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In this article, we consider the phenomenon of improvisation by small groups of middle years students while engaged in rich mathematical tasks in a classroom setting. Working from the premise that improvisation comprises a spectrum of behaviour, we propose that there is a range of improvisational behaviours that may be observed as the students work together. We discuss four levels along the spectrum -- interpretation, embellishment, variation, pure improvisation -- and offer vignettes from our research to illustrate each of their characteristics. We argue that considering improvisation as a spectral behaviour is a valuable way to view students' mathematical problem solving as it highlights how students draw on their own experiences and understandings while problem-solving, and how they vary in how far from the "script" they may venture in their discussions. Our results also show the importance of giving students time and space to "stay with" a mathematical task.

This article explores innovative praxis in Environmental and Sustainability Education (ESE) in four preservice teacher education programmes in Canada. ESE is finding its way into teacher education in a variety of innovative and interdisciplinary ways, as both part of mainstream programmes and in their co-curricular margins. Using a case study approach, each case builds on unique connections to Indigenous education, art education, cultural learning or educational gardening, which supports a variety of differing aspects in relation to ESE. These cases share a common theme of building relationships at the heart of ESE teaching and learning in the mainstream and the margins of the academy. Brought together through a Canadian network of faculty, researchers, policy-makers and community educators that was formed in 2016, these cases demonstrate a deep commitment and imaginative capacity for embedding ESE in Canada's teacher education systems.

We come together, five mathematics teacher educators from varied cultural backgrounds and diverse academic pathways, interested in teacher professional learning, and interested in exploring our understanding of colonial practices in mathematics education specifically and in education more generally. In this paper, we share our stories, drawing upon our own experiences in conversation with each other and in dialogue with academic readings. Our paper studies tensions encountered as we explore decolonizing educational practices within colonial structures, paying close attention to place/land-based pedagogies. In recognizing education as a colonial act, we examine some of the literature in decolonizing education and research before introducing arguments for the role of mathematics education in colonizing educational practices. As a collaborative research group, we met regularly over a period of 3 months, collecting audio recordings of our meetings, transcriptions, shared individual writings, and written responses to each other's writing and academic readings. We draw on narrative inquiry to structure our experiences of decolonizing mathematics teacher professional learning through living and telling stories and then through re-telling and re-living stories. We argue that this process is rewarding and challenging, and requires individual and collective ongoing dialogue. With our stories, we challenge the places where mathematics is performed, for example from classrooms to learning gardens, and what counts as mathematics. We conclude with questions to frame future directions and dialogues with an invitation to others to respond through critical dialogue and practice.

The co-authors, collaborators in garden-based teacher education, question the hegemony of grids in Western time, space and relationship structures in education, while also delving into our own complicity and entanglement with these grids. We ask: (1) Can we reject the grid in environmental education and in garden-based learning when it is an intimate part of our way of being in the world? (2) Can we be teachers who are at once within and not-within the regime of the Western Enlightenment/Modernist grid? (3) How can we take a ludic approach to the grid? Can we 'swing' and 'parkour' the strict grid of schooling? Pleasures and failures of the grid and experiences of alternatives to the grid are documented and exemplified through stories from garden-based teacher education. Considering parallels with principles of the alter-global movement and with the performative, embodied practices of swing dance, parkour and clowning, we meditate on becoming ecological teachers together beside the grid -- neither within nor without it, but with a deep awareness of its presence and structure. In uncertain, unchanging times, we want to take a playful, artistic approach to the old certainties and structures, swinging and parkouring from them rather than accepting or rejecting binary formulations.

Drawing from sources in gesture studies, cognitive science, the anthropology of religion and art/architecture history, this article explores cultural, bodily and cosmological resonances carried (unintentionally) by mathematical graphs on Cartesian coordinates. Concepts of asymmetric bodily spaces, grids, orthogonality, mapping and sacred spaces are discussed. The article argues that learners relate graphs to their own bodies through gesture, and this relationship extends to cultural associations with actual, imagined and built spaces. It is suggested that a more culturally-engaged mathematics pedagogy would benefit learners by acknowledging ancestral resonances already present in mathematical graphs and allowing for deeper learner participation in mathematics.

In recent years a great deal of work on mathematical word problems has focused on efforts to bring more of "real life" into the problems themselves and students' uptake of these problems. Following on from earlier studies of the word problem as a pedagogical and literary genre, the author argues that we cannot unproblematically assume an ability to know and to represent or match "real life" in word problems. The author uses the work of contemporary theorists Bakhtin, Lacan, Zizek and Baudrillard, including concepts like the chronotope, the Lacanian Real, Imaginary and Symbolic, simulacra, and impossible exchange, to argue that conditions of our postmodern world render transparent matchings of word problems with "real life" unsustainable.

This qualitative study explores the potential for metaphor, movement, gesture, and vocalization in helping learners notice mathematically important features of graphs, and in making mathematics more accessible for learners with visual impairment. Two elementary school students with visual impairment were introduced to several multimodal activities related to the graphs of mathematical functions, using a pre-/post-assessment methodology. Video recordings of the session were coded for qualitative changes in engagement with graphs through multimodal representations. After the activity intervention, both students showed improvements in their ability to voice, gesture, and describe details of mathematical graphs with accuracy and understanding. The findings demonstrate the potential of multimodal methods for teaching mathematics and enhancing other skill areas through movement, metaphor, voice, and gesture. The findings suggest that full-bodied experience with graphs can provide foundational support for learners with visual impairment to work with print or tactile graphics. We propose that purposeful selection of materials and collaboration between teachers of students with visual impairment, mathematics educators, and teachers of dance and physical education can enhance the design and implementation of effective lessons using multimodal means. [ABSTRACT FROM AUTHOR]

By their nature, gardens embody diversity. This article explores the cultural significance and value of school gardens for diverse communities in restoring and reclaiming their physical, mental, emotional, and spiritual health and resilience through stories, myths, and practical examples. It highlights details for experiential dimensions of garden based learning. Grounded in the research-based, seven-fold benefits of garden-based sustainability education, this article is the international collaborative effort of garden researcher-practitioners from Indigenous, multicultural, urban, biocultural, and STEM perspectives from over a half dozen different diversity-intensive urban learning gardens in the Pacific Northwest. It also describes dynamic experiential teaching approaches for sharing stories and engaging with hands-on approaches to garden-based learning at multiple scales and modes. Vivacious, research-based, garden-based learning from regional learning gardens activates urban areas as sites of diversity-enhancing sustainability education, nurturing the resilience and collaborative creativity required for biocultural flourishing. [ABSTRACT FROM AUTHOR]

The article offers a theoretical argument for the inclusion of mathematical performance in school mathematics. The author says that mathematical performance can be a medium for learning and inquiry as digital technology is reconfiguring the culture of mathematics teachers. She emphasizes that mathematics teacher should consider the nature of performance. She says that the best experiences of a democratic music that works involves mutual listening and expression.