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Answer: I love that this question brings focus back to the practical workings of teaching mathematics in classrooms; the real work of education. While my thesis is structured to focus on the policy and the fundamental power struggles at play within development of policy rhetoric, but also formal curriculum – there does need to be that tangible link between what is found through the research, and what this means for practice.

I think as a first step, and based on some of the teachings I have heard Prof. Matthews speak about and others, having the awareness that there do exist robust mathematical understandings and knowledge systems that belong to Indigenous peoples – not just here in Australia but globally, is a foundation that we can build from. It all comes back to the questions of, what purpose does the mathematics serve? What does it enable us to do? Historically and in the now; how is this understanding/system useful, and what kind of world view does it reflect?

Even through the teaching of the history of our formal mathematics curriculum and ‘dominant’ understandings (I.e. the development of the hindu-arabic counting system we use and why/how that came about etc)., by shifting some of the focus to ‘how’ and ‘why’ these systems were developed, we can help students to understand for themselves that the discipline we know as ‘mathematics’ did not just materialize because it is ‘truth’ (although some may argue differently); it came about because it served a society a purpose and a function that was useful at that time, and continues to be so. I guess this is what is meant by the expression I’ve heard, which goes something along the lines of “finding the humanity in mathematics” (Hursh).

This historical perspective can be one way of linking into other knowledge systems, such as those of Indigenous Australians, whose mathematics may appear quite dissimilar to what we think of as ‘formal’ mathematics as they served entirely different purposes for societies that were based on very different world views.

Take for example the development of the number system. To my knowledge in some Aboriginal languages number names were used for numbers up to four, and then a term for ‘many’ would refer to quantities greater than four. You could quite easily find the relevant language resources for the naming words for numbers. In our vicinity of Newcastle, Awabakal, the Miromaa language centre founded by Daryn Mckenney (https://www.miromaa.org.au/) have a host of resources for integrating language into teaching practice and I’ve found the organization extremely approachable and helpful in the past. I think the biggest thing to be wary of however when integrating language in this way, is that in past times, Indigenous counting and number systems have been perceived as inferior by non-Indigenous onlookers, who lacked understanding of Indigenous world views and systems and were not positioned to make judgement around their usefulness. To some, this was reason enough to consider that Indigenous peoples therefore did not possess sophisticated mathematical processes or thought – and therefore what could we learn from such communities and their ways of being about maths?

In such a case, what needs to be considered more broadly is the reason and function of numbers and quantifying – functions which are largely linked to the development of a universal trading system and in our current times, economic dominance. In societies such as pre-colonised, traditional Indigenous community, what purpose would there have been for exactness in calculating and quantifying ‘amounts’, when the very notion of ‘possession’, and indeed ownership of resources and land, was not such a priority. Aboriginal communities lived in synchronicity and harmony with the land, ‘on country’, and so the need to quantify, dissect, calculate, and account for was not such a ‘thing’ as it was in the western world post agricultural revolution; whose whole societal and economic structure was built upon the basis of land ownership and possession.

What we can begin to do then, is to introduce to students some of the understandings that Indigenous peoples have used, and continue to use, which reflect their world views and ways of being – both in the past, and currently. Professor Matthews explains, with endorsement from Aboriginal elders of the Yolgnu peoples, how the concept of ‘Moiety’ works and how it serves Aboriginal communities across various regions of the country – a unifying system that seems to be ‘in common’ across nations. For a more detailed and accurate explanation of what Moiety, and other concepts such as Gurrutu, means, refer to this wonderful article written by Chris, which was published in Teacher magazine

(https://www.teachermagazine.com/au_en/articles/indigenous-perspectives-in-maths-understanding-gurruu ). I have shown this article to fourth year primary teaching university students and they were completely blown away. Imagine the kind of impact this type of awareness could have on our primary and high school students, simply ‘knowing’ that such systems exist, and serve a very important function in the maintenance and continuance of all life forms on country. Lessons we could all learn from in this state of global climate emergency I am sure.

