In this essay, we explore how empty space can create possibilities for change in mathematics teacher education. We draw on ideas from physics, theater, architecture, and graphic design to understand emptiness as a space where meaning, relationships, and creativity can emerge. We suggest that intentionally leaving spaces within teacher education programs open (i.e., spaces in which the content remains undefined, and the trajectory remains flexible) can invite processes of co-creation, critical reflection, and pedagogical imagination. To develop this argument, we turn to Mikhail Bakhtin’s notion of the carnivalesque together with speculative design, using these perspectives to understand such spaces as dialogic arenas through which established hierarchies loosen and new meanings emerge. From this viewpoint, emptiness can be conceptualized as productive through the creation of conditions for interaction, experimentation, and evolving roles of teaching and learning. Building on this understanding, we also propose that such forms of openness invite a reconsideration of academic rigor in mathematics teacher education, suggesting that rigor may be understood through ethical responsiveness, relational depth, and the capacity to imagine alternative possibilities for teaching and learning. To illustrate these ideas, we present two speculative examples that show how such open spaces may unfold in practice and that offer provocations for more reflective, responsive, and creatively oriented forms of mathematics education.

This perspective paper examines the prevailing presumption of linearity in school mathematics - evident in curriculum design, lesson planning and conceptions of learning - and the limitations this imposes. While linear structures offer coherence and clarity, their dominance can marginalise diverse learning paths and reduce pupil engagement. Rather than rejecting structure, we propose a reconceptualisation: an educational fabric where linear and non-linear elements interweave to create more inclusive, responsive and dynamic pedagogies. This woven metaphor frames disruption not as disorder but as the creative rethreading of established forms, enabling layered, recursive and context-sensitive learning. Such a perspective extends to teacher education and professional development, where collaboration exemplifies the nonlinearity we seek to legitimise. We invite teachers, teacher educators, researchers, and policymakers to reconsider how time, knowledge, and learning are structured in order to reimagine mathematics education as a living, interconnected practice grounded in both rigour and relevance.

Reflecting on a soundart installation one year later troubles the question, “What are we going to do?,” decentering objectives with justification. Nested “laters” anticipate and evoke memories of past futures and future pasts, challenging direct study of crises and catastrophic predictions that reproduce an anthropocentric fixation on “What knowledge is of most worth?” Knowing and sharing things is magical, empowering, and individually self-fulfilling. Yet we can learn from those indigenous traditions in which magicians and shamans simultaneously offer prayers and ritual gestures to other animals, and to the powers of the earth and sky. The obligation is to ensure from the edges of our village that boundaries between human culture and the rest of nature stay porous and overlapping. Sharing what we know will not save the world. Developing forms of leadership, entrepreneurship, and civic engagement that enrich and expand reconnection with stewardship has a chance to do so.

Equitable teaching practices encompass evidence-based strategies utilised by teachers to support the learning of all pupils, while recognising each pupil’s unique background. This study delves into the perspectives of Scottish teachers regarding equity in mathematics, as evidenced by their self-reported practices. By analysing semi-structured interviews with 29 teachers from various school levels (early years, primary, and secondary) through Variation Theory, we were able to uncover common practices, such as ability grouping versus mixed-ability grouping, as well as practices specific to each level. We also identified overlaps between early-years/primary and primary/secondary, highlighting the importance of transitioning from one school level to another. It is imperative for teacher education programmes to address these transitions more explicitly in order to ensure greater continuity and consistency in equitable teaching practices.

Mathematics teaching usually occurs in a “public space”—e.g., a classroom is a public space with rules defining student and teacher roles that cultivate hierarchic relationships. Working in prison with young inmates (18–23 years old) from marginalized populations, we promoted alternative “common spaces” (physical and symbolic) challenging established hierarchies through equal contribution. Such mathematics curriculum employs an antideficit model (Louie et al., ZDM Mathematics Education 53(1):95–107, 2021) as prisoners co-research their lives with mathematics. Common spaces empower through recognition while enabling identity transformation. Qualitative data from the CoSpIRom [Common Spaces for Integration of Roma Project ( http://cospirom.sed.uth.gr ).] project explores: (a) whether common spaces, recognition and equal participation can support mathematics literacy while empowering and emancipating young prisoners; and (b) if and how this framework might prepare prospective teachers within contemporary diversity. Bringing together young prisoners and prospective teachers affects both populations. Mathematics curriculum developed through collaboration meets a consistent request from prisoners that educational programs bridge with the outside world as re-entry preparation. Contrasting life stories from prospective teachers operate as alternative models motivating prisoners to rethink their own life stories and self-concepts. Prospective teachers demonstrated significant intercultural communication skill development, empathy and solidarity for marginalized people, reporting important professional identity growth.

Unexpected lists that propel your teaching into refreshingly new directions! From lesson planning and assessment strategies to ideas for changing the world, there is something for everybody at every level and age of mathematics – entertaining humor, deeply serious provocations to push you out of the box, and good, clean wholesome tips for creative experiments in classroom organization.