Introduction

Indigenous sciences, in all their diversity, depth, and vitality, are actively developed by First Nations practitioners and increasingly respected by mainstream scientists. Global recognition of and advocacy for Indigenous sciences and technology have enriched fields such as ecology (Bardsley et al., 2019), geography (Ioris et al., 2019), geology (Larkin et al., 2012), astronomy (Hamacher et al., 2020), hydrology (Moggridge et al., 2022), and climate change (Leal Filho et al., 2022; Whyte, 2018). As scientific approaches to nature, technologies, and human society seem cross-culturally universal, thinking and communicating mathematically also is likely to be a universally human practice (Verran, 2000).

Yet Indigenous mathematical knowledges often have been declared deficient, or non-existent, and their study and teaching questioned and denied validity (Pais, 2011). Justifications have included assertions that some Indigenous cultures had, or have, no history or traditions of scripted language and systematic numeral systems, and usage of ‘rule of thumb’ measurement (Harris, 1987). Chevallard (1990) denied that mathematics developed inside African, Asian, and American-Indian cultures, and labeled Indigenous physics and mathematics as proto-physics and proto-mathematics, respectively, in contra to ‘fully-fledged’, presumably Western, science and mathematics. Rowlands and Carson (2002) regard mathematics of the people — or vernacular mathematics — as non-rigorous and informal, being distinctly different from ‘elite’ mathematics, i.e. academic or pure mathematics. Pais (2011) proposes that school should only provide students with decontextualised universal knowledge by excluding cultural components.

In response to a (then) draft proposal to include Indigenous mathematical perspectives in a revised Australian national school curriculum, Deakin (2010) argued that ‘the topic of Indigenous mathematics does not exist…attempts to discover an Indigenous Mathematics are…ill-directed…There is no indigenous tradition of mathematics, properly so-called, in this country [Australia]’. Deakin’s views represent a strong strand of active or passive resistance to Indigenous mathematics in Australian education and academia, influenced by complex historical, sociopolitical, and cultural factors and reflecting the role of mainstream mathematics in maintaining cultural imperialism in a colonial-settler state (Bishop, 1990).

Michael Deakin (1939–2014) was a senior lecturer and an educator of pre-service mathematics teachers at Monash University, Melbourne, from 1973 to 2003. He founded, edited, and was the main writer for the magazine Function,Footnote 1 which was aimed at mathematics teachers and high school students. In these capacities and activities, Deakin undoubtedly impacted generations of students and teachers (Polster & Ross, 2014). Although more than a decade has passed since publication of his article in The Australian Mathematical Society Gazette, the views expressed therein have been reiterated for generations (Thomas, 1996) and have persisted in mainstream mathematics, by default or indifference if not actively, and as a result have been reflected in school and university mathematics curricula, activities and positions taken by mathematics professional bodies, and educational policy and ideology. We focus on Deakin’s article not to demonise Deakin, who was mostly a positive influence on mainstream mathematics education in Victoria (although his negative influence on Indigenous education in mathematics was pervasive in Australia and Papua New Guinea), but simply because it is exemplary in that it encapsulates and expresses the most commonly held misconceptions about Indigenous mathematical knowledge.

In this article, we advance the epistemology of both Indigenous and non-Indigenous mathematics through our replies to Deakin’s five main critiques, which are tabulated in the ‘The Gatekeeper Critiques of Indigenous Mathematics’ section. In doing so, we review some of the most relevant literature on Indigenous mathematics. We refer to the critiques generically as ‘the gatekeeper critiques’ for three reasons:

  • They are representative expressions which reflect a dogma of exclusiveness of mathematics to a British-European tradition that pervades the community to this day.

  • This dogma is defended actively by guardians, or gatekeepers, against any perceived illegitimate incursions or subversions.

  • The ‘gatekeeping method’ of controlling access and inclusion is used to disallow the legitimacy of Indigenous mathematical knowledge.

