在教室中的大混乱

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　　在数学教室，数学技术的最有趣的应用之一是：它让老师带来许多新的和令人兴奋的主题进入课程。尤其，技术让老师带来对研究数学的兴趣也使当代的一些主题进入中学和中学教室.大混乱的数学主题和不规则碎片形在这方面特别地适当。他们是适时的，在这些领域的许多主意首先在学生期间被构思。他们是可应用的，应用的领域有像变化多端的医学，经济学，地质学，艺术和音乐，它们已经采用来自这些领域的的方法。而且他们很美丽---在计算机中所产生的物体的图像是华丽的 , 像是 Mandelbrot 组，Julia组，Koch雪片，和其它的一些吸引学生兴趣和狂热的图象。

　　然而，其中也存在问题。许多数学家不愿意看见另外的不规则碎片。时常，大混乱和不规则碎片的讨论堕落到只是用简单的漂亮照片表示, 全无任何数学的内容。最终, 学生认识到：现代的数学和一个电视游乐器类似，是计算机产生的许多活动 , 但是是不小心所产生的活动。这态度是既不幸的且不必要的。在漂亮的照片后面的是数学方法, 并且, 许多的方法能够被中学学生所理解。此外, 在图像后面的数学时常是比它们所产生的照片更加美丽!在这种情形下，学生是很容易了解认识一些兴奋 , 数学的同时代的主题, 仍然在最后错过，这是一种悲剧。我们的目标是在这一篇文章中介绍一些比较简单的分形方法给人们认识，从而，纠正这种悲剧。

大冒险游戏

One of the most interesting applications of technology in the mathematics classroom is the fact that it allows teachers to bring many new and exciting topics into the curriculum. In particular, technology lets teachers bring some topics of contemporary interest in research mathematics into both middle school and high school classrooms.

The mathematical topics of chaos and fractals are particularly appropriate in this regard. They are timely---many ideas in these fields were first conceived during the students' lifetimes. They are applicable---fields as diverse as medicine, business, geology, art, and music have adopted ideas from these areas. And they are beautiful---there is something in the gorgeous computer generated images of objects such as the Mandelbrot set, Julia sets, the Koch snowflake, and others that capture students' interest and enthusiasm.

Therein, however, lies the problem. Many research mathematicians cringe at the sight of ``still another fractal.'' Most often, discussions of chaos and fractals degenerate to simple ``pretty picture shows,'' devoid of any mathematical content. As a consequence, students get the idea that modern mathematics is akin to a video game---lots of computer-generated action, but mindless activity at best.

This attitude is both unfortunate and unnecessary. There is mathematics behind the pretty pictures, and moreover, much of it is