16,515 research outputs found
Learning by Seeing by Doing: Arithmetic Word Problems
Learning by doing in pursuit of real-world goals has received much attention from education researchers but has been unevenly supported by mathematics education software at the elementary level, particularly as it involves arithmetic word problems. In this article, we give examples of doing-oriented tools that might promote children\u27s ability to see significant abstract structures in mathematical situations. The reflection necessary for such seeing is motivated by activities and contexts that emphasize affective and social aspects. Natural language, as a representation already familiar to children, is key in these activities, both as a means of mathematical expression and as a link between situations and various abstract representations. These tools support children\u27s ownership of a mathematical problem and its expression; remote sharing of problems and data; software interpretation of children\u27s own word problems; play with dynamically linked representations with attention to children\u27s prior connections; and systematic problem variation based on empirically determined level of difficulty
The projector algorithm: a simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs
The Voronoi diagram is a certain geometric data structure which has numerous
applications in various scientific and technological fields. The theory of
algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich
and useful, with several different and important algorithms. However, this
theory has been quite steady during the last few decades in the sense that no
essentially new algorithms have entered the game. In addition, most of the
known algorithms are serial in nature and hence cast inherent difficulties on
the possibility to compute the diagram in parallel. In this paper we present
the projector algorithm: a new and simple algorithm which enables the
(combinatorial) computation of 2D Voronoi diagrams. The algorithm is
significantly different from previous ones and some of the involved concepts in
it are in the spirit of linear programming and optics. Parallel implementation
is naturally supported since each Voronoi cell can be computed independently of
the other cells. A new combinatorial structure for representing the cells (and
any convex polytope) is described along the way and the computation of the
induced Delaunay graph is obtained almost automatically.Comment: This is a major revision; re-organization and better presentation of
some parts; correction of several inaccuracies; improvement of some proofs
and figures; added references; modification of the title; the paper is long
but more than half of it is composed of proofs and references: it is
sufficient to look at pages 5, 7--11 in order to understand the algorith
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