A Non-Post on Indigenous Mathematics

A few weeks ago, ANU Science published a puff piece:

Maths has no borders: Professor Rowena Ball brings Indigenous mathematics to ANU

Professor Ball is an ANU mathematician and the article was a promotion of Mathematics Without Borders, a “research and teaching initiative” led by Ball. In particular, the article promoted a special topics course at ANU created and taught by Ball and fellow ANU mathematician, Hongzhang Xu, the course being “based on the Mathematics without Borders research program”. The article is bad in all the ways one can imagine.

At the time, I considered writing about this. The synopsis published by Ball and Xu, however, did not provide sufficient detail to properly understand what they taught, or how. A university puff piece in itself was also not worth the effort, not being in the same league as an article-editorial in Nature. So, and for other reasons, I decided to leave it. Then Claire Lehmann somehow happened upon the puff piece, and the fun began. Lehmann wrote an opinion piece for The Australian (Murdoch, paywalled), and the predictable bombs were lobbed. The best line was courtesy of an IPA clown during a Rita Panahi smugfest:

“Pythagoras theory, regardless of where you are, holds true.”

I was soon approached by a journalist-friend to write on the fuss. I semi-declined, since I figured it didn’t matter what I wrote during the fog of a culture battle. In any case, the culture battle didn’t last long, presumably for the same reasons I ducked the battle: beyond the silly university puff piece, it wan’t clear what was being fought over. I’m not going to go over the battle here; undoubtedly there will be next times when one can jump on, and I might then. I will, however, add a little background.

In 2010, Monash mathematician Mike Deakin had an article published in the Gazette of the Australian Mathematical Society:

Aboriginal and Islander Mathematics: Comments on one aspect of the proposed National Curriculum

As the title indicates, Deakin considered the treatment of Indigenous mathematics in the draft mathematics curriculum, the final version of which was subsequently approved in 2012.* Deakin was skeptical:

“… I will argue here that these [quoted] passages [of the draft curriculum] envisage the introduction into the National Mathematics Curriculum of a topic (Indigenous Mathematics) which, strictly speaking, does not exist.”

It is fair to say that Ball, among others, does not think highly of Deakin’s paper. In 2022, Xu and Ball posted a preprint,**

Is the study of Indigenous mathematics ill-directed or beneficial?

Xu and Ball’s paper is a direct if much belated response to Deakin’s critique:

“In this article we advance the epistemology of both Indigenous and non-Indigenous mathematics through our replies to Deakin’s five main critiques …”

Readers can decide for themselves what they think of Deakin’s paper and of Xu and Ball’s response. Most readers will have little doubt of my general opinion, and many readers will be aware that Mike Deakin, who died in 2014, was a friend of mine and the Mathologer’s. Burkard and I thought extremely highly of Mike, as a colleague and as a scholar. I cannot pretend to be objective.

I have my thoughts but I will leave it there. I might write on this in the future, depending upon how the promotion of “Indigenous mathematics” in Australia continues.


*) At the end of his paper, Mike thanks me for my “detailed comments on an earlier draft of this paper”. I cannot remember this, but I am sure my contribution was minor and was almost certainly centred on the draft curriculum more generally. I had no particular sense of “Indigenous mathematics” at that time.

**) I have no idea whether or where Xu and Ball’s paper may have been submitted for publication.


UPDATE (16/05/24)

A few days ago, an odd and lazy report on all this appeared in something called Times Higher Education. The report consists mostly or solely of grabbing quotes from the original ANU report and Twitter and the like. No big deal, and I had pondered ignoring it, but there are a couple reasons to note the report.

First, the report quotes our blog-commenter friend, Ed Barbeau, lifting sentences from the final paragraph of Ed’s comment, below. Secondly, the article quotes Professor Jessica Purcell, current President of the AustMS:

“We’re already excluding lots of people,” said Professor Purcell, president of the Australian Mathematical Society. “People…are disengaged because they just don’t see themselves in it.”

