1 Introduction

This paper exemplifies and critiques the use of the cultural symmetry model to design mathematics tasks, for a Māori-medium initial teacher education (MM-ITE) programme, that promote the acquisition of mātauranga (Māori ways of knowing and being), Māori language and school mathematics curriculum content concurrently. Mātauranga, a term that has featured in academic literature since the 1970s and is in Māori manuscripts dating back to the 19th century (Smith et al., 2016), has been popularised as a means of addressing the boundless and continually evolving Māori knowledge systems (Mead, 2013). Mātauranga encompasses Māori ontologies, epistemologies, and axiologies (Smith et al., 2016). The term mātauranga is particularly relevant to the body of knowledge that draws on Pacific ancestry and has continued to evolve in Aotearoa New Zealand [NZ] (Royal, 2006). Thus, mātauranga can be understood as the knowledge and applications of knowledge that have evolved through generations and are still applied, adapted, and have meaning for Māori communities today (Smith et al., 2016). Mātauranga is synonymous with te reo Māori (the Māori language).

The promotion and revitalisation of mātauranga have been foundational goals to Māori-medium schooling since its emergence in the 1980s. Underpinning these goals, graduates being able “to live as Māori and to be citizens of the world” (Smith et al., 2021, p. 10). Durie (2003), a prominent Māori academic in the fields of education and health, asserted that education for Māori must provide preparation for participation in Māori society and ensure readiness to engage with the wider world. The challenge then for MM-ITE is to prepare teachers to facilitate the acquisition of mātauranga with state-mandated curriculum areas, such as mathematics, to support Māori-medium students to achieve success as Māori (Ministry of Education [MoE], 2021).

According to the MoE (2022), there are presently 305 schools providing Māori-medium programmes (51% or more Māori language instruction), catering to a student population of 23,161, from Year 1 (age 5) to Year 13 (age 18) (MoE, 2022). Māori-medium schools in NZ are generally small and tend to work in isolation due to their geographic spread. They provide instruction primarily in the Māori language, with a strong focus on mātauranga. In this way, Māori-medium schools aim to position mātauranga alongside curriculum areas such as mathematics (Allen, 2023; Trinick, 2015).

Initial teacher education institutes have existed since 1862 in Aotearoa NZ (Openshaw & Ball, 2006). Initial teacher education was exclusively English language until the 1990s. Since the emergence of MM-ITE, incorporating mātauranga into teacher education curricula has presented a challenge, as many trainee teachers have had limited opportunities to explore mātauranga during their own education—a legacy of over 100 years of colonisation. To effectively teach Indigenous practices, it is crucial to ensure that teachers possess the necessary knowledge, skills, and understanding (Hōhepa et al., 2014). This necessitates a commitment to indigenising initial teacher education, which requires an Indigenous-led approach that directly benefits Indigenous peoples and knowledges (Gaudry & Laurenz, 2018). Strategies for indigenising MM-ITE include re-evaluating and redesigning school curriculum frameworks, providing opportunities for MM-ITE students to engage with mātauranga, involving Māori communities in decision-making processes, and allocating adequate resources to support the inclusion of mātauranga in school curriculum. Thus, MM-ITE face two broad challenges impacting on the nature and design of mathematics tasks. One challenge is epistemological (positioning Western mathematical concepts and methodologies next to Indigenous ways of knowing and practicing mathematics), the other sociolinguistic (the small pool of applicants with the necessary Māori language fluency and knowledge to teach mātauranga and mathematics curriculum content).

This research aimed to address the ongoing epistemological and sociolinguistic challenges in designing MM-ITE mathematics tasks that simultaneously address language, mātauranga and teaching of mathematics. To support this process, the cultural symmetry model developed by Meaney et al. (2013) was used to guide the design of MM-ITE mathematics tasks. The investigation aimed to answer the following research questions:

  1. 1.

    How useful is the cultural symmetry model in guiding the design of Māori Medium Initial Teacher Education (MM-ITE) mathematics tasks that demonstrate the simultaneous teaching of mātauranga and mathematics curriculum content?

  2. 2.

    To what extent do Māori wayfinding artefacts and practices provide opportunities for simultaneously addressing mātauranga and mathematics curriculum content in MM-ITE mathematics tasks?

