Geometry is a wonderful area of mathematics to teach. It is full of interesting problems and surprising theorems. It is open to many different approaches. It has a long history, intimately connected with the development of mathematics. It is an integral part of our cultural experience being a vital component of numerous aspects of life from architecture to design (in all its manifestations). What is more, geometry appeals to our visual, aesthetic and intuitive senses. As a result it can be a topic that captures the interest of learners, often those learners who may find other areas of mathematics, such as number and algebra, a source of bewilderment and failure rather than excitement and creativity. Teaching geometry well can mean enabling more students to find success in mathematics.
These aspects and considerations also tend to make geometry a demanding topic to teach well. Teaching geometry well involves knowing how to recognize interesting geometrical problems and theorems, appreciating the history and cultural context of geometry, and understanding the many and varied uses to which geometry is put. It means appreciating what a full and rich geometry education can offer to learners when the mathematics curriculum is often dominated by other considerations (the demands of numeracy and algebra in particular). It means being able to put over all these things to learners in a way that is stimulating and engaging, and leads to understanding, and success in mathematics assessments.
The study of geometry contributes to helping students develop the skills of visualization, critical thinking, intuition, perspective, problem-solving, conjecturing, deductive reasoning, logical argument and proof. Geometric representations can be used to help students make sense of other areas of mathematics: fractions and multiplication in arithmetic, the relationships between the graphs of functions (of both two and three variables), and graphical representations of data in statistics. Spatial reasoning is important in other curriculum areas as well as mathematics: science, geography, art, design and technology. Working with practical equipment can also help develop fine motor skills.
Geometry provides a culturally and historically rich context within which to do mathematics. There are many interesting, sometimes surprising or counter-intuitive results in geometry that can stimulate students to want to know more and to understand why. Presenting geometry in a way that stimulates curiosity and encourages exploration can enhance student’s learning and their attitudes towards mathematics. By encouraging students to discuss problems in geometry, articulate their ideas and develop clearly structured arguments to support their intuitions can lead to enhanced communication skills and recognition of the importance of proof. The contribution of mathematics to student’s spiritual, moral, social and cultural development can be effectively realized through geometry. As mentioned above, some ideas for using geometry to support spiritual and cultural development can be found in the references . Useful sources of material for supporting moral and social development can be found in the publications of the Stapleford Centre in Nottingham (for example, the Charis Mathematics resources for key stages 3 and 4), the “Summing up the World” series from Development Education in Dorset, and the Maths and Human Rights Resources Book published by Amnesty International.
Geometry is a rich source of opportunities for developing notions of proof. While more is said about this in a later section, it is worth emphasizing that visual images, particularly those, which can be manipulated on the computer screen, invite students to observe and conjecture generalizations. Proving conjectures requires students to understand how the observed images are related to one another and are linked to fundamental ‘building blocks’. In dynamic geometry software understanding observed images means working with points, circles, and parallel and perpendicular lines.
We live on a solid planet in a 3D world and, as much of our experience is through visual stimulus, this means that the ability to interpret visual information is fundamental to human existence. To develop an understanding of how spatial phenomena are related and to apply that understanding with confidence to solve problems and make sense of novel situations has to be part of the educational experience of all students. Geometry offers a rich way of developing visualization skills. Visualization allows students a way of exploring mathematical and other problems without the need to produce accurate diagrams or use symbolic representations. Manipulating images in the head can inspire confidence and develop intuitive understanding of spatial situations. Sharing personal visual images can help to develop communication skills as well as enabling students to see that there are often many ways of interpreting an image or a written or spoken description.
Much of our cultural life is visual. Aesthetic appreciation of art, architecture, music and many cultural artifacts involves geometric principles – symmetry, perspective, scale, orientation, and so on. Understanding many scientific principles and technological phenomena also requires geometric awareness, as do navigation, orienteering and map reading. Recognition of the familiar and the unfamiliar requires an ability to characterize and note key features.
Numerous current applications of mathematics have a strong geometric component. In many cases, the problem includes getting ‘geometric’ information into a computer in a useful format, solving geometric problems, and outputting this solution as a visual or spatial form, as a design to be built, as an action to be executed, or as an image to entertain. Solving these problems requires substantial geometric knowledge.
No comments:
Post a Comment