Selected Writings on Mathematical Education by Zhong Shanjie

Author: Li Zhonglai / Country: Mainland China
Publisher:
Publish Date: 2006-03-29
Features:
The topic concerns parametric equations. Since some commonly used curves, such as cycloids and the evolutes of a circle, are often expressed more conveniently using parametric equations, and parametric equations are frequently used when studying higher mathematics and other disciplines, it is both appropriate and necessary to introduce some knowledge of parametric equations (even if it is relatively simple) in secondary school plane analytic geometry courses. Of course, it is advisable to arrange this topic toward the later part of the course. To better achieve teaching objectives, in addition to correctly selecting and organizing the curriculum, it is also essential to appropriately determine teaching methods. As mentioned earlier, in terms of educational objectives, there are significant differences compared to other disciplines. This means that the teaching methods should be chosen to fully reflect the concepts of connection, movement, and change, thereby achieving the educational goals.
Of course, mental training cannot be carried out independently of the curriculum. However, in practice, if mental training is conducted sufficiently, it can indeed provide great help in teaching the curriculum. Therefore, the most prominent feature of teaching methods for analytic geometry lies in the reflection of these three concepts. Without a doubt, fully utilizing visual elements in teaching is an effective method to reflect the concept of change.
Of course, in secondary school mathematics teaching, the proportion of visual teaching is much lower for upper grades compared to lower grades. However, in reality, it is difficult to observe changes in graphs without observing them. It is also difficult for students to discover the laws of graphical changes without demonstrations using teaching aids (movement). Therefore, using visual elements promptly and appropriately is still something that deserves special attention in teaching.
Additionally, to reflect the concepts of connection and change, it is also very necessary to pay close attention to the correct use of analogy and comparison in the teaching process. For example, after studying ellipses, if the process of studying hyperbolas is compared to that of ellipses, it will not only help students understand hyperbolas better but also make it easier for them to grasp the similarities and differences between the two from a fundamental perspective. For instance, when studying systems of lines and circles, if specific numerical values are given to the parameters in the systems of lines or circles, allowing students to observe and compare, this will not only help them understand such abstract concepts better but also enable them to better understand how to recognize objects with common properties in a unified way. Therefore, analogy and comparison should also be considered as major teaching methods in analytic geometry.
Finally, since analytic geometry is typically taught in secondary school upper grades, it is studied only after students have mastered algebra, geometry, and trigonometry. Therefore, through the teaching of analytic geometry, it is appropriate to reinforce the knowledge of algebra, geometry, and trigonometry. Of course, this does not mean simply restating old knowledge at appropriate times, but rather refers to teaching in a way that allows students to gain a deeper understanding of old knowledge from a higher perspective, thereby achieving further reinforcement.
Only through the comparison of different perspectives can students gain a new understanding of the essence of knowledge, leading to a more profound impression and thus a more solid grasp. Conversely, when applying knowledge, based on this new understanding, students can also achieve more efficient application methods. P6-7