethnomathematics that has propped up by different scholars citing to different times. It then discusses various aspects of ethnomathematics and how mathematical concepts have been driven by cultural contexts before understanding the impact of thestudy of ethnomathematics on the author’s thinking, and ultimately concluding the paper.

As ethnomathematics is the study of culture and cultural groups from the perspective of mathematics, it is defined differently by different scholars based on their understanding of cultural groups and domains of their study. Most of the definitions emerged in the latter half of the 20^{th} century within anthropological literature (Hammond, 2000). Ascher and Ascher (1997) define ethnomathematics as “the study of mathematical ideas of non-literate people”. As the domain of study of both the researchers was African counting cultures, their understanding of the culture was attached with the illiteracy of natives. This definition is limiting the understanding of culture to illiterate people only, hence eliminating other cultural groups which show the evidence of formal and informal education system with them.

On the opposite end, the University of Idaho website provides a very open-ended definition by stating ethnomathematics as the study of different cultures in which mathematics arises (University of Idaho, n.d.). This definition is connecting cultures, in general, with the evolution of mathematics within them without specifying the definition of the culture. This allows for a broad range of groups to be considered as cultural groups.

## Discussion on Ethnomathematics

D’Ambrosio (1997) provides somewhat broader definition than that of Ascher duos, yet maintaining ethnicity centric approach by defining ethnomathematics as “a practice among the identifiable cultural group, like children of certain age, national-tribal societies, professional classes, labor groups, and so on” (D’Ambrosio, 1997). This definition expresses cultural groups as a group of people classified on the wide range of underlying parameters, may it be economic, professional, age-related, interest based, or any other.

Pompeu (1994) defines “ethnomathematics as a cultural knowledge or social activity which is recognised by western anthropologists as mathematical knowledge or mathematical activity” (Pompeu, 1994). This definition vest the authority with western anthropologists for attesting a body of knowledge as mathematics, on the basis of similarities found in their own version of mathematics. This raises the crucial question of the ownership of mathematics, and should the understanding of mathematics by western civilisation be considered supreme vis-à-vis less explored bodies of mathematical knowledge.

There exist a plethora of definitions around ethnomathematics, yet there is no concrete and unanimously accepted one due to the difference in understanding of culture and cultural groups, the domain of study, stage of evolution etc. Lack of consistency in defining ethnomathematics calls for further research in this domain with greater emphasis on less explored areas of mathematics that are identified in varied cultural groups.

Ethnomathematics has evolved itself over a period of time. The cultural context has given several dimensions to the otherwise technical subject (Favilli, 2007). Several streams of mathematics came forth as a result of the requirements of varied cultures. There are two schools of thought. One school of thought advocates that mathematics was always present and different cultures have discovered and nurtured it in their own context. Unlike other fundamental sciences like physics and chemistry, no tangible product exists in the case of mathematics. For example, the invention of the spacecraft is a tangible product when it comes to physics. But in the case of mathematics, scholars have developed theorems and corollaries that do not translate into anything tangible and it appears that mathematics was always there and it was just a matter of time that people got to learn about this subject. Thus the argument of discovery leverages itself on the lines of intangibility.

In contrast to this, the other school of thought suggests that cultures and civilisationshave developed the body of knowledge of mathematics. This argument discards the pre-existence of mathematics. Evidenceof emergence of the fraction to augment the requirements of trade validates the fact that context and circumstances led to the evolution of mathematics. More such examples can be found out in Mayan civilisation where religious and astronomical beliefs gave birth to the creation of the calendar.

The research work on ethnomathematics very marginally subscribes to any particular viewpoint (Knijnik, 2002). Ethnomathematics rather has been constructed as an interdisciplinary entity encompassing both coremathematical and contextual cultural elements. The core mathematical components have been coated with contextual cultural dimensions. This establishes the fact that culture and mathematics are not devoid of each other and rather both have been instrumental in shaping up each other. It is the cultural validation that mathematics have survived the test of times. For example, the conceptualization of zero is believed to have its roots in Hindu mythology. Correlation of notion of zero with the prevailing cultural traditions helped it to survive in its nascent stage and safeguard itself from being discarded as non-contextual.