The research questions were explored within a MM-ITE programme. The programme is considered dual-medium as students receive instruction in both te reo Māori and English, with the goal of promoting social and academic proficiency in both languages (see Linquanti & Cioè-Peña, 2016 for discussion on features of dual medium programmes). Graduates of the MM-ITE programme can choose to teach either in te reo Māori instruction (Māori-medium) or English language instruction (English-medium) schools. Most of the MM-ITE programmes are in private training establishments and Wānanga. Wānanga are Māori-led institutions of higher learning in Aotearoa NZ. While they are not universities in the traditional Western sense, they play a crucial role in preserving and promoting Māori culture and education (Trinick, 2019).

2 Cultural symmetry model

The cultural symmetry model developed by Meaney et al. (2013) has built on the work of earlier ethnomathematics researchers, for example, D’Ambrosio (1985)—mathematics depends on cultural experience; Gerdes (1985)—unfreezing Indigenous mathematics; Bishop (1991)—mathematics as a product of cultural activities, and Barton (2008) —mathematics in Indigenous languages. The cultural symmetry model situates mathematics curriculum content as one of many ways to examine a cultural artefact and allows for critique of how Western mathematical understandings can dominate such discussions. Thus, it opens socio-political discussions about whose knowledge is valued in what contexts.

Exploring cultural practices through the cultural symmetry model allows a more nuanced understanding of the value of artefacts and practices within a society, and problematises the colonisation of knowledge. Meaney (2002), the researcher primarily responsible for conceptualising the cultural symmetry model, argued there was always the potential for a cultural clash between the culture of school mathematics in colonised countries and Indigenous students’ cultures. Meaney (2002) described this clash as a hindrance to successful symbiosis rather than a harmonious integration. There are increasing applications of the cultural symmetry model in Indigenous mathematics education literature; Trinick et al. (2017)—Māori spatial orientation concepts and the importance of symmetry in wharenui (Māori meeting houses)—Meaney et al. (2021) reviewed applications of cultural symmetry in Norway and Aotearoa NZ.

Epistemology, ontology, and culture are intricately linked concepts that shape individuals’ and groups’ perceptions, understanding, and interactions with the world. Different cultural backgrounds influence how knowledge is acquired, justified, and interpreted. For example, Western cultures often prioritise empiricism and rationalism, while Indigenous cultures may emphasise holistic, experiential ways of knowing. Mika and Stewart (2017), proponents of Māori philosophy, position epistemology and ontology, not as separate, but as a linked process where ontology precedes epistemology. For Mika and Stewart (2017), ontology provides the lens through which epistemological experiences are grouped and new knowledge is created. Hauser et al. (2009) argue that ontological pluralism provides the opportunity for epistemological pluralism to occur. Whereby, the historical emergence of epistemologies is explored, and contemporary concepts or challenges are examined from multiple perspectives (Andreotti et al., 2011; Miller et al., 2008). Therefore, ontological or cultural pluralism, enables the co-occurrence rather than the hybridisation or abrogation of different cultural resources (Pöllmann, 2021). Arguably, the latter has been the case historically for mātauranga in mathematics curriculum and classrooms.

Ontological pluralism provides an opportunity to reposition mātauranga and Western knowledge-based school mathematics curriculum as distinct and complementary. For example, Indigenous academics, including those in MM-ITE, effectively walk in two worlds and speak to two audiences, the academic audience and their own Indigenous communities (Galla & Goodwill, 2017; Ka’ai, 2008). Therefore, MM-ITE academics are required by the pluralistic expectations of these two audiences, to approach mathematics task design through multiple lenses or ontologies. In this paper, we argue that Māori-medium mathematics teachers must navigate this ‘culture clash’, through meaningful mathematics task design that concurrently addresses mātauranga and mathematics curriculum content.

Although Western and Indigenous mathematical practices often stem from diverse epistemological foundations, Western mathematics education has been a dominant paradigm in most educational systems, including Aotearoa NZ for over 150 years. Much of the recent focus of Māori education research is based around a thesis of culture and identity, excluding any particular focus on Māori knowledge and epistemology (Cooper, 2012). This is attributed to the presumption that mātauranga is inferior to Western Science (Smith, 2012).

