In this paper, we explore the possible roles of craftwork at the origins of geometry. Since shapes are the characters of geometry, our focus is on crafting shapes. We review two approaches to the nature of making: hylomorphism and hylonoesis. Related to these, we distinguish between ‘shape’ and ‘shaping’. From studies on prehistoric pottery, we explore a conception of ‘shaping’ as thoroughly implicated by the manifold social, biological, and material life of a community. Then we explore the significance of ’shaping’, which animate shapes, immerse them in the vagaries of materiality, and fill them with secrets longing for their open realization; a realization that is at once — inseparably — material and imaginary. We make shapes as they shape our bodies, and then we become capable of imagining and gesturing them.

In this study, we develop a perspective on the diverse aesthetics historically associated with mathematics, inspired by Rancière's approach to aesthetics and politics. We call “Silencing Aesthetics” a dominant aesthetic that Rota has characterized as a “copout (…) intended to keep our formal description of mathematics as close as possible to the description of a mechanism”. The challenge this study attempts to explore is how to question silencing aesthetics to make space for inclusive ones. Our efforts have focused on setting up and studying inclusive and pluralist “Studios”, gathering craftworkers, anthropologists, mathematics educators, and mathematics enthusiasts. We include here a case study based on a conversation amongst basket weavers, anthropologists, and mathematics educators focused on the artisanal and mathematical nature of knots. We discuss the implications of aesthetical entanglements, such as those in our case study, for mathematics learning.

This study examines the entanglement of affects that occurred during a short episode at a science museum. The episode involved a small number of children and a teacher who had come to the museum in the context of a school field trip. It took place inside an exhibit called 'Hmong House', which reproduced various components of a traditional house of the Hmong people. A key aim of this paper is to trace, via the microethnographic analysis of a brief video recording, an affective journey meshing mathematical tessellation and Hmong shamanism. In addition, we elaborate on ways in which disparate themes, such as tessellation and shamanism, became interwoven in the life of those visiting the Hmong House at the time. The episode of the Hmong House may inspire other activities in which students or visitors, with life trajectories partially rooted in Indigenous cultures, can share practices that are foreign to other students. The most important qualities of these activities, we suggest, are the respectful dignity with which they are demonstrated and engaged with, and the freedom to undertake interdisciplinary journeys - without subjection to artificial disciplinary boundaries - in which improvisation and surprising turns are expected and ever-present.

Aesthetical Entanglements in Mathematics Learning Conference or Workshop Item How to cite: Nemirovsky, Ricardo; Kathotia, Vinay and Mégrourèche, Charlotte (2023). Aesthetical Entanglements in Mathematics Learning. In: 13th Congress of the European Society for Research in Mathematics Education (Shvarts, Anna ed.), 10-14 Jul 2023, Budapest, Hungary, European Society for Research in Mathematics Education.

This article reviews contributions to teaching reading and writing of Myriam Nemirovsky, whose conceptualization foreshadowed an emancipatory pedagogy. To do so, we have reviewed her work and interviewed three of her colleagues: Elena Laiz Sasiain, Liliana Tolchinsky Brenman and Francesco Tonucci. In the first part we recount key moments in Nemirovsky’s life and set forth ideas that helped her to develop her approaches to teaching reading and writing. The text explores the development of her pedagogical thinking based on her teaching and research experiences. Later, we present parallels between Myriam Nemirovsky’s work and ideas of Jaques Rancière and Joseph Jacotot to highlight core elements in an emancipatory pedagogy and illustrate their presence in Myriam Nemirovsky’s practices and thinking. To conclude, we reflect on how Nemirovsky’s teachings helped to mobilize innovative ideas among educators. Her legacy includes a conception in which learning how to read and write is a contextualized process already underway, always unfinished, in constant transformation and largely unpredictable. It is a vision which no longer prioritizes the measurable, neutral and standardizable.

Museum-based mathematics exhibitions are increasingly prominent but under-theorized learning environments. In this study, we analyse the curriculum of United States mathematics exhibitions developed in the early 21(st) century in terms of their complex suggestions about the nature of mathematics and mathematical sense-making. We apply Ranciere's notions of politics and aesthetics to explore what we describe as dissensus present in the texts, images, and multi-sensory exhibits of several major mathematics exhibitions. Our analysis characterizes this dissensus as a paradoxical mix of alternative and familiar mathematical aesthetics. On the one hand, we identify an alternative aesthetic emphasizing everyday ubiquity, sensuality, and informal sense-making. At the same time, we identify a countervailing emphasis on dominant notions of mathematics as esoteric, immaterial, and formal-symbolic. Museum mathematics efforts sometimes describe themselves as expanding how the public views and defines mathematics. A close examination of the exhibitions in this study reveals a complex picture, in which dominant and alternative forms of mathematics are co-present. The analysis suggests that museum-based mathematics researchers and practitioners view their work as containing political and aesthetic dimensions that can disrupt or reify what society counts as mathematics.

In this article we reflect on a line traced by Julia. Julia is an undergraduate student in a class that includes a project entitled ‘Lives of Lines’. As part of the activities of this project, the students were asked to draw continuously for a minute with a white marker on a black page, without lifting the marker, and without trying to represent anything in particular. We analyse Julia’s tracing of the line as a kind of improvisation – the same type of improvising that occurs in conversations, music playing, hiking, dancing and countless other activities. We characterize the improviser as a daydreamer immersed in a reverie: an open field of reciprocating forces, desires, surprises and recollections playing themselves out as some of them encounter their way forward free to proceed, and others do not. The improviser becomes an arena in which body, hand, pen, paper, chair, other bodies, traces, words and sounds mutually displace and attract on their own.

