<b>From Mathematical Micro- to Macro-identities: How Mathematical Authority Emerged from Moment-to-moment Positioning </b><b></b>

How authority operates in mathematics classrooms greatly influences students’ understanding of mathematics and the identities related to mathematics students develop. Teachers instrumentally inform how authority operates in classrooms, particularly how students are positioned as mathematical authorities, yet understanding of teachers’ perspectives on authority in classrooms is limited. This article aims to describe teachers’ perspectives of authority in mathematics classrooms by reporting prospective secondary mathematics teachers’ beliefs about mathematical authority. Drawing on interviews, lesson plans, field reflections, and teacher generated diagrams, I report four prospective teachers’ beliefs about teachers and students as mathematical authorities in classrooms. I report the prospective teachers’ similar, yet distinct beliefs about mathematical authority and how their strongly held beliefs helped explain variations across the four teachers’ descriptions of practice. Additionally, I discuss how the results support a dynamic, relational conceptualisation of mathematical authority. Implications for mathematical authority and teacher beliefs research are also discussed.

In this mixed-methods study, we introduce an activity-based perspective on mathematical authority that considers who leads mathematical activities in a classroom. Our framework for mathematical authority extends existing research that focuses primarily on Authoring mathematical ideas to include the activities of Visualizing and Speaking. Using data from 129 lessons, we identified six authority structures that reflect consistencies in how mathematical authority was distributed across Authoring, Speaking, and Visualizing. These authority structures provide greater clarity and nuance to prior work on shared authority. We also used loglinear models to identify relationships among co-occurring authoritative activities and found that when students had authority for Visualizing, Speaking, and using Previously Generated Student Work, they were more likely to have authority for Authoring.

(2004). Newton, Maclaurin, and the Authority of Mathematics. The American Mathematical Monthly: Vol. 111, No. 10, pp. 841-852.

Students’ ability to reason for themselves is a crucial step in developing conceptual understandings of mathematics, especially if those students are preservice teachers. Even if classroom environments are structured to promote students’ reasoning and sense-making, students may rely on prior procedural knowledge to justify their mathematical arguments. In this study, we employed a multiple-case-study research design to investigate how groups of elementary preservice teachers exercised their mathematical authority on a growing visual patterns task. The results of this study emphasize that even when mathematics teacher educators create classroom environments that delegate mathematical authority to learners, they still need to attend to the strength of preservice teachers’ reliance on their prior knowledge.

Questions regarding the construction of mathematical authority haveimplications for learning, specifically for students’ views of themselves as mathematics learners and doers with valuable contributions. We consider the ideas proposed by eight first-grade students who had the most airtime during 12 lessons of a classroom teaching experiment.We noted when ideas were proposed and how those ideas were responded to. We observed one student, Tia, be positioned and trusted as a source of mathematical authority by their peers and found that Tia had more airtime, made more contributions, and proposed more ideas than their peers. Our theoretical contribution is the link we make between work on mathematical authority and work on epistemic trust by making sense of our findings in terms of research on fact checking and trusting inaccurate informants.

PurposeThis paper examines how mathematics teacher educators (MTEs) learn to enact equitable mathematics instruction using technology through lesson study (LS).Design/methodology/approachA LS team with three MTEs conducted three iterations of LS on teaching the Pythagorean Theorem in an in-person, technology-mediated environment. Many forms of data were collected: Desmos activities, videos of research lessons (RLs), videos of MTE RL debriefings, artifacts of student learning in the Desmos Dashboard, and MTEs' written self-reflection. The authors investigate the teacher educators' learning through LS by analyzing the MTE debriefings of the RLs using Bannister’s (2015) framework for teacher learning in communities of practice.FindingsThe MTEs learned to enact equitable mathematics instruction using technology through addressing emerging issues related to intellectual authority and use of student thinking. Throughout the LS, the MTEs sought ways of promoting students' mathematical authority and using student thinking through features of the Desmos platform.Research limitations/implicationsThis study focuses on MTEs' learning without examining participating preservice teachers' learning. It demonstrates the benefits of LS for MTEs' professional learning.Practical implicationsThis study showcases how a research-based Desmos activity is used and refined to promote MTE learning how to implement equitable mathematics instruction.Originality/valueThe study contributes to better understanding of how LS could be used to develop MTEs' professional learning. Moreover, the dual process of participation and reification was concretized through diagnostic and prognostic frames in the LS context, which enriches the concept of community of practice.