In the ‘Reply to the Gatekeeper Critiques’ section, we examine the gatekeeper critiques critically through the questions: What is mathematics? and Who can do mathematics? In the ‘Discussion’ section, we propose a new rapport whereby Indigenous and non-Indigenous mathematics may inform each other and advance together.

The Gatekeeper Critiques of Indigenous Mathematics

In this work, we adhere to the United Nations’ definition of Indigenous peoples, presented in Box 1. We also use the accepted synonym, First Nations peoples, and in the Australian context, Aboriginal and Torres Strait Islander peoples.

Box 1. United Nations’ definition of ‘Indigenous’

José R. Martínez Cobo, the Special Rapporteur of the Sub-Commission on Prevention of Discrimination and Protection of Minorities for the United Nations (2004, p. 2), defined Indigenous communities or peoples as:

     Those which, having a historical continuity with pre-invasion and pre-colonial societies that developed on their territories, consider themselves distinct from other sectors of the societies now prevailing in those territories, or parts of them. They form at present non-dominant sectors of society and are determined to preserve, develop and transmit to future generations their ancestral territories, and their ethnic identity, as the basis of their continued existence as peoples, in accordance with their own cultural patterns, social institutions and legal systems.

In discussing published evidence of mathematics practised by Australian First Nations, Deakin questions and dismisses or impugns several reports on Indigenous mathematical knowledge, concluding that the examples described most definitely are not mathematics. These ‘gatekeeper critiques’ of Australian First Nations mathematics are presented in Table 1.

Table 1 The gatekeeper critiques (Deakin, 2010) (page numbers in the table refer to those of the cited article)
Full size table

It is surprising that the gatekeeper critiques have not to date been challenged in the literature. Among our mathematical colleagues, most recognise that there are well-documented non-Eurocentric traditions of mathematics, such as Inca (Ascher & Ascher, 1969), Mayan (Peat, 2002), Chinese (Tiles, 2002), and Indian mathematics (Joseph, 2009). For some, however, the notion of ‘Indigenous mathematics’ as systems of knowledge utilising symbolism, abstraction, classification, and quantitation is going a step too far.

Reply to the Gatekeeper Critiques

Some denigratory statements on Australian Indigenous mathematics have been rebutted by Stokes (1982) and Harris (1987). In the following replies, we review selected mathematical knowledges of Australian and non-Australian First Nations peoples, as a step towards advancing wider understanding and appreciation of Indigenous mathematical knowledge.

Critique 1. No Symbolically Scripted Language, Ergo, No Mathematics

It is often assumed that mathematics cannot be done without a system of written language expression. This is because mathematics is believed to be an abstract, context-free pastime tied to a formal system which relies upon canonical sets of symbols that have to be scripted onto an external medium. This idea has led to a view of mathematics as a set of ultimate truths existing in the æther, outside ordinary human activity and devoid of social and cultural considerations (Millroy, 1991).

However, it is just as often forgotten or ignored that mathematical expertise, for example in arithmetic and geometry, was originally developed to solve real problems that arise in daily life (Ascher & Ascher, 1986; Sizer, 2000). Concepts of number and geometry are linked with social structures, ceremonial practices, artwork, and interaction with the environment (Lloyd et al., 2016). In any culture, there is an agreed, structured, and self-consistent system to classify, transform, quantify, communicate, and predict patterns and cycles of importance, whether it be in unwritten or written forms.

An illustrative case of sophisticated unwritten mathematics is that of the Yolŋu peoples of north-eastern Arnhem Land in Australia. Yolŋu mathematics systematises the relationships, cycles, and patterns that govern human and natural activities and is deeply intertwined with daily life and cultural practices (Lloyd et al., 2016).

Much of ordinary day-to-day arithmetic and geometry carried out on the job, formerly by non-literate and unschooled workers, to this day are idiosyncratic to particular trades, unwritten and even unspoken (Wood, 2000). The apprentice learns by watching carefully then doing the mathematics themselves. The use of tools — an unwritten approach — to support arithmetic has a long history; there are different media for recording and computing with numbers, including stones, twigs, knots, and notches (Hansson, 2018).

