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IUM1: Indigenising University Mathematics Symposium, September 20–21, 2021

Christopher Tuffley

What does it mean to indigenise mathematics education and research, and how and why should we do it?

Addressing these questions in a university setting especially was the subject of the first Indigenising University Mathematics Symposium, which took place over September 20–21, 2021. It was jointly hosted by the University of Newcastle’s Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA), and its Wollotuka Institute of Indigenous Engagement and Advancement. Judy-anne Osborn (University of Newcastle), cochair of the organising committee and a major driving force behind the symposium, described it as having been sparked in part by a challenge issued to academic staff at the university from across the disciplines: “We have to indigenise the curriculum. The government says we’ve got to do it, Universities Australia says we’ve to do it, we’ll lose funding if we don’t do it, and it’s morally the right thing to do.” The symposium grew out of her grappling with what this meant and how to do it in the context of mathematics.

The symposium took place entirely online, with an opportunity for the participants to gather virtually between sessions. The online format may have been forced on the organisers by the ongoing covid19 pandemic, but it almost certainly enabled more people to attend: well over 100 people participated, connecting in from locations including Australia, Canada, Hawai’i, New Zealand, the United Kingdom, and the continental United States.

The symposium was designed within a framework of mutual sharing and learning, where we would all benefit from the experiences and knowledge of all present. To this end each session was followed by a yarning session (yarning in the sense of sharing a story), where we could reflect on and share our ideas about what had been said so far. These drew upon the principles of the Yarning or Talking Circles seen in Australian Aboriginal and North American Indigenous cultures, adapting them as best possible to the online space we were meeting in and the time constraints of the symposium. The meeting chat offered another channel in which people could respond, and was lively with the exchange of ideas throughout.

The symposium was organised around the six interconnected themes of Re-imagining the Living Present; Country and Mathematics; Language and Oral Traditions; Indigenous Mathematics; Love and Pedagogy; and Traditional Knowledge. Each theme was chaired by a partnership of indigenous and non-indigenous speakers, who gave a presentation and then led a yarning session. A seventh keynote session discussed decolonisation, and there were five shorter community contributions. The presented portions of the symposium were all recorded but the yarning sessions were not, so that participants could freely exchange thoughts and ideas without fear of any half-formed ideas being recorded for posterity. The videos of the sessions can all be viewed on the symposium webpage IUM1.

Some Key Ideas

The symposium was fascinating, thought-provoking, and challenging. I learnt a lot and was very glad I’d taken part. Here are some of the key ideas I took from it. Space limitations preclude me from mentioning everything that I would like to.

In the process of writing this article I reviewed the videos of the presentations to refresh my memory of what had been said. In doing so I was struck again by the depth of knowledge and thinking of the presenters, and the impossibility of doing them justice in a short article. I encourage you to visit the symposium webpage IUM1 and view the videos yourself, so that you can hear their thoughts in full and in their own words.

Indigenous knowledges are grounded in evidence and observation, are still living and relevant today, and should be respected and valued as such

In Theme 1 Reimagining the Living Present Mark Mac Lean (University of British Columbia) addressed a fallacy he sees prevalent in Western education of presenting indigenous peoples as historical, and as either an object of study, or as needing to be brought into the world of “modern mathematics and modern mathematical thinking.” He spoke of the need to “replace that fallacy with the truth, that indigenous peoples have knowledge and pedagogies, including mathematical knowledge and pedagogies, rooted deep in these long histories, in some cases tens of thousands of years, and that are still vibrant and still relevant in the present, and still a living thing in these peoples around the world.”

Kathleen Butler (Director, Wollotuka Institute, University of Newcastle; Aboriginal, of the Bundjalung and Worimi people) then spoke of the ways in which indigenous knowledge and oral traditions are all too often seen as secondary to Western knowledge, and regarded as metaphor, or accepted as true only when evidence is found to support them. She sees this as problematic. It is debated how the Australian megafauna went extinct, with some arguing that they were hunted to extinction shortly after humans arrived in Australia, while others argue that all evidence points to Aboriginal people having coexisted with the megafauna for over 20,000 years. What no one seems to be doing, Butler says, “is going back to the oral traditions, which tell us very clearly that we coexisted with megafauna.” Butler says “I’m really interested in how we could think about indigenous knowledges as something which have their own inherent value, rather than only having a value when evidence supports their existence. Because I think that indigenous knowledges do come from a really clear evidence base.”