Professor Purcell said mathematics was broader than many people realised and, while it contained universal truths, there were many different ways of articulating them.

“It may not be somebody typing on a calculator [but] these more indigenous cultures [are producing] similar conclusions. We may not understand the methods…but it’s still mathematics.”

I’m less than thrilled if the President of AustMS wrote or said such things, but without further context I’m not going to dwell upon it. There is no indication that the reporter interviewed Purcell, and I have no idea of the origin of the quotes. So, while there is no reason to doubt the accuracy of the report, I cannot verify the quotes and know nothing of the context in which they might have been uttered.

22 Replies to “A Non-Post on Indigenous Mathematics”

  1. About 50 years ago, my university established a program to enable aboriginal students to study science and there arose the issue of appropriate mathematics. I wrote to a guy in Arizona who worked with the Navajo and also one in London Ontario who was generally interested in indigenous knowledge. The Navajo are a more settled society with some commerce, so in keeping with other societies developed strategies for dealing with arithmetic involved. However, how the mathematics develops depends on the efficiency of the notation and the algorithms, and this is where the location of its origins counts. However, keep in mind that every great idea is due to some particular individual or small group, and this can happen anywhere. To the extent that any group is to systematize notation and algorithms, it has the ability to begin to ask more general and abstract questions, as we have seen with Chinese, Indian, Greek, Arabic as well as European contributions to the subject.

    A lot of North American indigenous “nations” are hunter gatherers which in particular means that they are not in organized agricuture or living in urban areas which requires the keeping of and processing arithmetic information, as well as transmitting it over space and time. Knowledge is transmitted orally, and skills are passed directly from one person to another.

    However, this does not mean that one can simply ignore the indigenous needs in a mathematics class. However, it can be subsumed under the general principle that it you want to teach any kid you have to meet them where they are. One place where mathematics and indigenous culture meet is in the arts and games, both of which involve mathematics structure and combinatorics in some form. However, there is a commonality in both across cultures, so that you can have diversity and commonality in the same situation.

    The history of mathematics teaches us that when a superior method of doing something comes along, it is generally adopted on its own merits no matter who you are. Our number system goes back to India (through the Arabs); the algebraic notation invented in 16th century Europe is universally used. To pretend that our present mathematics is an offshoot of colonialism is not only silly, but potentially harm the very people that we claim to help.

    1. Thanks, Ed. Of course questioning the merits of Aboriginal-Islander mathematics in no way implies that special concern should not be made for Aboriginal-Islander students, whose education (and health and everything) is a general, undeniable mess. The decolonisation people seem to argue that the promotion of Indigenous mathematics is a natural, and perhaps essential, element of improving matters. Independent of the questions of intellectual integrity, I’m skeptical.

      In Australia, I am aware of two prominent Aboriginal-led movements for improving Aboriginal education, that led by Chris Matthews and that founded by Noel Pearson. In very brief summary, Matthews, who has a PhD in mathematics, is strongly in the Aboriginal culture camp, and Pearson is strongly in the buildings-and-basics camp. I have a much higher regard for Pearson.

      In terms of a hunter-gatherer society being less concerned with and less able to transmit formalised knowledge, Australians will be aware of the very strong push, led by Bruce Pascoe, to recast Aboriginal-Islander society as more geographically stable and agriculture based. I believe that the book by Sutton and Walshe has thoroughly demolished Pascoe’s work, but of course others disagree.

  2. Some of the examples of indigenous mathematics that I have seen are not really about indigenous mathematics. They are more about ideas in the traditional lives of indigenous peoples that have sparked an idea about the mathematics in the curriculum. For example, a game played in indigenous communities might spark an idea for an exercise on probability. Such an exercise might enhance the knowledge that students have about the lives of indigenous peoples, or demonstrate to students that mathematical ideas can be found in many contexts.

    1. Thanks, Terry. I wrote in detail hereabout Aboriginal-Islander games as they appear in the Australian Curriculum. Whatever might be possible and reasonable, the elaborations in the Curriculum are absurdly contrived.