Mathematics teaching resources, in initial teacher education and compulsory schooling in Aotearoa NZ, have primarily promoted the use of mātauranga to aid the acquisition of mathematics concepts without addressing Māori epistemologies. For example, Māori visual representations such as kōwhaiwhai (painted scroll ornamentation) and narrative representations such as karakia (oral chants) can represent valued environmental or wayfinding information (Allen, 2023). Comprising the recognition of visual elements, the arts (the signs, images, and iconography immediately recognisable as representing that culture) and books of Māori myths (generally written by Europeans) (Allen & Trinick, 2022), in both educational discourse and school curricula, a delimited view of mātauranga has dominated.

By prioritising Western epistemologies, those designing ITE mathematical tasks can marginalise Indigenous knowledge systems and mathematical practices. This is demonstrated in NZ curriculum resources where Māori culture and language are promoted as vehicles to improve the teaching and learning of mathematics (Allen & Trinick, 2022). Therefore, we decided to explore the efficacy of the cultural symmetry model in guiding the design and delivery of MM-ITE mathematics tasks that drew on Māori wayfinding artefacts in a way that exemplified ontological pluralism. Thereby providing opportunities for epistemological pluralism to occur.

The cultural symmetry model includes the tripartite foci of language, culture, and mathematics curriculum content Meaney et al. (2013). The first step of the model addresses the cultural practices connected to the artefacts being utilised. For this study, this first step involved researching the historical emergence of Māori wayfinding practices. The MM-ITE mathematics tasks designed for this study drew on Māori wayfinding representations, both visual and oral narrative artefacts. Opportunities and challenges of using Māori wayfinding artefacts in MM-ITE mathematics task design were illuminated in the research and task design process. The identification and utilisation of the artefacts is discussed in subsequent sections.

The second step of the cultural symmetry model is using the appropriate Indigenous language to examine the characteristics and/or design of the wayfinding artefacts. Edmonds-Wathen (2011) pointed out that cross-linguistic research has identified variations in how diverse groups of people discuss and perceive space and location. This spatial frame of reference serves as the conceptual foundation for representing spatial positions. As noted by Levinson (2001) and more recently, Shusterman and Li (2016), spatial representations can be linguistically or culturally derived. Various typologies have been developed to describe spatial frames of reference, including intrinsic (based on the speaker’s viewpoint), relative (involving the perceiver’s viewpoint and the position of another object), and absolute (using fixed points of reference, such as landmarks or cardinal directions) as proposed by Pederson et al. (1998). For example, to say ‘Kei te haere ki te muri (going to the back or north) is ungrammatical in the Māori language if referring to ‘going to the back’ (intrinsic). The use of ‘te’ for a local noun is grammatical if referring to ‘going to the north’ (absolute, using a fixed system that is larger and external to the described scenario).

The MM-ITE mathematical tasks designed for this study incorporated activities that required MM-ITE students to examine Māori terminology and syntactic structures required to communicate about wayfinding practices. In many minority Indigenous education contexts the language—the repository of cultural knowledge—is marginalised or in the case of te reo Māori, endangered. Therefore, the MM-ITE tasks design necessarily required MM-ITE students to learn and use te reo Māori terms and phrases that were new to them. In this way, the language focus of the cultural symmetry model supports the language revitalisation goals of Māori-medium education.

The third step is using mathematical principles to discuss the design elements and/or characteristics of the artefacts. Thus demonstrating how mathematics can add value to understandings about the artefacts without detracting from cultural understandings. For the purposes of this study, wayfinding was defined as finding your way to a destination using environmental cues (Farr et al., 2012). Beside locomotion, wayfinding is a skill required for navigating from one place to another (Wiener et al., 2009). Wayfinding encompasses embodied experiences (mind, body, cognition, and emotions) in the context of sociocultural information systems (Meaney et al., 2021).

In a mathematics classroom, wayfinding can be a spatial problem-solving task (Casakin et al., 2000). Solving the task and reaching the desired destination is dependent on human and environmental factors (Farr et al., 2012). Spatial memory is one human factor, involving the ability to remember and recall spatial information accurately (Tine et al., 2018; Van de Weijer-Bergsma et al., 2015). It includes remembering the arrangement of objects, spatial configurations, or locations of geometric figures. The method of loci, like other memory strategies, consists of two phases: encoding and recall (Anderson & Brewer, 2014). During the encoding phase, it is crucial to visualise a familiar area and create a vivid spatial representation of a path connecting memorable locations within that area. Each item to be remembered is then associated with a specific place along the path, making use of easily recallable landmarks. The MM-ITE mathematics tasks, designed for this study, presented Māori wayfinding artefacts as oral and visual mnemonic devices aiding in the coding and recall of valued spatial information.