Peoples of many Indigenous Pacific and Australian nations used — and can use still — parts of the body to count quickly and accurately (Goetzfridt, 2007; Owens & Lean, 2018a; Wood, 2000), communicating methods, operations, and results through speaking, listening, and gesture.

Weaving skills were taught orally and by demonstration to train apprentice weavers in constructing the numerical relationships that give rise to the desired complex geometrical designs with symmetries (Hansson, 2018). Knotted quipus were used by non-literate Inca people of South American Andes regions to allot land and levy taxes (Ascher & Ascher, 2013). The quipu with its columns of base 10 numerical data encoded as knots can be thought of as a spreadsheet, and it seems likely that the Inca knew and applied some array and matrix operations.

Dan, an Indigenous language of central Liberia, is non-written but Dan speakers can carry out arithmetic operations orally, including addition, subtraction, and division; play games that require fast counting, tracking, and calculating skills; and practice geometric principles in constructing buildings (Sternstein, 2008).

Fractal geometry, developed to a high art in Western mathematics from the late 1960s and executed in silico, has non-Western antecedents that were implemented in the built environment in Africa (Eglash, 1998). Chaology and fractal geometry have also been a part of traditional Chinese architectural and garden design for thousands of years (Li & Liao, 1998).

Moreover, Latin-European mathematics — which became mainstream — originated and was carried out in unwritten forms. Verbal or non-written approaches played important roles in ancient Greek mathematics, which relied on diagrams, sculpture, and verbal echoes. References and expansions were chanted orally (Netz, 2005).

We need to understand that mathematics is socially constructed in the context of a community, where meaning is negotiated, and conventions are agreed upon certain groups of people. Owens (2020) and Paraide et al. (2023) have recorded ways in which Indigenous peoples use non-written methods for systematic spatial reasoning about patterns and relationships in the environment. Writing or scripting mathematical operations and processes is the dominant approach today from Western mathematics but it is not the only one.

Critique 2. No Coherent System of Numerals

Pursuant to the above discussion, we have diverse and well-documented evidence that societies nominally without writing systems have well-developed and useful numeral systems. Is that mathematics? Not so, according to Deakin, who uses the terms ‘proto-numerals’ and ‘proto-mathematics’ to claim that Australian First Nations numeral systems are not mathematics because they cannot support basic arithmetic manipulation.

In fact, the ‘turtle egg arithmetic’ practised by Yolŋu people of north east Arnhem land, selected by Deakin as an example of imprecise or ambiguous use of number counting, gives perfectly accurate results; rather, it appears that Deakin’s understanding of this system is inaccurate. Cooke (1991, p. 12) and Lloyd et al. (2016, p. 7) explain the word ‘rulu’, which means a group or bundle in general. However, prefix and suffix will be added to scope down its meaning to a specific number. For example, ‘wangany-rulu’ means five. A closer look at the Yolŋu turtle eggs sharing strategies finds that subtractive division in base 5 is the method applied, as illustrated in Fig. 1.

Fig. 1
figure 1

Visualisation of Yolŋu people’s turtle eggs distribution process in a base 5 numeral system

Full size image

We mention also the following examples that are well-documented:

  • The Anindilyakwa people of Groote Eylandt, Northern Territory, Australia, use systematic numerals up to 20 in their spoken language (Harris, 1987), i.e. a base 20 system.

  • Glendon Lean’s research on nearly 900 counting systems of Papua New Guinea (PNG), Oceania, and Irian Jaya provides a wide range of counting and tallying systems with diverse cycle systems, including cycles based on 2, 5, and 10, and discrete systems including (1, 2), (2, 5) (2, 5, 20) cyclic number systems (Owens, 2001). The cyclic number systems such as (2, 5, 20) use only the cycles of 2, 5, and 20 to generate the number words. For example, in this system, the numbers 3 and 4 might be represented as difference between 2 and 5, and combinations of 2, respectively; the numbers 6 to 9 might be represented as differences of 4, 3, 2, and 1 from 10 or additions on 5 or doubles of 2 to 4 or doubles plus 1 while numbers between 2 and 5 might be combinations of 1 and 2 or include a specific numeral name depending on the people’s language and village (Owens & Lean, 2018b).