  3. It seems that Ball has a different definition of what mathematics is. Does knowing the physical methods to create twin counter-rotating vortices in smoke signal mean you are doing the mathematics associated with it? Ball seems to believe that doing the physical process is mathematics. Or that drawing a shapes in art is mathematics. It is strange, as when I throw a ball I am not doing any projectile motion calculations in my head (even though I know how to do them on paper). Throwing a ball is a physical process, nothing to do with understanding the mathematics of it.

    1. Thanks, Potii. There is no question that the decolonisation people are redefining “mathematics” (and “science”). I think they are mostly explicit about doing so, with the claim that the accepted, narrow definition excludes too much. Of course I disagree.

      One can be open to a broader notion of “mathematics”, and the history of mathematics contains many examples of mathematical concepts being applied, usually clumsily, before later mathematicians better understood and better expressed the ideas. But that does not mean it is helpful or honest to characterise any activity that contains a homeopathic element of mathematics as being “mathematics”.

  4. I have wondered a lot about this but know very little. So here’s where I’ve got to. It may not be very informed, but here goes ….

    I have almost no knowledge of Aboriginal culture, just that it is the oldest continuous human culture on the planet. And that Australia under First Nations stewardship was biodiverse and environmentally rich (perhaps after Aboriginals having caused the extinction of mega-fauna tens of thousands of years ago). So First Nations culture knows how to do something rather extremely important that Western culture can’t do and deserves a great deal of respect in its own right as containing great wisdom and knowledge. Europeans have things to learn ….

    I read the excellent book Songlines, part of the First Knowledges series, and was completely blown away by the sophistication of such an oral culture. This included the ability to navigate across the country where people had never been before through oral transmission of knowledge. There are many other skills Europeans can only marvel at. Bill Gammages’ book ‘The Biggest Estate on Earth’ sets out the sophistication of the First Nations landscape management regime.

    What was striking about to me in reading ‘Songlines’ was the complete alien-ness to my Western upbringing of Aboriginal knowledge systems. That knowledge was developed in a completely different manner (oral culture) for totally different purposes in an utterly different context from a European written based culture.

    My sense is that fitting Aboriginal knowledge into Western knowledge ‘buckets’ such as mathematics doesn’t work. It seems to then become rather tokenistic (like calculating the square km of an Aboriginal homeland). It may denigrate the sophistication and power of First Nations knowledge, which is sophisticated in achieving the goals it is designed for. Europeans find it very hard to appreciate that knowledge since it’s so different from what we are used to.

    And of course mathematics is fundamentally a written subject (we as maths teachers know such limitations well as opposed to those who advocate lots of IT in the maths classroom). Maths is not a subject basic to an oral culture, nor likely to achieve many of its goals (as evidenced by the fact that they didn’t use what Europeans call maths as far as we know).

    Surely more appropriate than token studies would be the rigorous study of First Nations culture in its own terms as a subject in its own right.

    However so much Aboriginal knowledge has been lost (particularly here in the south) because Europeans drove First Nations people off the land and killed them. So we seem to have neither the knowledge nor the teachers to be able to convey a deep understanding of the culture and knowledge systems.

    Instead of pursuing such rigorous development or acknowledging the limitations, the whole thing seems to become a go through the motions exercise of ticking a box. For example as mentioned above we get the maths incidentally occurring in ‘twin counter-rotating vortices in a smoke signal’ as Aboriginal maths, rather than a deep study of why these vortices were used, what was achieved and the techniques and level of sophistication involved in creating such vortices.

    Ironically one of the things that we have lost in recent maths education is use of techniques based in oral culture such as rote learning and chants – so I don’t hold out much hope!

    I hope this makes sense – happy to be corrected.

    1. Thanks very much, JJ. Your knowledge of Aboriginal culture may be weak, but it’s way stronger than mine. I’m rushing now, and will try to get back later in more depth (or at least more length), but a quick comment.

