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In order to transform mathematics classrooms for students with disabilities, teachers must design based on learners at the margins. Universal Design for Learning (UDL) is a framework built on these principles (Meyer et al., 2014), but lacks a robust process through which educators can reimagine mathematics. We theorize that when educators merge the principles of Universal Design for Learning in mathematics with the process of Design Thinking, more accessible designs for curriculum, routines, tools, and spaces would result. Using design research methodology, we analysed a 6-week summer course for mathematics educators on UDL and Design Thinking. Data sources included class video recordings and transcripts, class artefacts, as well as pre- and post-participant surveys. Findings indicate shifts in participants' knowledge of UDL from a passive to an active role, empathy as a central guiding principle, and the potential of using Design Thinking as a process to enact UDL.

Neurodiversity refers to the idea that all brains—no matter their differences—are valuable and should be accepted. Attitudes toward the neurodiversity perspective can have real-life impacts on the lives of neurodivergent people, from effects on daily interactions to how professionals deliver services for neurodivergent individuals. In order to identify negative attitudes toward neurodiversity and potentially intervene to improve them, we first need to measure these attitudes. This article describes the development and initial validation of the Neurodiversity Attitudes Questionnaire (NDAQ), including item revision based on expert review, cognitive interviews, systematic evaluation of participants' response process, and analysis of the instrument's internal factor structure using exploratory structural equation modeling. Pilot analysis with 351 individuals mostly living in the United States who were currently working in or intending to pursue helping professions indicates that the NDAQ has construct validity, is well understood by participants, and fits a five-factor structure. While the NDAQ represents the first instrument designed to specifically assess attitudes toward the neurodiversity perspective, further validation work is still needed. Neurodiversity refers to the idea that brain differences (including disabilities) are valuable and should be accepted. Attitudes toward neurodiversity can have real-life impacts on the lives of neurodivergent people (those whose brains do not fit society's "standard"). These impacts can include effects on daily interactions, as well as how professionals such as teachers and doctors deliver services to neurodivergent people. In order to identify negative attitudes toward neurodiversity and potentially improve them, we first need to measure these attitudes. This article describes the development of the Neurodiversity Attitudes Questionnaire (NDAQ). NDAQ development included revision of questionnaire items based on feedback from experts and neurodivergent people, systematically evaluating the way participants responded to questionnaire items, and analysis of how the NDAQ items are grouped into different factors. A preliminary analysis with 351 individuals mostly living in the United States who were currently working or planning to work in a helping profession (e.g. doctors, teachers, therapists, and so on) indicates that the NDAQ measures attitudes toward neurodiversity, is well understood by participants, and fits a five-factor structure. While the NDAQ represents the first instrument designed to specifically assess attitudes toward the broad idea of neurodiversity, further work is still needed. [ABSTRACT FROM AUTHOR]

Randomized incremental constructions are widely used in computational geometry, but they perform very badly on large data because of their inherently random memory access patterns. We define an insertion order which removes enough randomness to significantly improve performance, but leaves enough randomness so that the algorithms remain theoretically optimal.

We consider a problem which is part of the process of determining the three-dimensional structure of a protein molecule using X-ray crystallography: given an estimated map of the electron density of the molecule as a function on three-dimensional space, we identify regions which are likely to belong to a-helices. Our approach is to compute a new kind of skeleton - the power shape - and then identify the helical substructures within the power shape with a variant of geometric hashing.

The Delaunay triangulation of a set of points in 3D can size Th (n2) in the worst case, but this is rarely if ever observed in practice. We compare three production-quality Delaunay triangulation programs on some 'real-world' sets of points lying on or near 2D surfaces

The power crust is a construction which takes a sample of points from the surface of a three-dimensional object and produces a surface mesh and an approximate medial axis. The approach is to first approximate the medial axis transform (MAT) of the object. We then use an inverse transform to produce the surface representation from the MAT. This idea leads to a simple algorithm with theoretical guarantees comparable to those of other surface reconstruction and medial axis approximation algorithms. It also comes with a guarantee that does not depend in any way on the quality of the input point sample. Any input gives an output surface which is the 'watertight' boundary of a three-dimensional polyhedral solid: the solid described by the approximate MAT. This unconditional guarantee makes the algorithm quite robust and eliminates the polygonalization, hole-filling or manifold extraction post-processing steps required in previous surface reconstruction algorithms. In this paper, we use the theory to develop a power crust implementation which is indeed robust for realistic and even difficult samples. We describe the careful design of a key subroutine which labels parts of the MAT as inside or outside of the object, easy in theory but non-trivial in practice. We find that we can handle areas in which the input sampling is scanty or noisy by simply discarding the unreliable parts of the MAT approximation. We demonstrate good empirical results on inputs including models with sharp corners, sparse and unevenly distributed point samples, holes, and noise, both natural and synthetic. We also demonstrate some simple extensions: intentionally leaving holes where there is no data, producing approximate offset surfaces, and simplifying the approximate MAT in a principled way to preserve stable features.

The medial axis transform (or MAT)is a representation of an object as an infinite union of balls. We consider approximating the MAT of a three-dimensional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewise-linear approximation to the object surface, which we call the power crust. We assume that we are given as imput a sufficiently dense sample of points from the object surface. We select a subset of the Voronoi balls of the sample, the polar balls, as the union of balls represenation. We bound the geometric error of the union, and of the corresponding power crust, and show that both representation and topologically correct as well. Thus, our results provide a new algorithm for surface reconstruction from sample points. By construction, the power crust is always the boundary of a polyhedral solid, so we avoid the polygonization, hole-filling or manifold extraction steps used in previous algorithms. The union of balls representation and the power crust have corresponding piecewise-linear dual representation, which is some sense approximate the medial axis. We show a geometric relationship between these duals and the medial axis by proving that, as the sampling density goes into infinity, the set of poles, the center of the polar balls, converge to the medial axis.

The problem of computing a piecewise linear approximation to the surface from a set of sample points is important in solid modeling, computer graphics and computer vision. A recent alorithm [1] using the Voronoi diagram of the sample points gave a guarantee on the distance of the output surface from the original sampled surface assuming the sample was 'sufficiently dense'. We give a similar algorithm, simplifying the computation and the proof of the geometric guarantee. In addition, we guarantee that our output surface is homeomorphic to the original surface; to our knowledge this is the first such topological guarantee for this problem.

We give a simple algorithm for surface reconstruction from a set of point samples in R3, using only one three-dimensional Voronoi diagram computation. We also give a fairly simple proof that the reconstruction is topologically correct when the input is a sufficiently dense sample from a smooth surface.

This article explores how to ensure visitor engagement with art objects at exhibition spaces in art museums through relational aesthetics, which focuses on the intersubjective relationship that art objects arouse in visitors. In the 1990s, Bourriaud coined the term "relational aesthetics" in reference to interactive installation art, but the concept has yet to be actively examined in terms of visitor engagement. In this context, this study examines the theoretical framework of relational aesthetics in terms of its implications for art museum education. As a result, this article explains how constructing relational aesthetics depends on intervening in visitors' experiences with art objects through participatory acts to build individual visitors' creative agency as well as intersubjectivity between visitors and objects in the in-between space.