Mathematics stands apart as a subject in the school curriculum, universally regarded as important, the only subject taught in practically every school in the world, and apparently sublimely impervious to the constraints of the cultural environment and social value-system within which it is being taught. One way in which mathematics education can convey values to children is through relating mathematics to society in a more comprehensive way than simply through individual examples or themes. J. Henry's thesis also shows why a competitive style of mathematics teaching is relatively ineffective in cultures which are based on cooperation rather than on competition, for example American Indians, and many African and Asian societies. Mathematics is important, and schools and parents world-wide give it high priority. Many Local Education Authorities in England have issued their own 'guidelines' for primary mathematics. At a higher level, of course, it can be argued that the authority of mathematics is illusory.

Space, Evidence, and the Authority of Mathematics

Ever since the great revolution which produced modern science there has been a hope that a science of society would be created on a par with the sciences of nature. Two early heroes of the Scientific Revolution, Galileo and Harvey, created radical transformations of science — respectively, a physics of motion and a physiology based on the circulation of the blood — which became paradigms for a new social science.1 Scientific precepts of Bacon and of Descartes were available as guides in this new venture. A primary challenge was to accommodate a new social science to mathematics: either to use classical mathematics for a non-traditional purpose or to introduce a kind of mathematics other than geometry on the Greek pattern. Would-be social scientists could thus find novel ways of dealing with their subject that would transfer to their endeavors the authority of mathematics and the new natural sciences.

To consider more fully the contextual complexities of living ethically as curriculum scholars, we wish to attend to the various discursive regimes that effectively delimit and circumscribe research projects initiated in partnership with Indigenous peoples and their communities. The habitual disregard of Indigenous peoples stems from the colonial frontier experience. The overriding assumption at work in these colonial frontier logics is that Indigenous peoples and Canadians inhabit separate realities. The inherent intention is to deny relationality. Within the research community there is an increased awareness of the importance of including Indigenous people in the development of research programs related to their communities. We were invited by an Indigenous community to work with the community and school leadership to develop a research program related to student performance in mathematics. Through our work, we have come to wonder about the authority of researchers, the authority of mathematics, and the authority of culture. We have come to understand how easy it is to replicate colonial logics as authoritative and have encountered conflicts when resisting these stances. In this paper, we offer some reflections and insights regarding how, and in what ways, we attempted to disrupt colonial logics. Through our listening to the teachings of children and teachers, we have come to conceptualize cultural relationality as an ethic guiding our participation in a research project with an Indigenous community.

In the early 1540ies G.J. Rheticus wrote an anonymous treatise entitled both Epistola deTerrae Motu and Dissertatio de Hypoth[esibus] Astron[omiae] Copernicanae. In thisletter he discusses why proclaiming the motion of the earth does not need to beconsidered as an impious act incompatible with the words of Holy Scripture. Based onan analysis of authorities mentioned by the author in this letter, I conclude thatRheticus’ strategy on the one hand consists in playing down the importance of thetraditional Aristotelian-Ptolemaic notions on the universe in the field of astronomy andby emphasizing the indirect character of Biblical authority in these matters. On the otherhand, he claims the absolute, immediate authority of mathematics in astronomy bywhich he consequently challenges the traditional medieval hierarchy of sciences.Rheticus considers the achievements of Copernicus to be part of divine providence.

THE "NEWTONIAN STYLE."Sir Isaac Newton revolutionized physics and astronomy in his book Mathematical Principles of

Newton, Maclaurin, and the Authority of Mathematics

In this paper, I will argue that with the emergence of digital virtual worlds (in video games, animation movies, etc.) by the animation industry, we need to rethink the role and authority of mathematics, also from an ontological point of view. First I will demonstrate that the application of mathematics to the creation and description of the digital, virtual worlds behaves in many respects analogously to the application of mathematics to the description of real-world phenomena from the viewpoint of the history of science. However, from other aspects, the application of mathematics significantly differs in this virtual world from the application to real-world fields. The main thesis of my paper is that the role of mathematics in the digital animation industry can be ontologically different from its usual role. In the application of mathematics to digital virtual worlds, mathematical concepts are no longer just modelling tools, forming a subordinated, computational basis, but they can direct and organise, and even create non-mathematical theory, something that we can call, for example, digital physics and biology. I will study this new, creative role of mathematics through some concrete phenomena, specifically through gravity. Our conclusion is that the animation industry opens an entirely new chapter in the relationship between (digital) sciences and mathematics.

The calendar is designed to speak directly to students, giving them open-ended questions to engage their intellect through their interests. Students ore encouraged to work on the activities individually, in pairs, or in small groups. No answers ore provided for the calendar questions so that students ore encouraged to look to themselves as the mathematical authority, to develop the confidence and critical-thinking skills necessary to validate their thinking.