To engage MM-ITE students in tasks that addressed both Māori wayfinding practices and mathematics curriculum content, the MM-ITE mathematics tasks needed to encourage students to utilise multiple representations of wayfinding information concurrently. The term representation refers to process and to product—in other words, to the act of capturing a mathematical concept or relationship in some form, and to the form itself. Mathematical ideas can be represented externally and internally, and learning is an interplay between the two (Goldin & Shteingold, 2001; Lingefjärd & Ghosh, 2016). Internal representations refer to how students construct the mathematical image in their minds. External representations include manipulatives, pictures, diagrams, spoken languages, and written symbols (Goldin & Shteingold, 2001; Pape & Tchoshanov, 2001) and internal representations include mental models and cognitive representations of the mathematical concept (Putnam et al., 1990). The MM-ITE mathematics task designed for this study incorporated external spatial representations such as stories, maps, models, photographs, pictures, and metaphors. The tasks design promoted the use of internal spatial representations such as mental maps formed from external representations, but which can then in other circumstances be recreated as external representations.

Bruner (1964, 2006) described three categories of representations: enactive, iconic, and symbolic. Bruner (1964, 2006) also stated that learners should go through three levels of engagement with each category to build a complete understanding of a mathematics concept. While we did not present the tasks in the order proposed by Bruner (2006), through the MM-ITE mathematics task design we endeavoured to illustrate the various representations.

Dreyfus (1991) and Alkhateeb (2019) argued that learning happens when students transition between multiple representations. Their research found students should use multiple representations in parallel, exploring connections between the representation forms and fluently moving between them (Alkhateeb, 2019; Dreyfus, 1991). Thus, it is important children have access to different representations of the same mathematical idea or concept and can translate between the different representation forms (Goldin & Shteingold, 2001; Lesh et al., 1987). If students can translate between the different representation forms, students can be said to understand the mathematical idea (Lesh et al., 1987). Gentner and Ratterman (1991) suggested that students need to understand how different representation forms are related to one another for effective mathematics learning to take place. Thus, the MM-ITE mathematics tasks designed for this study gave students opportunities to utilise a range of representations and to represent the same information in multiple ways.

Emphasising the importance of using multiple representations in the MM-ITE mathematics task design reflected recommendations in the literature for explicit teaching of the language of mathematics to aid in the acquisition of mathematical knowledge (Erath et al., 2021; Radford & Barwell, 2016). Whereby the value of communicating mathematically using multiple representations (Herbel-Eisenmann, 2002; Kuntze et al., 2018) was highlighted in the MM-ITE mathematics task design to support the acquisition of Māori language, mātauranga and school mathematics simultaneously (Allen, 2015).

In summary, the theory underpinning ethnomathematics is that mathematical ideas and practices are not universal or culture-neutral but are influenced by the cultural context in which they develop. Different cultures may have unique ways of approaching mathematical problems, representing mathematical concepts, and using mathematical knowledge in daily life. Language is considered a crucial aspect of mathematical understanding. The ways in which mathematical concepts are expressed, discussed, and communicated are influenced by linguistic structures.

By positioning Indigenous knowledges and mathematics curriculum content as distinct and complementary (Guerzoni, 2020), the cultural symmetry model provides opportunities to examine concepts from different perspectives. The cultural symmetry model acknowledges the potential for cultural clashes to arise when the culture of school mathematics collides with the students’ cultural backgrounds, particularly those from disempowered Indigenous communities. Thus, the cultural symmetry model promotes epistemological pluralism rather than abrogation to address conflicts between epistemologies.

The model recognises the interconnectedness of language, culture, and mathematics within education, which can lead to a more balanced and meaningful approach to mathematics education and research, particularly benefiting Indigenous students and their communities. Moreover, it underscores the importance of using multiple representations for mathematical communication, as this supports language proficiency and the acquisition of school mathematics concepts (Allen, 2015). It serves as a valuable tool for educators and researchers seeking to bridge cultural gaps and enhance the educational experiences of Indigenous students while preserving and strengthening their cultural heritage.