The common understanding of mathematics as one of the subject in school curriculum, which is indifferent to cultures and values, is strongly contrasted by the work of a group of researchers and authors, led by Bishop, who claimed that culture plays a formative role in how learning of mathematics is imparted among children with that cultural background (Lerman, 2002). The prevailing understanding of mathematics as an academic subject is because of the dominance of western curriculum in schools all over the world. This has resulted in the replacement of the study of mathematics as a cultural activity with a need-based study to remain functional in the globalising world. This widespread understanding is negated by Bishop and D’Ambrosio, who brought forward the case of such cultural groups who still use their systems of mathematics in their day to day functioning. For example, according to the studies, children in rural areas of India or tribal communities of many African countries are taught mathematics using the local methods and colloquial examples (Gerdes, 2001).

It can be understood that there exist numerous understanding and beliefs around ethnomathematics among different segments of society. Such variance is the result of factors like economics and cultural dominance of some groups over others, and lack of extensive research. With ongoing research, in due course of time, ethnomathematics will evolve as a well-developed discipline of study encompassing all the existing schools of thought and related theories.

How Cultural Context Delivers Mathematical Concepts

The understanding of any subject matter is driven by the cultural context and mathematics is no different. Ever since the stone age, humans have gathered knowledge from their surrounding and shaped it using their own intellect. In the primitive societies, human used to keep a track of its livestock (Kuper, 2005). Although the subject matter at that time was not institutionalised as mathematics but the initial understanding was on the similar lines. Later, as the civilisations evolved, the subject matter of mathematics begun to find prominence and today it is shaping up the new inventions of the 21^{st} century by solving some of the most complex problems.

The knowledge of mathematics has always been grounded in the local cultural context. This knowledge over the period of time has evolved across different societies. Each society in its own way has been benefitted of mathematics (Earnest & Treagust, 2006).Intellectual exchanges across civilisationshave a large share of mathematics. Such cross-cultural exchanges have enriched the intellectual ecosystem through their respective body of contextual knowledge.

In Egyptiancivilisation, humans settled along the banks of river Nile. For agricultural and religious purposes, they began to observe different patterns of lunar phases and seasons. Ten fingers of human hand gave birth to the decimal system in Egypt. Symbols for communications were derived from local articles. In ancient times, repetitive counting was carried out to multiply (Boyer & Merzbach, 2011).Initially, mathematics was devoid of the concept of fractions.The need for the development of fraction came from trade and accordingly concepts of fraction were developed. Egyptian culture has a huge influence of pyramids wherein dead bodies are preserved. Area, perimeter, volume and other features of the pyramid as an entity was developed in response to the cultural requirements of the times.

Evolution of geometry can also be traced in ancient culture (Boyer, 2012). Human began to identify a pattern in the shapes and sizes of objects in the natural surroundings. This led to the foundation of the subject matter of geometry. Later on, the mathematicians developed different theorems to theorise the subject matter. The Roman administration and trade used a different set of number systems to conduct itself. This is widely popular even till date. The need to calculate larger astronomical distances lead to the birth of trigonometry. This subject matter still continues to evolve.

In china, bamboo rods were used to signify different numbers which later on gave birth to what today is known as an abacus (Anjing, 2005). The development of mathematics in China was an outcome of the need of mathematically competent administrators. Several schools to learn and develop mathematics were opened in China. Chinese used approximations to find out solutions to complex equations. The Mayan civilisation settled in America used mathematics for their astronomical and calendar calculations.

The last few centuries have seen tremendous evolution in mathematics. This is largely attributed to thetechnological developments and the contribution of mathematics to that development. Mathematics has played a huge role in the discovery of new places and laid the ground for landmark events like an Industrial revolution. Contexts have changed over the period of time and so has the subject matter of mathematics which is largely influenced by the context. In today’s time, systemic theories of mathematics have been developed which are applied in the domains of physics, finance, cryptography, logics and computation among others. A lot has changed, what has certainly remained constant is the deep influence of culture on the subject matter of mathematics.

The subject matter is an interesting research topic. A clear and precise understanding is required to understand the nuances of mathematics in relation to the cultural context. The author got deep insights about the evolution of the subject matter under the cultural influence. For the author, Mathematics so far was a technical subject but the understanding of the journey of the subject through different civilisations in various societies has enriched the intellect of the author. It was learned that Mathematics has played a critical role in intellectual cross-cultural exchanges. These exchanges over a period of time have entirely changed the landscape of human civilisation. Mathematics by virtue of its nature finds relevance in the application in several other subjects. It is this feature of Mathematics that makes its utility inevitable in almost all technical inventions.