  • Research on body-part tally systems in PNG finds more counting systems, ranging from base 27 to 35 (Dwyer & Minnegal, 2016). When communities engaged in trade, they often adapted their numerical systems, integrating features from neighboring trading partners. For instance, some base 10 systems adopted characteristics from base 5 systems, while others with (5, 20) cycle systems incorporated a single 10 instead of two sets of five (Owens & Muke, 2020).

Diverse number systems have been developed across different cultures, and they offer unique and effective ways of understanding numbers and mathematical relationships (Owens & Lean, 2018b).

Critique 3. No Measurement of Time and Space

Although Deakin acknowledges that ‘astronomy has historically been seen as a branch of mathematics’, he shuts the gate against Indigenous knowledge on astronomy by claiming it makes no measurements of time and space and therefore is not mathematical. Such misconceptions are still prevalent, and have been challenged by Hamacher and Norris (2011). The connections between Indigenous astronomy and mathematics are becoming clearer. Much richer Aboriginal and Torres Strait Islander knowledge of astronomy has been revealed, through what Deakin refers to as the ‘elaboration of stories’ (Noon & Napoli, 2022). Indigenous knowledge has shown an insightful understanding of the earth-moon-sun system and its correlations with terrestrial events, including seasonal cycles and weather forecasting (Bureau of Meteorology, 2022; Hamacher et al., 2019; Norris & Hamacher, 2009). The Yolŋu people always have understood the relationship between the Moon cycles and the ocean tides. Yolŋu Elders can predict the time and height of the next tide after seeing the position and phase of the Moon, which contrasts with Galileo’s incorrect explanation of the tides (Norris, 2016).

But Deakin has devised an infallible test for the existence of Indigenous mathematics. This is that there must be ‘an Aboriginal method of predicting eclipses’. Very well, let us assume this is a legitimate test, rather than the distorted and fallacious argument that it is. To predict an eclipse, one needs clear and accurate understanding of the relationships between the motions of the Sun and Moon, and there is evidence that this knowledge was held by some Indigenous groups. Hamacher and Norris (2011) report a prediction by an Aboriginal woman in Roebuck Bay, Western Australia, of a solar eclipse that occurred on 22 November 1900, which was described in a letter dated in December 1899.

Spatial coordinate systems are well-defined in most First Nations societies. The use of cardinal directions (north, south, east, and west) is common among Aboriginal language groups in Australia (Norris & Harney, 2014). Guugu Yimithirr language speakers use cardinal directions rather than concepts of left, right, behind, or front to determine objects spatially and in time (Norris, 2016). Thus, the mathematical concept of vectors is strongly embedded culturally. Strong directionality and astronomical knowledge enabled Aboriginal people to undertake long-distance travel and develop extensive trading networks (Forster, 2021; Norris & Harney, 2014).

It is questionable simply to deny the existence of mathematics by ignoring, or attributing to chance, the mathematical thought and practice evidenced by these capacities.

Critique 4. No Mathematics in Kinship Rules

Research on Aboriginal kinship rules is one of the first bridges between Indigenous and mainstream mathematics. Kinship rules were first described in Western mathematical terms by Mathews (1900), and many studies since have found the existence of group theoretical knowledge in such systems (Keen, 1988; Laush, 1980; Radcliffe-Brown, 1930). A probabilistic approach by Field (2021) has revealed the intergenerational social trade-offs and benefits of the Gamilaraay kinship system, and the simpler aspects of the mathematics of Yolŋu kinship have been elucidated for a teaching audience by Matthews (2020).

Broadly, group theory is a suite of largely non-numerical mathematics involving canonical classifications of sets of objects with defined mathematical properties. A group is a set G with an operator that takes two elements of G and combines them to produce a third element of G. Groups appear throughout mathematics and the sciences, and in human relationship systems.