3 Methodology

The research was conducted in a Māori-medium initial teacher education programme with students who were embarking on a significant educational journey, often the first in their families to do so. The MM-ITE programme spans three years. This project focused on two cohorts allowing for reflection and iteration. What made this MM-ITE programme context particularly intriguing was the diverse backgrounds of the students and their unique motivations to pursue higher education. Most had learnt te reo Māori as adult students with a few graduates from Māori-medium schools, reflecting a strong commitment to preserving and revitalising their Indigenous heritage. The majority had grown up in urban areas, often dislocated from their tribal areas. Some of the MM-ITE students saw teaching not only as a means of personal growth and cultural reconnection but also as an opportunity to actively participate in their children’s Māori-medium education. This dual motivation, driven by personal and community interests, made their educational journey not only a pursuit of knowledge but also a deeply rooted commitment to preserving and strengthening their cultural heritage.

This MM-ITE programme had been in existence since 1998, however, this and other teacher education programmes generally have had declining numbers in the past 10 years, placing further pressure on teacher supply. One of the key components of the first year mathematics education programme was to examine strategies to address mātauranga and mathematics curriculum content concurrently. However, historically and for the reasons described previously, few MM-ITE students have knowledge of Māori mathematical practices (Trinick, 2019). There has also been a tendency for some Māori students, like many Indigenous and minority groups globally, to develop a negative attitude toward mathematics due to historical, cultural, and educational factors. The colonial history of NZ included the suppression of Māori culture and language, which has created a sense of disconnection and alienation among Māori students from the mainstream education system, including mathematics (Te Maro, 2018). Stereotypes, bias, and low expectations from teachers have contributed to a negative self-perception and attitude toward mathematics (Turner-Adams & Rubie-Davies, 2023).

Drawing on the cultural symmetry model, the data collection was conducted in two phases. In the first phase, the authors conducted a literature search of Māori wayfinding narratives, practices and visual representations. Initially, a search of existing and relevant mathematics resources was also conducted on the NZ Maths website and a Google search was used to find further examples of digital artefacts widely available to teachers. As there were few widely available teaching resources supporting the teaching of cultural and linguistic understandings of Māori wayfinding practices, a wider literature search was required. The literature search was conducted through searching online databases, museum websites, visiting museum exhibitions, sourcing literature and artefacts held in art museum retail shops, and searching the authors’ own collections of wayfinding literature. Although European reinterpretations of Māori wayfinding narratives as myths and legends has often minimised the practical information within, Taonui (2015) asserts that exploration narratives (such as uruuru ao and uruuru whenua) suffered less alteration than other Māori narrative forms. Therefore, the English language versions of the narratives can still be useful for exploring the characteristics of Māori wayfinding practices.

Due to the paucity of literature on Māori wayfinding practices specifically, a wider search of Pacific literature was also conducted. Māori wayfinding practices evolved from Pacific ancestors and have been adapted for use on land and at sea here in Aotearoa NZ. As suitable literature and artefacts were located during the literature search, further literature, and artefacts were sourced through examining references used by authors or any stated influences for visual and narrative artefacts.

The literature search findings supported the first and second steps of the cultural symmetry model which involve identifying cultural understandings and appropriate Indigenous language for discussing the Indigenous artefacts, in this case Māori wayfinding narratives. The second phase of the data collection included the design and delivery of the wayfinding tasks in the MM-ITE programme. Utilising the literature and Māori wayfinding artefacts retrieved in phase 1 of the research process, a series of wayfinding tasks and stimuli were developed and introduced to the MM-ITE students over a university semester for each of the two cohorts in phase 2.

The MM-ITE mathematics tasks were designed to highlight the three steps of the cultural symmetry model. The Māori wayfinding artefacts were introduced to MM-ITE students and the artefacts’ characteristics were discussed. Students were then engaged in activities where there was explicit teaching and discussion of identified terms and syntactic structures required to communicate about the characteristics of the artefacts. Finally, MM-ITE students created their own wayfinding artefacts, utilising the characteristics and language that had been explored previously. During this final step, the MM-ITE students were encouraged to use multiple representations of wayfinding information including those exemplified in mathematics curriculum content. In the following sections, examples of the three step approach to task design and delivery are offered with the reflections of the authors on the process of researching, designing and delivering the tasks. Each section provides a brief discussion of the authors’ perceptions on the efficacy of using the cultural symmetry model in MM-ITE mathematics task design to address epistemological and linguistic challenges.