This article relates a case study on how a conversation with materials and diagrams – the actual use of materials and diagrams to think, imagine, explain, collaborate, design and build – featured a certain kind of interplay between material and digital components. The physical components present in this setting included a water wheel, which is a wheel driven by flow of water whose rotational motion is a classic example of chaotic dynamics regulated by Lorenz equations. Digital components allowed for real-time graphical displays corresponding to the turning of the water wheel. We selected for this article a sequence of episodes from an interview with Jake, an undergraduate student majoring in engineering. Through a micro-ethnographic analysis, we reflect on how Jake combined the responsiveness of the digital displays with the tangibility of the water wheel to gain insight into some of the intricacies of oscillatory motion.

This part conclusion presents some closing thoughts on the key concepts discussed in the preceding chapters. The part explores "what children's bodies do in places, paying attention to what happens beyond words. In particular, movement through and in place is the foremost way children experience museums". A place-assemblage is a configuration of relations and noteworthy corners that is altered on an ongoing basis by, say, children hiding, currents of air, sunlight across windows, a family entering into the museum and so on. The part provides at least one example of a line of flight that opened, or could have opened, a transfigured place. It presents an "informal buggy park" that had spontaneously been created by visitors allowing adults "to bring children out of their buggy to the creative play zone." The part describes a staircase that "held a fascination for the young visitors" to the point that one of the children went down the stairs bumping on her bottom.

This paper focuses on the emergence of abstraction through the use of a new kind of motion detector—WiiGraph—with 11-year-old children. In the selected episodes, the children used this motion detector to create three simultaneous graphs of position vs. time: two graphs for the motion of each hand and a third one corresponding to their difference. They explored relationships that can be ascribed to an equation of the type A – B = C. We examine the notion of abstraction on its own, without assuming a dualism abstract-concrete according to which more of one is less of the other. We propose a distinct path for the attainment of abstraction, which involves navigating a surplus of sensible qualities. The work described in this paper belongs to early algebra, we suggest, because it involves the elementary symbolic treatment of unknowns and generals. More broadly, it advances a perspective on the nature of mathematical abstraction.

This paper focuses on the emergence of abstraction through the use of a new kind of motion detector — WiiGraph — with 11-year old children. In the selected episodes, the children used the sensor to create three simultaneous graphs of position vs. time: two graphs for the motion of each hand and a third one corresponding to their difference. They explored relationships that can be ascribed to an equation of the type A – B = C. We propose two distinct paths for the attainment of abstraction, one focused on working with unknowns lacking sensible qualities, and another that involves navigating a surplus of sensible qualities. This study is a case study for the latter, which we portray as a process of opening channels of flow and exchange among sensible qualities, such that these cease to be self-enclosed and start to configure a plane of unity, which, far from denying their differences, brings them into mutual circulation.

We distinguish emergent learning from “teleological” learning, which is learning for the sake of passing pre-defined tests and goals. While teleological learning may succeed or fail, emergent learning is always going on in ways that move pass disciplinary boundaries and anticipated results. To advance a perspective on pedagogies of emergent learning we analyze selected episodes from a program for children who volunteered to enroll. The sessions alternated between the after school club they attended and an art museum. The program engaged the children in basket weaving, in the analysis of baskets exhibited at the museum, and with ways in which flat materials can be shaped in 3D space along distinct surface curvatures. These experiences have inspired us to outline two streams of pedagogical ideas that seem to nurture and go along with the unforeseeable paths of emergent learning.

We present a video-based study of family visits to Taping Shape, an immersive exhibition that allows visitors to explore the inside of geometric objects. The exhibition was designed to support embodied sense-making, intimacy, and material encounter with mathematical objects. This study builds on research on walking and movement as forms of place- and sense-making. We draw on the notion of a meshwork to examine how children and their families co-produce, develop familiarity with, and assemble meanings for the exhibition space. This case study focuses on 4-year-old Easton and his extended family, exploring how Easton's talk and movements are part of an intergenerational meshwork that weaves together an emergent and distributed sense of place within the built geometries of Taping Shape. Our findings further considerations of embodiment and materiality in children's geographies.

En este trabajo presentamos un estudio de casos con ninos de 11 anos para explorar el concepto de diferencia algebraica. Este estudio forma parte de un trabajo mas amplio realizado en el proyecto “Maths and Motion” en la Manchester Metropolitan University (UK), con el objetivo de incrementar nuestra comprension sobre como los estudiantes usan las nuevas tecnologias para dar sentido a conceptos matematicos. En trabajos previos se ha disenado un software que permite la exploracion de varios conceptos matematicos a traves del uso de sensores de movimiento basados en la tecnologia de Nintendo Wii, esta tecnologia nos permite registrar los movimientos de los participantes. De acuerdo con numerosos autores (Arzarello, Paola, Robutti, & Sabena, 2009; Elia, Gagatsis, & van den Heuvel-Panhuizen, 2014; Nemirovsky & Ferrara, 2009; Nemirovsky, Rasmussen, Sweeney, & Wawro, 2012) consideramos que el conocimiento matematico esta corporeizado, en el sentido de que el movimiento del cuerpo juega un papel central. Esta idea esta sustentada en una serie de hallazgos empiricos que relacionan cuerpo, conceptos y cognicion en un amplio rango de disciplinas.