Deakin denies the group theoretical structure of kinship rules on the premise that, since group theory was believed to be invented by European mathematicians, mathematical studies of kinship rules are merely an application of European theory. Algebraic analyses of kinship Cargal (1978) are bridges between Aboriginal knowledge system and modern mathematics but are misinterpreted by Deakin (p. 235) as ‘pseudo-mathematics’.

However, Deakin’s assertion largely ignores other research, such as that of Boyd (1969), which reveals and analyzes intricate structures within Arunta and Kariera kinship systems by using group theory methods. These kinship rules often include complex relational dynamics, analogous to the permutations and operations carried out on algebraic structures (Boyd, 1969; Rauff, 2016). de Almeida (2019) translated the kinship terms and rules of Cashinahua people into algebra and compared the structure to modern mathematical theories like the dihedral group in group theory. A more thorough exploration and recognition of such scholarship could have potentially reframed Deakin’s understanding.

Not all rules within a given kinship structure are classification rules and have a group theoretical interpretation though. Such irregularities are associated with other cultural considerations, such as totemic ancestors, and are the ‘doubts’ and ‘exceptions’ mentioned by Deakin. But Aboriginal knowledge is an integrated system not a discipline-segregated one such as mainstream mathematics (Lloyd et al., 2016; Sizer, 2000). Understanding kinship structures needs mathematics but cannot be done by using only mathematics. Denham (2013) uses both mathematics and biology to illustrate how kinship systems would have helped Australian Aboriginal societies, prior to colonial history, respond to a food crisis. Field (2021) uses biological principles and mathematics to demonstrate how the Gamilaraay kinship system has minimised incidence of recessive diseases.

Rather than simply denying identified connections between group theory and kinship rules, as Deakin has done, we should welcome the likely expansion of mathematical knowledge by studying these connections. Kinship rules are a clear indication that First Nations societies mastered some powerful principles of mathematics and biology to live and manage the community better. An interdisciplinary or even transdisciplinary approach enables us to explore how these kinship structures interact with various aspects of life — social, cultural, environmental, biological, and more. Studying these ‘exceptions to the rule’ may offer new lessons and insights for various disciplines, including mathematics.

Critique 5. Inability to Predict Complex Weather Patterns and Events

Gatekeeping is again the main forte of the fifth critique, which relies on the belief that only Western culture has developed an advanced system to understand weather patterns. This fallacy often stems from the assumption that meteorological understanding is exclusive to Western science due to its systematic collection and analysis of data. Deakin wrongly perceives Indigenous knowledge as primitive or inferior because it does not follow the same methodologies, is not recorded in scientific literature, and is typically disseminated through oral traditions, stories, or ceremonies. This misconception dismisses the wealth of intricate, place-based, and time-tested knowledge that Indigenous cultures have developed over thousands of years of close interaction with their environments.

Indigenous meteorology relies on observational knowledge of the skies and plant and animal cycles (Green et al., 2010), and uses pattern recognition and probabilistic reasoning, which are crucial mathematical thinking skills.

Pattern Recognition, Probabilistic Reasoning, and Prediction

Qualitative, or non-numerical, mathematical knowledge involved close observations of patterns in the behaviour of animals and plants and in the appearance of celestial bodies due to atmospheric moisture, and linking these patterns to imminent changes in weather and the variable advent of seasons such as monsoonal rains. Stellar scintillation and colours, due to moisture and ice crystals in the atmosphere, have been used by Torres Strait Islanders to predict weather and seasonal changes (Hamacher et al., 2019). Scintillating stars that appear sharp to the eye indicate dry, clear skies. When most of the visible stars are blue, atmospheric moisture is indicated as the red and green wavelengths of light are absorbed by water vapour. Animal and plant proxy indicators also were, and are, used: Torres Strait Islanders can predict windy and rainy days by observing the movements of sharks (Beizam) (Green et al., 2010); Walabunnba people predict heavy rains when the mirrlarr (rain bird) calls out; the flowering of the boo'kerrikin (Acacia decurrens) indicates for the D'harawal people the end of the cold and windy weather; it is well-understood that if certain plants flower early there is a high probability of a wet season (Bureau of Meteorology, 2022).