4 Identifying characteristics of Māori wayfinding narratives

As outlined above, the first step of the cultural symmetry model addresses cultural understandings of the artefacts under study. For this study, the historical emergence of Māori wayfinding practices as represented by the cultural artefacts became a key focus of this first step. The first phase of the research process involved a literature search to identify Māori wayfinding practices and artefacts that could be utilised in a MM-ITE mathematics programme. The Māori wayfinding narratives, including their visual and oral representations, identified in the literature search provided practical information for utilising environmental cues when navigating within a landmass (urururu whenua) or between landmasses (islands) (uruuru ao), as devised by Māori ancestors who regularly traversed the Pacific ocean. Often originating in the Pacific and later brought to Aotearoa NZ by Māori ancestors, these narratives emphasise the identification of land, sky and sea markers as wayfinding aides.

Often uruuru ao (for navigating between landmasses) took the form of chase stories, where a navigator chased an octopus (wheke), whale, fish or other creature from island to island, in the process identifying route markers (Allen, 2023; Trinick & Allen, 2023) as is the case with the narrative of Te Wheke o Muturangi (The Octopus of Muturangi; see Grace, n.d.; Meihana, n.d.). Another common construction of Māori wayfinding narratives is love stories, where a suitor follows a love interest from place to place, identifying route markers along the way. The Legend of Poutini is an interesting example of an urururu whenua (for navigating within a landmass) as both a love story and a chase story (Toitū te Whenua LINZ, 2018). In the narrative, Waitaiki is abducted by Poutini and Waitaiki’s partner gives chase, stopping at various locations on the North and South Islands of Aotearoa NZ. The stopping points indicate the route taken and the location of valued resources (Google Maps, n.d.). As identified in Sect. 2, the use of the love/chase narrative construction in Māori wayfinding narratives can be considered a mnemonic device, providing an interesting and memorable story that aids in the coding and recall of valued wayfinding information.

Both Te Wheke o Muturangi and The Legend of Poutini narratives have been reimagined in visual and audio-visual formats in Māori and English. The digital map representations of the narratives, used in the MM-ITE mathematics tasks, emphasise the wayfinding content of the narratives (see Google Maps, n.d.; Meihana, n.d.). To further develop MM-ITE students’ understanding of wayfinding narratives’ characteristics, including their use as mnemonic devices, MM-ITE students were tasked with researching narratives from their own tribal areas and then creating their own wayfinding narratives. Engaging with the wayfinding narratives provided MM-ITE students with opportunities to explore the routes followed and examples of how wayfinding information was coded and disseminated through the narratives. Most of the MM-ITE students had grown up in the city, a result of the substantial urban migration of their grandparents (Trinick, 2015). Consequently, many had become dislocated from their tribal roots. Exploring Māori wayfinding narratives devised by their ancestors within their own tribal areas provided opportunities for MM-ITE students to connect with their own ancestral wayfinding practices and their own tribal communities. The MM-ITE students acknowledged a sense of pride at recognising and celebrating their communities’ mathematical contributions. This encouraged a sense of agency and ownership in the learning process. Understanding how mathematical concepts are applied in everyday cultural practices also provided opportunities for the MM-ITE students to see practical and real-world applications of mathematics.

5 Language learning opportunities

When devising their own wayfinding narratives, MM-ITE students were required to use Māori linguistic representations for orientating themselves in time and space alongside universal direction and location terminology. The narratives presented to the MM-ITE students as examples were provided bilingually. Often, MM-ITE students found the reo Māori versions more complex and difficult to understand than the English language versions. This demonstrated to the teacher educators the need for the linguistic focus in the MM-ITE mathematics task design as exemplified in the cultural symmetry model. As had been the case for other MM-ITE students in a previous research project (see Meaney et al., 2021), many of this group had a minimal idea of the origins of Māori spatial frameworks, therefore which framework was at play in various parts of the narratives, and in turn how these frameworks influenced the syntactic structure, and vice versa.

In order to decode the cultural and linguistic concepts underpinning the Māori wayfinding narratives, MM-ITE students had to grapple with terminology for key geographic points, directions between these points and the general terms for direction and scale. We would certainly argue that Māori wayfinding concepts are multi-layered. Some directions are derived from winds, some from the passage of the sun across the sky, others as historical memories from their Pacific origins (Trinick & Allen, 2023). Thus, the authors noted the importance of MM-ITE mathematics task design including explicit teaching of the various linguistic structures and terminology derived from a range of spatial frameworks. For example, R–Rāwhiti (where the sun rises) was introduced alongside E–East providing MM-ITE students with opportunities to learn Māori language, mātauranga and universally used mathematics terminology and symbols concurrently.

