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School mathematics tends to have developed from the major cultures of Asia, the Mediterranean and Europe. However, indigenous cultures in particular may have distinctly different systematic ways of referring to space and thinking mathematically about spatial activity. Their approaches are based on the close link between the environment and cultural activity. The affinity to place strengthens the efficient, abstract, mathematical system behind the reference and its connection to the real world of place and a holistic worldview. This paper sets out to present an overview of various approaches to aspects of space and geometry by drawing on linguistic and cultural literature, my collaborative research in Papua New Guinea, and from personal communications with indigenous colleagues in Australia and elsewhere. This diversity provides a challenge by which teachers can deconstruct their thinking about mathematics and subsequently to review the content of teaching and to be more responsive to the diversity of cultural backgrounds of students. To assist with recognising ecocultural mathematics on space and geometry, 4 principles are established and discussed on language structures, reference lines and points, measures of space and worldviews and interpretations of space as place.

Identity as a mathematics teacher is enhanced when a teacher explores the cultural setting of their mathematics. The reports of projects that link culture and mathematics were analysed to explore the impact of sociocultural situations together with affective and cognitive aspects of self-regulation on identity. The reports were written by pre-service and in-service, mainly secondary, teachers at the University of Goroka, Papua New Guinea. While all 239 reports were read and considered in terms of the thesis of this article, 60 (25%) were analysed in detail developing the argument. The results indicate the strengths of such projects to take account of cultural knowledge when colonised education systems are further modified through reforms that emphasise culture. The significance for teacher education is the role that an activity which links culture and school mathematics plays in building values and identities.

Teaching mathematics in children's first language has both cognitive benefits and assists with developing cultural and mathematical identity. In Papua New Guinea, many different Indigenous languages are used for instruction in elementary schools and teachers often need to identify or develop mathematics terminology themselves. Building on prior research, guidelines were developed to assist teachers to find the mathematics in their own vernacular languages. The guidelines were adaptable to different language ecologies and covered mathematical areas such as counting systems, measuring and comparing, location and shapes. The guidelines were applied by teachers in professional development workshops. The paper shows examples from some of the workshops of the challenges that the teachers faced in trying to identify appropriate mathematical language, and that contributed to the refining of the guidelines. [ABSTRACT FROM AUTHOR]

Transforming teachers and their approach to teaching Indigenous students requires partnerships with the Indigenous community, planning, small steps, and funds. This article illustrates how teachers can change when funds are available to assist schools and communities to implement appropriate and effective professional development, to establish partnerships between school and community, to revise teaching approaches and curriculum, and to value family and Aboriginal cultural heritage. The larger study involved four schools in a Stronger Smarter Learning Community in a small rural city, the whole city community, and the interaction among the schools. Interviews with principals, teachers, Aboriginal students, and their community highlighted the increasing interaction between the Aboriginal community and the schools, the increasing warmth and welcome extended both ways, and the impact that these approaches are having on curriculum, teaching, and learning. This article presents the impact in one of the schools involved in the mathematics project. The findings illustrate how the projects facilitated changing teachers' perceptions, skills, and practices and implemented curriculum, and resulted in a culturally responsive, place-based mathematics education. [ABSTRACT FROM AUTHOR]