Indigenous understandings, measurement, and predictions of the weather and season change include comparisons and syntheses of these varied, often independent, observations (Clarke, 2009). Rather than denying and ignoring the merits of First Nations meteorology and mathematics, co-production of weather and season knowledge with Indigenous knowledge holders and experts may add new insights into our existing forecasting system (Bureau of Meteorology, 2022). May the algorithm be improved, and computing speed be accelerated by accounting for First Nations knowledge? Could non-meteorological observations, such as plants and animals’ behaviour changes, improve existing weather models? These potentials cannot be realised if we disrespect the knowledges of First Nations cultures.

Discussion

A Dilemma

Our responses above indicate that the gatekeeper critiques on Indigenous mathematical inadequacy are not rooted in either sound evidence or clear logic.

The mathematical ideas of Indigenous people have been disregarded as part of the colonial strategy of gatekeeping the periphery of structured scientific knowledge (Ball, 2015; Goetzfridt, 2007). The Doctrine of Discovery, sanctioned by Pope Alexander VI, absolved Spain and Portugal’s colonisation of the Americas, asserting their Christian superiority and legitimising the appropriation of non-Christian territories. This decree not only enabled European colonial expansion but also marginalised the Mayans’ mathematical and astronomical knowledges. In an act of strategic cultural suppression, European invaders destroyed the Mayan libraries and destroyed most of the codices (Rosa & Orey, 2008). But the surviving texts reveal the sophistication of Mayan knowledge of astronomy and mathematics, especially their advanced understanding of zero, and the greater cycles of time and numbers (Batz, 2021).

A legacy of European colonialism is the predominant belief that mathematics is defined by Western mathematics, which is the privileged manifestation of rationality of the human species (Goetzfridt, 2012). To exclude research on Indigenous mathematics from ‘real mathematics’, the terms pseudo-mathematics, proto-mathematics (Thomas, 1996), and ethnomathematics (Ascher, 2017; Ascher & Ascher, 1986) were coined.

Ethnomathematics has emerged as a scholarly response to dismissive perspectives that relegate Indigenous mathematical systems to the realm of pseudo-mathematics (Ascher & Ascher, 1986). Ethnomathematics accentuates the inextricable links between cultural contexts and mathematical practices, thereby challenging the monolithic and Eurocentric view of mathematics as a singular, universally applicable discipline in its current presentation (Ascher & Ascher, 1986). However, ethnomathematics has attracted controversies. Ethno-mathematicians believe that mathematics is culturally specific and each cultural group develops its own ways and styles of explaining, understanding, and living with the help of their mathematical knowledge (Ascher, 2017). However, this focus on ‘culture-specific mathematics’ has unintended consequences, deepening beliefs that studies of Indigenous mathematics add nothing to mainstream mathematics (Vithal & Skovsmose, 1997). Ethnomathematics is regarded by many mainstream mathematicians as a counterpoint to the universalism and internationalism of mathematics (Pais, 2013), and has been denigrated in international opinion media as a ‘fringe absurdity’ (Brooks, 2021).

A Way Forward

If we revisit Deakin’s critiques of Indigenous mathematics, we see that his overall argument is deductive and follows a top-down approach in which his conclusion is based on a premise that is assumed to be true (Weddle, 1979). The premise relies on answers to the questions: What is mathematics? Who can do mathematics? Deakin does not answer these questions but assumes that the reader knows and agrees with him. This lack of clear premise makes his critiques dubious and subjective at the outset. Here we clarify our understandings of these two questions.

  • What is mathematics?