6 Highlighting mathematical representations

The final step of the cultural symmetry model is to use mathematics to add value to the cultural and linguistic artefacts. It was decided that this value could be added by providing opportunities for the MM-ITE students to develop their understanding of mathematical concepts of wayfinding and spatial orientation representations. Initially, we introduced iconic representations, including visual or spatial representations such as the digital representations of the wayfinding narratives. While the wayfinding narratives do contain elements of language and symbols, they are primarily used to create mental images of events, characters, and situations. These mental images can be considered iconic in nature (Bruner, 2006), as they provide a visual or spatial component to understanding the narrative. Within the enactive category of representation forms (Bruner, 2006), MM-ITE students were given opportunities to use manipulatives and hands-on objects. The MM-ITE students were also taken outside to discuss the various sky and land signs used by Māori ancestors to help with wayfinding. This type of embodied learning and hands on experience can be considered enactive.

The MM-ITE mathematics task design included the use of interactive digital tools. For example, the MM-ITE students were encouraged to use Google Maps to create visual representations of their own wayfinding narratives. The Google Maps software allows users to drop pins at important locations and annotate these with a written or recorded narrative. The pinned points can be ordered in a list to create a route or journey. The MM-ITE students were also encouraged to create drawn representations using symbolic notation to bridge the written and digital formats. In doing so the MM-ITE students, referred to and manipulated spatial symbols (co-ordinates, distance, compass directions, and scale) which supported the transfer of the spatial information from wayfinding narratives to the digital format.

The use of interactive digital representations and the discussion of sky and land markers provided examples of the use of enactive representations while simultaneously reinforcing the terms and structures for communicating about wayfinding information in te reo Māori (the Māori language) promoted in step 2 of the cultural symmetry model. Because the Google map software is web-based and utilises multiple modes of communication, it allowed the MM-ITE students to show their thinking and easily share the digital representations and recordings with peers and their communities. Several researchers (e.g. Gates, 2018) argue that we need to think differently about communication and suggest there is an over-reliance on linguistic and textual modes at the expense of visual and spatial modes of communication particularly in STEM subjects. Levinson (2001) and Majid et al. (2004) noted that learners may need to construct conceptual representations of space in a language-dependent manner from cultural and linguistic input in their communities. Having the MM-ITE students’ work stored in a digital format ensured they could easily share and edit it, perhaps through interactions with knowledgeable community members, during their ongoing language and mathematics learning journey (Allen, 2017).

Research has indicated that students benefit when their learning is connected to their practical and real-world knowledge (Baranes et al., 1989; Rittle-Johnson & Koedinger, 2005). Lesh et al. (1987) highlighted five representation forms that teachers should use in their teaching: real-life experience, manipulative models, pictures or diagrams, spoken symbols, and written symbols. The narratives that MM-ITE students created for this study originated from where they lived or some other place they found interesting. The characters the MM-ITE students created went on journeys or chased something, naming the landmarks and directions as they went. The MM-ITE students then created digital or drawn maps, adding in and naming geographic and noteworthy features. When creating hand drawn maps MM-ITE students were encouraged to add symbolic representations of measurement such as scale, and any other relevant mathematics symbols and notations. Symbolic representations are the most complex representation form and stretch students to consider their mathematics understanding differently. Research has found that symbolic representations strengthen students’ conceptual knowledge (e.g., Ainsworth et al., 2002; Nason & Woodruff, 2005). Based on the observations of student’s work, we argue that uruuru ao and uruuru whenua (narratives for wayfinding at sea and on land respectively) can be added to the list of representation forms that Māori-medium teachers should use in their teaching.

The MM-ITE students’ use of different representations in creating their wayfinding narratives and digital maps provided opportunities to discuss how the socio-cultural contexts of cultural artefacts can contribute to some representations being more suited to specific situations. This provided opportunities for socio-political discussions about what kinds of representations come to be valued and why, exemplifying epistemological pluralism. Māori wayfinding narratives are linguistic (interesting story) and conceptual (mind map). Thus, being able to discuss the value of wayfinding narratives as internal representations and the value in different circumstances of external representations can contribute to decolonising schooling.