So, in brief (and I apologize for the extended response), I think the first step is having ALL Australian students realise a few things, like:

1. Mathematics as we know it came about to serve human interests and needs (historically); and that these needs and interests were largely based on Western world views and patterns of interaction with the environment. Mathematics always has served a function. It has a ‘humanness’.
2. This ‘function’ of mathematics looks different in different societies, based on very different world views and ways of being. Therefore, the mathematics inherent within communities does not all look the ‘same’ – I.e. we cannot simply look for examples of what we perceive as ‘formal’ mathematics within different societies and their systems of knowing, as they just don’t exist in the same way.
3. We can find examples of the strengths of Indigenous knowledge and understandings, such as the example I referred to around the concept of ‘Moiety’, and more importantly, the function these knowledges served, and continue to serve, Indigenous communities. We need to consider, and help our students consider, the kinds of world views and ways of being and doing on country, that these systems reflect.
4. Simply teaching from the perspective that there is not just one way of knowing mathematics. Begin to broaden our perspectives and recognize that the perceived limits around what we consider to be legitimate, or valid mathematics are merely perceptions. There is always the possibility to expand upon our definitions.

I hope this helps; much of this I am sure you have already considered and or done before. Some of it is also up for considerable debate – I don’t have all the answers but this is how I have made sense of things thus far.

I also wouldn’t want to make any recommendations further to this without input from Indigenous community. The AECG and their extensive workshops and PL programs are also a first port of call – although I have found that finding ‘content’ on Indigenous mathematics locally is harder to find, where resources and info for other KLA’s is generally better covered.

The N. Qld students certainly enjoy the rugby league on Mars activity, for example! (I guess in NT communities it might be preferable to play AFL on Mars!) Your question and this answer also highlight the importance of “two-way education” in remote communities, which was pioneered by Mandawuy Yunupingu and is very much central to the philosophy of ATSIMA.

Answer: To answer your question, no I had not come across De Morgan’s work previously in this space – it does sound very relevant however to the question around how certain knowledges come to be more ‘privileged’ than others within what ‘counts’ as formal, and indeed essential mathematics.

I did allude to this aspect of the topic within my presentation; that the perceived legitimacy of certain knowledge’s over and above others is not by a chance occurrence, especially if we consider the rise of Scientific enquiry post-enlightenment; the increasing emphasis on objectifiable, verifiable ‘truth’, and the parameters around this kind of knowledge production (i.e. who can lay claim to ‘truth’ within the context of having to ‘prove’ everything scientifically? Who is thereby left out of the conversation? Whose knowledge claims can be perceived as legitimate, or non-legitimate, based on the premise of scientific ‘reason’ and rationality?).

My original avenue into this line of thinking was through the work of Michel Foucault – ‘The order of things’ (1970), and ‘The archaeology of knowledge’ (also 1970) – and at one point I wanted to focus my whole thesis on grappling with this concept within the Australian context of Indigenous epistemology and knowledges, and the imposed, if not mandated, Eurocentric paradigms that we have yet failed to challenge, which are hinged on the assumption of ‘terra nullius’. Another author who has seriously questioned the underpinnings of mathematics as we know it as a discipline today is Valerie Walkerdine. From the sound of De Morgan’s work, it would seem that they probably share a focus in unveiling the ‘whiteness’ behind what we now think of as the foundations of western knowledge constructs.

For the purpose of this work and this thesis, I will not be going more deeply into this topic as I simply don’t have the scope to do so – my focus has had to become much more narrowed on the question of ‘equity’ for Indigenous learners within the discipline, and how this is currently being framed. However, the backdrop to that, and the context is that we must be aware of some of those power dynamics at play around the construction, verification, and legitimacy of certain knowledge claims. This work you have suggested may help me reconcile that within one of my chapters or publications – so thank you! I will have a look for this work and see how it contributes to the conversation.