Butler spoke of the way in which our use of numbers can create something artificial that separates us from the natural world. She gave the Gregorian calendar as an example of this, with its twelve months that don’t align with the phases of the moon, in contrast to the lunar calendars used by many indigenous peoples. “I think that what indigenous knowledges bring to our understanding of mathematics is very much embedding us in what is happening in the world around us. That the numbers that we [use] to artificially create something, don’t change our natural world, and in fact, in doing that artificiality, we remove ourselves from a really important element of connection.” Butler says many indigenous academics and thinkers argue that this removal of ourselves from the natural world greatly contributes to wicked problems such as climate change.

Doing mathematics is a universal human endeavour

The idea that doing mathematics is simply part of being human is one that came through several times over the course of the symposium, in both the keynote presentations and the yarning sessions. Rowena Ball (Australian National University; Aboriginal and Irish descent) put it as follows in the Decolonisation Discussion:

Doing maths and mathematical thinking are universal human imperatives across all societies and cultures, as natural and intrinsic to our humanity as doing art, and probably as ancient.

Michael Assis (CARMA and the University of Melbourne) put a case for this idea in Theme 3 Language and Oral Traditions. Assis presented a series of mathematical texts from ancient civilisations, ranging from Babylonian and ancient Chinese texts on the Pythagorean Theorem, to a Mayan work on predicting lunar eclipses. Some of the cultures were geographically close enough to influence each other, but others were geographically well removed. This leads him to the conclusion that mathematics is a human endeavour that all cultures have engaged in — that wherever you go there is mathematical thought — and moreover that the development of mathematics doesn’t have a single linear history, with the same ideas arising multiple times in multiple places.

What is mathematics, and what is indigenous mathematics?

In Theme 4 Indigenous Mathematics, Edward Doolittle (First Nations University of Canada; Indigenous North American, of the Kanyen’kehaka people, commonly known in English by the exonym Mohawk) addressed the three questions What is indigenous? What is mathematics? and What is indigenous mathematics?

Doolittle sees the first as very simple: indigenous means connected to place, and importantly connected to the culture associated with the place. The second he sees as much more problematic, and in fact the wrong question to ask: rather than try to define the noun “mathematics” he thinks the better question is to define the adjective “mathematical”—just as we can more easily define the adjective “infinite” than we can the noun “infinity.” In answer to this better question Doolittle referred to a list of six fundamental activities that Bishop Bis88 argues are universal to all cultural groups, and necessary and sufficient for the development of mathematical knowledge: counting, locating, measuring, designing, playing, and explaining.

To answer the third question Doolittle puts these two answers together: indigenous mathematics is mathematical knowledge that is local, and acquired using the methodologies of the place.

Ball gave an example of indigenous mathematics in the Decolonisation Discussion. She described songlines transforms, an Aboriginal solution to the wayfaring problem of navigating a route of hundreds or thousands of kilometres which no one in the party has travelled before, in the absence of portable maps and satellite navigational aids.

The songline for a route encodes it in the winter stars, so that elders who know the route can teach it over winter to the party who are to travel it the coming summer. Ball argues that the songline is a mathematical transform, akin to the Fourier transform: “The songline transform is an approximation to the route, just as a truncated Fourier series is an approximation to the function. You remove distracting noise in the songline, just as you remove high frequency components of a Fourier transform signal.”

Ball believes that much Aboriginal and Torres Strait Islander mathematical knowledge is alive and can still be studied by those who are willing to do so; and that the discipline of mathematics stands to be broadened and enriched by including the mathematics of indigenous cultures.

Mathematics and colonisation

In the Decolonisation discussion, Ball argued that mathematics played a role in colonisation, and colonisation played a role in shaping mathematics. To support this claim Ball discussed the two major strands in the development of her field of applied dynamical systems and stability theory: the three-body problem and the stability of the centrifugal fly-ball governor.

Ball argues that these problems were crucial to the colonial enterprise, and this provided powerful motivation for solving them. The centrifugal fly-ball governor was used to control the steam engines of the Industrial Revolution, but the instability of the governor meant that they could become hopelessly inefficient, requiring more than the labour they replaced to control them. This, Ball says, posed a risk to the wealth of the industrialists seeking to profit from the raw materials such as cotton and whale oil obtained through colonialism.

The success of sea voyages of colonial trade and exploitation depended critically on being able to calculate longitude at sea, in order to navigate safely there and back. Major European sea powers including Spain, the Netherlands, and Great Britain all offered substantial rewards for simple and practical methods of computing longitude to sufficient accuracy. Ball says this motivated efforts to solve the three-body problem, because longitude could be calculated by the lunar distance method if the position of the moon were known sufficiently accurately, and led to the development of a great deal of sophisticated mathematics.