Spatial orientation ‘representation’ links both school mathematics curriculum content and Indigenous spatial orientation practices by providing a means to convey mathematical concepts in a way that aligns with the cultural and cognitive framework of Indigenous spatial orientation practices. Mathematical representations–diagrams, charts, and models–are visual tools that help students grasp abstract mathematical concepts. Indigenous spatial orientation practices often involve tangible and visual cues, such as the use of stars or landmarks for navigation. Both approaches facilitate a deeper understanding of mathematical and spatial concepts by providing concrete, visual representations. Therefore, discussing synergies between mathematical curriculum representations and Indigenous spatial orientation practices can serve as a bridge between school mathematics curriculum and Indigenous practices.

7 Conclusion

In conclusion, this research project explored the efficacy of the cultural symmetry model for supporting the design and delivery of MM-ITE mathematics tasks that concurrently addressed mātauranga and mathematics curriculum content. The Māori wayfinding artefacts and practices investigated for the study provided opportunities to simultaneously address mātauranga and mathematics curriculum content in MM-ITE mathematics tasks. The wayfinding narratives provided invaluable insights into the multifaceted nature of spatial cognition and linguistic diversity. The narratives served as sociocultural guides, emphasising the significance of landmarks found in the land, sea, and sky for navigation.

We observed that considering these narratives as mathematical representations was new to MM-ITE students. This reflects the shift in common use of wayfinding narratives over time, where the scientific knowledge has been obscured and the narratives transformed into myths and legends. Our research sought to rekindle and reclaim this latent wayfinding knowledge, shedding light on the rich traditions and scientific principles that underpinned Māori wayfinding. Engaging with these narratives provided insights into routes and locations but also highlighted the use of internal representations, mnemonic techniques, and interactive imagery. These techniques have been a fundamental part of Māori wayfinding practices, aiding in the construction of cognitive maps to navigate on land and at sea.

By contextualising mathematical representations within these narratives, we aimed to model for the MM-ITE students ways to promote mātauranga alongside mathematics curriculum content. Arguably, we focused on a topic universal to all cultures, wayfinding. The next challenge could be to focus on a less common area of mātauranga. In other contexts, we suggest focusing on how different cultures ‘represent’ concepts as a fruitful strategy to integrate diverse learning areas, such as mathematics, literacy, the arts and so on.

The cultural symmetry model served as a useful guide in designing MM-ITE mathematics tasks particularly when concurrently teaching mātauranga and mathematics curriculum content. However, adherence to the model during the task design process highlighted a particular challenge for Māori-medium education, a lack of resourcing. In line with the cultural symmetry model, we began our task design process by exploring the historical emergence of Māori wayfinding narratives. For MM-ITE academics it is not unreasonable to conduct an in-depth literature search, which involves physically visiting art and history museums alongside online database searches, to retrieve the information required for designing learning tasks. Although one could argue this is not a requirement for those teaching in English-medium ITE who have access to an international catalogue of English language resources. For classroom teachers, it is unlikely they would be allocated the time and resources to complete this task. Therefore, if the cultural symmetry model is to be fully utilised by Māori-medium classroom teachers and Māori-medium ITE academics, equitable resourcing is required. This includes making mātauranga based teaching resources more widely available.

Pursuing steps 2 and 3 of the cultural symmetry model provided insights into the linguistic challenges for MM-ITE students and the importance of using multiple representations to support students’ linguistic development, both mātauranga and mathematical. For example, we noted that MM-ITE students had minimal ideas of the cultural origin of Māori language associated with spatial orientation. This may be the way of the future as the language becomes standardised because of a common national curriculum, state-produced resources and so on. By using a range of mātauranga representations in discussing and communicating Māori wayfinding narratives–iconic, enactive, and symbolic forms–students were able to make connections between the Māori terms and their wayfinding origins. Designing tasks that focused on highlighting different mathematics representations of wayfinding information allowed us to emphasise the importance of understanding and translating between various representation forms. In essence, our research project demonstrated the value of a strategy of co-locating Indigenous spatial orientation practices and mathematical representations, thus concurrently addressing mathematics curriculum content and Indigenous knowledge and practices. This approach provided opportunities to enhance students’ mathematical proficiency and also supported the revitalisation of cultural knowledge including language. It is our hope that this project serves as a model for educators to embrace the richness of cultural diversity in mathematics education, enriching the learning experience and fostering a deeper appreciation for the intertwined nature of mathematics, culture, and language.