      I’m wary of some of the things you wrote, but I understand the point you’re making, which was also very much the theme of Sutton and Walshe’s book. Along with debunking many of Pascoe’s specific and general claims, Sutton and Walshe (mainly Sutton) were clearly furious at what they regarded as the denigration of Aboriginal society, of Pascoe attempting to shoehorn Aboriginal society and understanding of the world into Western measures of progress.

      1. I understand that Tyson Yunkaporta is regarded as a leading indigenous thinker. I’ve read his book Sandtalk and didn’t understand a great deal of it as it was complex and alien, but could see it had real value. I have also heard him on podcasts and he speaks a lot of sense.

        I’ve just read the attached paper, and they don’t think much of the national curriculum! So he shares at least come of this website’s concerns!

        The paper includes the following quotes:

        “The findings of this initial analysis raises serious questions about ACARA’s assertion that they intend to provide all students with the opportunities to develop a deep understanding of the histories and cultures of Aboriginal and Torres Strait Islander peoples.”

        and a brilliant quote here: “The only two [Mathematics] Content Elaborations in the secondary curriculum appear in the analysis of statistics and data in Year 10, where students are asked to mathematically compare Aboriginal and Torres Strait Islander populations with the Australian population as a whole. There is no embedded expectation on teachers to provide an opportunity for students to engage in any critical analysis that would lead to an informed understanding why such a statistical discrepancy exists between data sets for Indigenous
        and non-Indigenous Australians.”

        What a fantastic case study it would be to explain to Year 10s that data must be set in real world context. I had indigenous people explain to me when doing such data in Govt, that of course if your forebears had been affected by the Stolen Generations, as a mother in hospital, you would be rather reticent in nominating your baby as indigenous.

        “ACARA has fallen well short of its own stated goals, and responsibilities to provide the content needed by teachers to address the points of tension between the divergent positions of Indigenous and non-Indigenous Australians. Instead, ACARA has timidly opted to shift this responsibility to the often ill-resourced classroom teacher who, unsupported by the curriculum.”


        1. Thanks, JJ. I think it is perfectly reasonable and very good to try to give students a proper sense of Aboriginal culture and history and prehistory. The two problems I have are:

          1) Mathematics classes are for learning mathematics. Lowe and Yunkaporta may be right about the lack of any such critical analysis of Aboriginal issues, but it has no proper role that I can see in a mathematics curriculum.

          2) This has to be done with respect for truth, not for the overwhelming concern for championing the (genuine) underdog, which is the standard. I’d suggest that your call for the “rigorous study of First Nations culture in its own terms as a subject in its own right” is impossible, contradictory and a Pandora’s box. It’s impossible because “rigorous” is hardly the proper word for study in high school; recognise the limitations. It’s contradictory because the study of Aboriginal culture “in its own terms” makes no sense; the respect of Aboriginal culture on its own terms doesn’t change the fact that we are seeking to analyse it, in a modern “western” manner. It is a Pandora’s box because any serious study of Aboriginal culture is going to bring to light what was always, in aspects, a very tough and at times very brutal existence; and, it will bring to light that the colonial paternalism, which is now so much denigrated, was not all bad and certainly not all badly intentioned, and that the decline in this paternalism has had some grave consequences.

          On point (2), there is a massive issue here of who one should trust. Chapter 2 of Sutton and Walshe’s book was an eye-opener for me, presumably in the same way that Songlines for you. It gave me a sense of the otherness of Aboriginal’s perception of the world that I hadn’t really read clearly before. I believe there is something there. But what, exactly, I am very hesitant to suggest, and I have absolutely no clue who to trust to find out better. I don’t trust Aboriginal activists, for obvious reasons. I also trust modern anthropologists to the same extent that I trust modern maths ed academics, and for very similar reasons.

          I am open to reading something worth reading. But I’ll have to be reasonably convinced it’s worth reading before reading it, and I don’t know how you’ll do that.