Data sovereignty and self determination

Several speakers argued that each community should be able to determine its wants and needs and priorities, and the means by which to achieve them. This is closely connected to issues of data sovereignty. Here for example is Butler speaking in Theme 1 Reimagining the living present:

There are so many ways in which statistics have been used to present us [Aboriginal peoples] in deficit. When we talk about things like “close the gap” — a term used in Australia to try and have Aboriginal and Torres Strait Islander data for things such as health, education, housing and employment, to have those meet that of the mainstream Australian population. And again what that does for us in terms of language, is it says that the mainstream or the West is our benchmark. But that may not be what our communities see as our particular benchmark. When we look at things like NAPLAN, which is our national testing in schools, that may not be the measure that Aboriginal and Torres Strait Islander communities see as success in education. […] If we have different priorities, and our community priorities are different to what’s recognised in government reports, then government funding is not going to be shared equitably to meet our particular outcomes. So addressing issues around who’s gathering the data, for what purpose, and for what question, is really important.

Good pedagogy is good for everyone

In Theme 5 Love and Pedagogy Michael Donovan (Macquarie University; Aboriginal, of the Gumbaynggir people) told us about the findings of his 2016 PhD thesis Don16 What form(s) of pedagogy are necessary for increasing the engagement of Aboriginal school students? He asked 15–17-year-old Aboriginal students what made a good teacher, a good school, and a good curriculum; their responses mirrored what pedagogical theorists have argued for the last 50 years, showing that the best methods of engaging indigenous students are the same methods that work for all students, and are the things that students do want.

So how do we indigenise mathematics?

The symposium was never intended as the last word on this question, but rather as the first step of an ongoing process. At the close of Theme 1 Reimagining the living present Mac Lean offers a view of what this will involve.

The thing to remember, Mac Lean says, is to place indigenous peoples, and their knowledge and their thinking in the very centre of the conversation—and indeed, to let them be the ones to speak. For those of us who are part of colonising cultures—“that disrupted other cultures, and basically prevented them from continuing and creating their own future spaces that would be universities”—he says it’s a time for backing off, giving space for the conversation to happen, listening carefully, contributing our expertise where it can be of help—and importantly, working to ensure the necessary funding is made available.

Next Steps

An edited book organised around the conference themes is planned, and special sessions on indigenising university mathematics were held at the December 2021 meetings of the Australian and Canadian Mathematical Societies. A second symposium IUM2: Indigenizing University Mathematics 2 will be held at the end of November 2022, taking place at three venues: online, First Nations University of Canada, and the University of Newcastle’s Wollotuka Institute. For details see the symposium webpage IUM2.

More information can be found on CARMA’s Indigenising University Mathematics Project webpage CAR.

References

[Bis88]
Alan J. Bishop, Mathematics education in its cultural context, Educational Studies in Mathematics 19 (1988), no. 2, 179–191.Show rawAMSref\bib{bishop-1988}{article}{ author={Bishop, Alan~J.}, title={Mathematics education in its cultural context}, date={1988}, issn={00131954, 15730816}, journal={Educational Studies in Mathematics}, volume={19}, number={2}, pages={179\ndash 191}, url={http://www.jstor.org/stable/3482573}, } Close amsref.
[CAR]
CARMA, Indigenising University Mathematics Project webpage. https://carma.newcastle.edu.au/iumproject/. Accessed 2022-02-25.Show rawAMSref\bib{carma-ium}{misc}{ author={CARMA}, title={Indigenising {U}niversity {M}athematics {P}roject webpage}, note={\url {https://carma.newcastle.edu.au/iumproject/}. Accessed 2022-02-25}, } Close amsref.
[Don16]
Michael Joseph Donovan, What form(s) of pedagogy are necessary for increasing the engagement of Aboriginal school students? (English), Ph.D. Thesis, University of Newcastle, 2016.Show rawAMSref\bib{donovan-2016}{thesis}{ author={Donovan, {Michael Joseph}}, title={What form(s) of pedagogy are necessary for increasing the engagement of {A}boriginal school students?}, type={Ph.D. Thesis}, school={University of Newcastle}, language={English}, date={2016}, } Close amsref.
[IUM1]
Indigenising University Mathematics symposium webpage. https://carma.newcastle.edu.au/meetings/ium/index.html. Accessed 2022-02-25.Show rawAMSref\bib{IUM1}{misc}{ title={Indigenising {U}niversity {M}athematics symposium webpage}, note={\url {https://carma.newcastle.edu.au/meetings/ium/index.html}. Accessed 2022-02-25}, label={IUM1}, } Close amsref.
[IUM2]
Indigenizing University Mathematics 2 webpage. https://www.ium2022.org/. Accessed 2022-09-15.Show rawAMSref\bib{IUM2}{misc}{ title={Indigenizing University Mathematics 2 webpage}, note={\url {https://www.ium2022.org/}. Accessed 2022-09-15}, } Close amsref.

Credits

Author photo is courtesy of Christopher Tuffley.