The author got to realise how the concepts of mathematics were perceived differently in a different context in several parts of the world. While in Egypt, the culture of preserving dead bodies in pyramids influenced geometry, in China, the need to have efficient administrators resulted in the establishment of schools of Mathematics. The religious and astronomical beliefs of Mayan civilisation resulted in the formation of calendars. It was learned that landmark events of human history like the industrial revolution, Higgs boson experiment etc. could not have happened without the development of mathematics as a subject. Decimal system which is widely practised today is an outcome of the need of earlier times. Roman system of numerals has survived the times and is still practised across the globe. The author found the research topic abstract in the first place but later intense reading helped in diving deeper into the content.

A varied range of definitions are present and so author found it difficult to understand the definition of ‘Ethnomathematics’. It was figured out that the present body of research is largely a work of western scholars and thus only a particular lens has been used in dealing with the subject matter. This has limited the scope of the research topic and the reader might run the risk of accepting a particular perspective while completely neglecting the other side. The subject matter can be enriched by taking a holistic approach to dealing the subject matter and accordingly the author believes that there is a huge scope of research as the subject stills have a huge unexplored territory.

Conclusion

Ethnomathematics is an interesting area of research but its scope is limited by its present body of knowledge. There is no common agreement among the scholars about the working definition of ethnomathematics. The birth of mathematics dates back to several centuries and its evolution has been influenced by the context. Cross-cultural intellectual exchanges have largely been shaped by mathematics. It needs to be understood that mathematics is not just a technical subject which is devoid of organic aspect rather it is deeply influenced by the context in which it develops and conversely hugely impacts its context as well. Different centuries and civilisations have witnessed the development of mathematics through different phases.

The body of knowledge created by western scholars though restricts the scope of the subject matter but paves the way for other scholars to carry out cutting edge research and further compare it with the existing research. This shall result in enriching the subject of ethnomathematics which shall be instrumental in designing new pedagogies of learning mathematics. Clearly, there is a dire need of conducting more research through ever-evolving techniques of research to enrich thebody of knowledge of ethnomathematics.

References

Anjing, Q. U. (2005). Changing the Paradigm: Research on History of Mathematics in China [J]. The Chinese Journal for the History of Science and Technology, 1, 008.

Ascher, M. (2002). Mathematics elsewhere: An exploration of ideas across cultures (pp. 3-3). Princeton, NJ: Princeton University Press.

Ascher, Marcia and Robert Ascher. (1997). Ethnomathematics, Ethnomathematics: Challenging Eurocentrism In Mathematics Education. New York: State University of New York Press.

Boyer, C. B., & Merzbach, U. C. (2011). A history of mathematics. John Wiley & Sons.

Boyer, C. B. (2012). History of analytic geometry. Courier Corporation.

D’Ambrosio, U. (1997). Ethnomathematics and its place in the history and pedagogy of mathematics. Ethnomathematics: Challenging Eurocentrism in mathematics education, 13-24.

D'Ambrosio, U. (2001). Mathematics across cultures: The history of non-Western mathematics (Vol. 2). H. Selin (Ed.). Springer Science & Business Media.

Earnest, J., & Treagust, D. F. (2006). Education reform in societies in transition: International perspectives. Sense Publishers.

Eliot, T. S. (2010). Notes towards the Definition of Culture. Faber & Faber.

Favilli, F. (2007). Ethnomathematics and mathematics education. In Proceedings of the 10th International Congress of Mathematics Education, Discussion Group 15: Ethnomathematics.

Gerdes, P. A. U. L. U. S. (2001). Ethnomathematics as a new research field, illustrated by studies of mathematical ideas in African history. Science and Cultural Diversity: Filing a gap in the history of sciences. Cuadernos de Quipu, 5, 10-34.

Hammond, T. (2000). Ethnomathematics: Concept Definition and Research Perspectives. New York: Columbia University Press.

Knijnik, G. (2002). Ethnomathematics: Culture and politics of knowledge in mathematics education. For the Learning of Mathematics, 22(1), 11-14.

Kuper, A. (2005). The reinvention of primitive society: transformations of a myth. Routledge.

Lerman, S. (2002). Cultural, discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics. In Learning discourse (pp. 87-113). Springer Netherlands.

Pompeu Jr, G. (1994). Another definition of ethnomathematics. Newsletter of the International Study Group on Ethnomathematics, 9(2), 3.

Schoenfeld, A. (2009). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. Coleccio?n Digital Eudoxus, (7).

University of Idaho. (n.d.). Ehtnomathematics. Retrieved May 14, 2017, from University of Idaho: https://www.cs.uidaho.edu/~casey931/seminar/ethno.html

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