  5. Hi Marty

    I phrased the stuff about understanding more of Aboriginal culture very poorly. I was trying to say to take tokenistic things out of maths and put them in a proper subject – noting as you say that such a subject would be very difficult, especially in modern society where nuance and truth tend to take a back seat to tribalistic positions.

    But it’s not just that, we Europeans are taking a very very long time to appreciate the knowledge and wisdom in Aboriginal culture (not to mention our role in its destruction) so it’s not surprising there is very little out there. A school textbook would be ideal, but it would appear to be a long way off. But there is progress – I haven’t yet read the other books in the First Nations Knowledge series, but the Songlines one was excellent. And the numbers of Aboriginal university graduates are growing ….

    In the meantime, I think it would be interesting to have a genuine discussion with someone like Yunkaporta. From what I can tell, he is the sort of person who gets nuance and is open to reason.
    Indeed, Yunkaporta identified what I see as a current gap in maths education and I was impressed by his understanding of the need to explicitly teach students the notion that data has context – particularly since he is no statistician, nor data analyst. In fact this was something I only really learnt on the job.

    I would very happily drop the ‘left skewed’ and ‘right skewed’ stuff that I never saw used in the real data world and spend a period on this (and other) examples of how data is ‘organic’ and its origins really matter.

    1. Hi JJ. I’m not trying to pick on you, but all of this makes me wary. How much “knowledge and wisdom” is there to learn from Aboriginal culture? Honestly, I don’t know, but I’m not going to assume a lot any more than I’m going to assume nothing. And as I wrote, I have no idea whom to trust.

      When every writer in a discipline has an agenda, it is very very difficult for a novice to get a handle on it.

      1. Hi Marty – my thoughts – not particularly well expressed ….

        Firstly Aboriginal culture is the oldest living culture on the planet, estimated at over 60,000 years. That’s quite an achievement. And what we are now discovering is just how sophisticated it was. Aboriginals managed to have an environmentally biodiverse and rich landscape under their management and they developed it over tens of thousands of years. We have managed to significantly reduce biodiversity and environmental quality in less than 250 years and it’s not going to stop until the environment stops us I suspect. Western culture is destroying the very environment that sustains it (it’s clear that the environmental crisis is more than climate change). Under Aboriginal stewardship, the landscape was enriched and sustainable (whatever that means). That alone would indicate that there is something to learn. But from what I’ve heard there is much much more – perspectives that should make us think and reflect that there is not only one way of seeing/doing things …..

        Secondly I believe that students as citizens should learn about many things (not just those that are practical), and as the oldest living culture is a fundamental part of Australia, it is worthy of study by all Australians. It is a culture that has grown out of our landscapes and is fundamental to this place where we live. We need to tell the whole story of Australia and its human history, and Aboriginal culture makes up 98% or so of that. Bit hard to ignore ….

        Having the pendulum gone from a completely negative view of the culture (‘primitives’) to nearly perfect, at some point hopefully there will be a more balanced view (which is what people like Sutton, Yunkaporta and various others seem to be developing). This understanding hopefully reflects the fact that Aboriginal culture is a human one (not one of angels) and has strengths and weaknesses like all cultures and is adapted to an environment that shaped it.

        In the meantime, it appears that we need to muddle through and identify the thinkers who make sense to us and aren’t culture war warriors.

  6. Indigenous mathematics refers to the mathematical knowledge and practices developed by indigenous cultures around the world. It encompasses the unique mathematical systems, concepts, and problem-solving methods that have been passed down through generations within indigenous communities.
    Indigenous mathematics often differs from the traditional Western mathematics that is commonly taught in schools. It reflects the deep connection between mathematics and the cultural, social, and natural environments of indigenous peoples. Indigenous mathematical knowledge is rooted in the specific contexts, traditions, and ways of life of these communities. materi pelajaran sekolah

    1. Thanks very much, Ron. Yes, I also chased down the eclipse thing, and was (of course) similarly unimpressed. Hellier’s post about Ball also appears here, on his blog.

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