We guess that Deakin’s understanding of mathematics was close to the mainstream understanding of school mathematics, which is defined by Yeh et al. (2021, p. 200) as ‘a discrete set of topics taught in a linear progression that ultimately leads to calculus’. Yeh et al. define mathematics as ‘a living, ever-changing body of knowledge and set of cultural practices involving human activities of counting, locating, measuring, drawing, representing, playing, understanding, comprehending, and explaining’.

Building upon this definition, we propose an expanded understanding of mathematics as a classification science centred around the abstraction and exploration of concepts, relationships, patterns, and structures. For us, mathematics can be conceived as a disciplinary matrix for the systematic categorisation and analysis of real and abstract constructs. Its elements include the various human endeavours of identifying, categorising, and deriving insights from patterns and associations among quantities, geometrical figures, data arrays, and logical propositions. Humans develop mathematics primarily to appreciate and communicate the observed cycles and structures of nature and our relationships to each other and the world, process data and build technologies, and make working forecasts.

Moreover, much of mathematics is non-numeric (Kemeny, 1959). Numbers are not necessary in classification, which refers to the arrangement or sorting of objects into groups based on their common properties. Qualitative or semi-qualitative methods are often used in mathematics for classification and analysis. Set theory classifies objects, known as elements or members, into groups or sets based on shared properties. These properties may not be numerical; for instance, objects can be classified based on shape, colour, or other non-numerical attributes. Non-European systematic classification systems may allow for richer mathematical thought that goes beyond the numerical focus of much European mathematics. Buddhist cosmology classifies sentient beings and rebirth into six realms or paths, often named as 'Wheel of life' (Bhavachakra), including Deva (god), Asura (demigod), Human, Animal, Preta (hungry ghost), and Naraka (hell) (Gethin, 1998). This continuous nature of worldview (samsara) can be viewed as a topological space that is  deformed but retains its  structure, depicting how the cycle of birth and rebirth continues without a beginning nor an end. The Wuxing is the fundamental classification system in the School of Naturalists (Law & Kesti, 2014). The Wuxing system classifies phenomena into five categories: Wood, Fire, Earth, Metal, and Water. They describe a cycle of generation (where one element generates another, e.g. Wood feeds Fire) and a cycle of overcoming (where one element overcomes another, e.g. Water extinguishes Fire). This system represents cyclic processes, akin to cyclic graphs in graph theory, where each node (element) has both an incoming edge and an outgoing edge, symbolising the generating and overcoming sequences.

The Dreaming, for Indigenous Australians, provides a complex, multi-layered framework that informs the understanding of the world, life, and the cosmos (Neale & Kelly, 2020). The Dreaming informs kinship systems, which are intricate classifications of relationships within a community. These classifications determine not only familial roles but also social interactions, responsibilities, marriage practices, and more (Sutton, 1982). The Dreaming does not segregate the past, present, and future linearly. Instead, all time is seen as interconnected. This cyclical or ‘everywhen’ concept of time is a different classification from the linear one in many European-based cultures.

Mathematics in this definition, or statement of meaning, along with its associated culture-specific practices, is a dynamic body of knowledge, molded by the ongoing processes of mathematical discovery, problem-solving, and deductive or/and inductive reasoning, influenced by both historical discoveries and diverse cultural contributions (Raju, 2018). Our definition recognises and includes the rich diversity of mathematical activity across cultures and historical periods.

  • Who can do mathematics?

Based on our responses to the gatekeeper critiques and our definition of mathematics, our view is that mathematics is part of the cultural heritage of all peoples, as intrinsic to our humanity as art and probably as ancient. We need to acknowledge that mathematics is a pan-cultural phenomenon and has many faces (Millroy, 1991). The differences between cultures pertain to the ways in which mathematics is expressed. Mainstream mathematics is deductive and axiomatic, and active decontextualision of its content is viewed as worthy scholarship (de Almeida, 2019). In Indigenous cultures, by contrast, mathematics is dispersive in all aspects of life. This is why culture is central to studies in ethnomathematics.

