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This chapter focuses on the linking of geometry and algebra in the teaching and learning of mathematics - and how, through such linking, the mathematics curriculum might be strengthened. Through reviewing the case of the school mathematics curriculum in England, together with examples of how the power of geometry can bring contemporary mathematics...
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... the QCA report on algebra and geometry (Qualifications and Curriculum Authority, 2004) indicates, one way of linking geometry and algebra is to exploit the capacity of dynamic geometry software to provide novel ways of visualizing algebraic relationships. As an illustration, teachers in a Hampshire school (in England) worked on a project in which their students used dynamic geometry software to plot quadratic functions that match the flight of a basketball, providing their students with hands-on experiences of how the various algebraic coefficients affect the shape of the graph -as illustrated in Figure 1. Another way of linking geometry and algebra is to use different approaches to tackle the same problem. ...
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... A principal objective of mathematical studies is to develop mathematical thinking and to impart to students reasoning methods that will assist them in finding solutions both at higher mathematical levels and in other fields of learning and knowledge (Jones, 2010). While solving many different problems is one important means in developing sound reasoning habits (NCTM, 2009), we claim that reasoning development can be even more enhanced by using a variety of different methods for solving one individual problem. ...
Mathematics educators agree that linking mathematical ideas by using multiple approaches for solving problems (or proving statements) is essential for the development of mathematical reasoning. In this sense, geometry provides a goldmine of multiple-solution tasks, where a myriad of different methods can be employed: either from the geometry topic under discussion or from other mathematical areas—analytic geometry, trigonometry, vectors, complex number, etc. Employing multiple proofs fosters better comprehension and increased creativity in mathematics for the student/learner, enriching teachers’ pedagogical accomplishments and promoting lively class discussion. Given the important role of multiple-solution problems within and between mathematical topics, the evidence is astonishing that classroom teachers rarely introduce their students to multiple-solution tasks. Hence, one can conjecture that this gap between theory and practice could turn connecting tasks with the employment of technological tools into a powerful environment for the development of pre- and in-service mathematics teachers’ knowledge. For this reason the authors believe that exposing and providing mathematics teachers with an arsenal of specific tasks with a variety of solutions from different mathematical areas is essential. Based on a conducted case study, both teacher trainees and lecturers clearly indicated that solving problems in multiple ways is valuable in developing thinking ability for both students and teachers, encouraging creativity and increasing the quality of teaching—hence this technique should be included in the secondary school curriculum as well as in teacher training programs.
... This shows that more efforts are needed in mathematics education research if more students are going to be successful in mathematics. Such efforts may benefit from more attention being paid in the curriculum and in textbooks to the links between number, algebra and geometry (Jones, 2010). ...
The topic of sequences and series is important in mathematics and in a wide range of other disciplines. The mathematics of infinite series, in particular, has significant applications in physics, biology, economics, medicine and other disciplines. This paper provides an analysis of the topic of sequences and series in the curriculum and textbooks for schools in England. It shows that ideas about mathematical sequences begin to be taught in primary school and extent to cover sequences of natural numbers, often set in a geometrical context. Students in England who opt to continue with mathematics beyond age 16 study both arithmetic and geometric sequences, including the sum to infinity of a convergent geometric series. Further options at this age mean that some students study the summation of simple finite series and induction proofs for summation of series and for finding general terms. Overall, in the topic of sequences and series there are opportunities to link numeric, algebraic and geometric ideas.
This study aimed to empirically investigate preservice teachers’ learning experiences during a series of geometry-themed choreography and dance activities. These pedagogical tasks were designed to examine the challenges and solutions that participants experienced when exercising their spatial reasoning abilities, particularly during the transition between the choreography designs they had generated on a Lego-based grid worksheet and real-world dancing positions. The data collection for this study took place at an elementary school located within a bilingual metropolitan area along the southwestern United States border. The overall findings indicated that all of the participating preservice teachers’ dance teams encountered interrelated challenges, most of which were variations on disorientation, primarily due to transitioning between representations of space and the deprivation of the typically employed spatial navigation reference points that are used in normal activities.
Fecha de recepción: 23/03/2013 Fecha de aceptación: 15/11/2013 Resumen En este artículo presentamos un estudio en el que discutimos el aprendizaje de perímetro y área de figuras geométricas, y señalamos algunos caminos de tratar las dificultades de aprendizaje de estas nociones geométricas, caminos que producimos y describimos en algunas de nuestras investigaciones anteriores, en la cuales utilizamos el Modelo de los Campos Semánticos como marco teórico. Vamos a mostrar también algunos resultados de estas investigaciones, que nos han permitido elegir a las características deseables para el desarrollo de tareas educativas que involucran área y el perímetro, basado en el proceso de producción de significados. Palabras clave: figuras geométricas, modelo de campos semánticos. Abstract This article presents a study in which we discussed the learning of perimeter and area of geometric figures, and point out some ways to treat learning difficulties these geometric notions, such paths we created and described in some of our previous research, in which we use the Model of Semantic Fields as theoretical referential. We'll show also some results of these investigations, which allowed us to elect desirable characteristics to the development of educational tasks involving area and perimeter, based on the process of production of meanings.
This chapter focuses on digital technologies and geometry education, a combination of topics that provides a suitable avenue for analysing closely the issues and challenges involved in designing and utilizing digital technologies for learning mathematics. In revealing these issues and challenges, the chapter examines the design of digital technologies and related forms of learning activities for a range of geometries, including Euclidean and co-ordinate geometries in two and three dimensions, and non-Euclidean geometries such as spherical, hyperbolic and fractal geometry. This analysis reveals the decisions that designers take when designing for different geometries on the flat computer screen. Such decisions are not only about the geometry but also about the learner in terms of supporting their perceptions of what are the key features of geometry.
KeywordsDesign-Digital technologies-ICT-Learning-Geometry-Geometries
