This book presents current perspectives on theoretical and empirical issues related to the teaching and learning of geometry at secondary schools. It contains chapters contributing to three main areas. A first set of chapters examines mathematical, epistemological, and curricular perspectives. A second set of chapters presents studies on geometry instruction and teacher knowledge, and a third set of chapters offers studies on geometry thinking and learning. Specific research topics addressed also include teaching practice, learning trajectories, learning difficulties, technological resources, instructional design, assessments, textbook analyses, and teacher education in geometry.
Geometry remains an essential and critical topic in school mathematics. As they learn geometry, students develop essential mathematical thinking and visualization skills and learn a language that helps them relate to and interact with the physical world. Geometry has traditionally been included as a subject of study in secondary mathematics curricula, but it has also featured as a resource in out-of-school problem solving, and has been connected to various human activities such as sports, games, and artwork. Furthermore, geometry often plays a role in teacher preparation, undergraduate mathematics, and at the workplace. New technologies, including dynamic geometry software, computer-assisted design software, and geometric positioning systems, have provided more resources for teachers to design environments and tasks in which students can learn and use geometry. In this context, research on the teaching and learning of geometry will continue to be a key element on the research agendas of mathematics educators, as researchers continue to look for ways to enhance student learning and to understand student thinking and teachers' decision making.
Most real-world spectrum analysis problems involve the computation of the real-data discrete Fourier transform (DFT), a unitary transform that maps elements N of the linear space of real-valued N-tuples, R , to elements of its complex-valued N counterpart, C , and when carried out in hardware it is conventionally achieved via a real-from-complex strategy using a complex-data version of the fast Fourier transform (FFT), the generic name given to the class of fast algorithms used for the ef?cient computation of the DFT. Such algorithms are typically derived by explo- ing the property of symmetry, whether it exists just in the transform kernel or, in certain circumstances, in the input data and/or output data as well. In order to make effective use of a complex-data FFT, however, via the chosen real-from-complex N strategy, the input data to the DFT must ?rst be converted from elements of R to N elements of C . The reason for choosing the computational domain of real-data problems such N N as this to be C , rather than R , is due in part to the fact that computing equ- ment manufacturers have invested so heavily in producing digital signal processing (DSP) devices built around the design of the complex-data fast multiplier and accumulator (MAC), an arithmetic unit ideally suited to the implementation of the complex-data radix-2 butter?y, the computational unit used by the familiar class of recursive radix-2 FFT algorithms.
The innovative volume seeks to broaden the scope of research on mathematical problem solving in different educational environments. It brings together contributions not only from leading researchers, but also highlights collaborations with younger researchers to broadly explore mathematical problem-solving across many fields: mathematics education, psychology of education, technology education, mathematics popularization, and more. The volume's three major themes-technology, creativity, and affect-represent key issues that are crucially embedded in the activity of problem solving in mathematics teaching and learning, both within the school setting and beyond the school. Through the book's new pedagogical perspectives on these themes, it advances the field of research towards a more comprehensive approach on mathematical problem solving. Broadening the Scope of Research on Mathematical Problem Solving will prove to be a valuable resource for researchers and teachers interestedin mathematical problem solving, as well as researchers and teachers interested in technology, creativity, and affect.
In this monograph, the authors present a modern development of Euclidean geometry from independent axioms, using up-to-date language and providing detailed proofs. The axioms for incidence, betweenness, and plane separation are close to those of Hilbert. This is the only axiomatic treatment of Euclidean geometry that uses axioms not involving metric notions and that explores congruence and isometries by means of reflection mappings. The authors present thirteen axioms in sequence, proving as many theorems as possible at each stage and, in the process, building up subgeometries, most notably the Pasch and neutral geometries. Standard topics such as the congruence theorems for triangles, embedding the real numbers in a line, and coordinatization of the plane are included, as well as theorems of Pythagoras, Desargues, Pappas, Menelaus, and Ceva. The final chapter covers consistency and independence of axioms, as well as independence of definition properties.
There are over 300 exercises; solutions to many of these, including all that are needed for this development, are available online at the homepage for the book at www.springer.com. Supplementary material is available online covering construction of complex numbers, arc length, the circular functions, angle measure, and the polygonal form of the Jordan Curve theorem.
Euclidean Geometry and Its Subgeometries is intended for advanced students and mature mathematicians, but the proofs are thoroughly worked out to make it accessible to undergraduate students as well. It can be regarded as a completion, updating, and expansion of Hilbert's work, filling a gap in the existing literature.
This book describes how a key signal/image processing algorithm - that of the fast Hartley transform (FHT) or, via a simple conversion routine between their outputs, of the real-data version of the ubiquitous fast Fourier transform (FFT) - might best be formulated to facilitate computationally-efficient solutions. The author discusses this for both 1-D (such as required, for example, for the spectrum analysis of audio signals) and m-D (such as required, for example, for the compression of noisy 2-D images or the watermarking of 3-D video signals) cases, but requiring few computing resources (i.e. low arithmetic/memory/power requirements, etc.). This is particularly relevant for those application areas, such as mobile communications, where the available silicon resources (as well as the battery-life) are expected to be limited. The aim of this monograph, where silicon-based computing technology and a resource-constrained environment is assumed and the data is real-valued in nature, hasthus been to seek solutions that best match the actual problem needing to be solved.