Our intention is not to propagate a naïve, overwhelming, notion that everything is mathematics. Instead, we aim to propagate a wider recognition that mathematics is a universal human endeavour with diverse cultural expressions. From the 1970s, ethnomathematics researchers have attempted to re-define mathematics as a pan-culture phenomenon. Anthropologists, represented by Marcia Ascher and Robert Ascher, examined the ways in which non-Western cultures and traditional societies use non-numerical mathematics, by referring to logic and spatial configurations (Ascher & Ascher, 1969, 1986). Barton (1999) redefines mathematics from a culturally relativistic, philosophical, perspective and argues that mathematics is ‘neither a description of the world, nor a useful science-like theory…(but) a system, the statements of which are “rules” for making sense in that system’ (Barton, 1999, p. 56). Barton (2007) introduced concepts like ‘near-universal, conventional mathematics’ (NUC-systems) and ‘quantitative, relational, or spatial aspects of human experience’ (QRS-systems) to illustrate how mathematical understanding can vary across different cultures, emphasising that mathematical ideas and practices are dependent on the cultural and societal context in which they are developed and used.

Bishop (1988) asserts that all human cultures perform six universal mathematical activities, which are counting, locating, measuring, designing, playing, and explaining. To D'Ambrósio (2006a, 2006b), the universal human abilities of comparison, classification, quantification, measurement, explanation, generalisation, inference, and evaluation are mathematical in nature.

Indigenous knowledge may be one of the keys to understanding how best to add new insights and advance mainstream and Indigenous mathematics together, but the mathematics research and teaching communities have to earn the respect of Indigenous knowledge holders. With two-way learning (Purdie et al., 2011), Indigenous mathematics students, teachers, and researchers will continue to thrive and the growing awareness, recognition, and value of this knowledge will empower the communities to whom these stories belong.

Studies have shown that an Indigenous mathematical approach can improve students’ interests in studying mathematics and their performance (Fernández-Oliveras et al., 2021; Parra & Valero, 2021; Roza et al., 2020). The inclusion of Aboriginal and Torres Strait Islander mathematical elaborations in the Australian schools curriculum, from foundation year to Year 10, is therefore cause for optimism that mathematics teaching will succeed in becoming more inclusive (https://www.australiancurriculum.edu.au). However, to achieve equity and two-way communication between two different systems, inclusion should go further than teaching a colonial curriculum in Indigenous language (Gavarrete, 2015; Meaney et al., 2013; Paraide et al., 2023; Parra & Valero, 2021; Trinick et al., 2014).

Conclusion

In this article, we have replied to five common ‘gatekeeper’ critiques of Indigenous mathematics. Although our focus is on the Australian context of mathematics education and research, the dismissal and misrepresentation of Indigenous mathematical competence internationally calls for collective reformative action.

In our reply to the pushback on Indigenous mathematics, we also have highlighted non-Western knowledge systems that challenge the dominant, monolithic conception of mathematics. Our investigation goes beyond the mere validation of Indigenous mathematical practices — it seeks to question and expand the narratives around ‘What is mathematics?’ and ‘Who can do mathematics?’. By offering alternative, inclusive narratives, we aim to disrupt the status quo, illuminating the polyphonic nature of mathematics.

Nevertheless, we recognise that this journey towards an inclusive understanding of mathematics will not be easy. The resistance from certain education sectors, steeped in an exclusive Eurocentric mathematical heritage, stands as a formidable challenge. The ‘gatekeeping’ dynamics evident in Australia’s engagement with Indigenous mathematics underscore this difficulty. Yet, it is precisely these challenges that make the dialogue on Indigenous mathematical knowledge critical and timely.

We urge the academic and educational communities to recognise and embrace the richness and diversity of mathematical knowledge systems across cultures, to rethink their perceptions of mathematical adequacy, and to abandon restrictive, culturally biased gatekeeping. This transformational journey, though fraught with challenges, is essential in developing a truly inclusive landscape of mathematical knowledge from which students may honor and learn through the mathematical wisdom of all cultures, not just the dominant ones.