国内外数学教育改革的历史表明，数学课程历来是数学教育改革的焦点.我国新一轮基础教育阶段的数学课程改革正如火如荼的进行,数学课程标准对于课程内容和教学方式作出了新的要求.新课程的理念主要通过相应的数学课程内容来实现，笔者将符合新课程理念的数学课程内容归为理解性数学.本文主要以弗兰登塔尔和伍洪熙的数学教育思想为基础，对理解性数学进行界定，给出理解性数学设计的定义与方法,并以带饰群和面饰群为主题进行理解性数学设计.具体工作包括：(1)以带饰群和面饰群为主题，基于相应的文献资料建构带饰群和面饰群的一个自然的、直观的理解，并能识别与构造相应的带饰与面饰.主要内容包括平面等距变换只有四种类型的证明和几何上只有七种带饰群、十七种面饰群的证明.(2)基于以上数学内容的理解,进行以带饰和面饰为主题的探究性课题学习.包括带饰与面饰的识别与构造，只有七种带饰群的探究性证明.(3)几何变换在中学数学相关教学中的应用.包括三种平面等距变换与三角形全等的判定法则，旋转变换与复数几何意义的引入.附录是理解性数学的另一例子.以期为中学数学探究性课题学习提供案例、为中学数学相关内容教学提供启示.

Mathematics curriculum is the center of the mathematics education task.The history of mathematics education reform at home and abroad shows that mathematics curriculum has always been the focus of mathematics education reform.A new round of mathematics curriculum reform of basic education stage in our country is in full swing.Mathematics curriculum standards has made new demands for course content and teaching methods.The concept of new curriculum implemented mainly through corresponding mathematics curriculum content,I will put the mathematics curriculum content conforming to the concept of mathematics curriculum standard classified as understanding mathematics.This article mainly based on H.Freudenthal and Hung-Hsi Wu professor's mathematical education thought to define the understanding mathematics,put forward the define and method of designing of understanding mathematics,and with Frieze groups and wallpaper groups as the theme,carrying out the design of understanding mathematics. Specific work include(1)With Frieze groups and wallpaper groups as the theme, based on the related literature material. Teasing out a natural way of thinking to understand Frieze groups and wallpaper groups. The main content including plane isometric transformation only four types of certificates and there are exactly seven geometrically different types of friezes and seventeen different types of wallpapers.(2)Based on the understanding of the above mathematical content, to carry on the investigative study design.including the recognition and creation of friezes and wallpapers.(3)The application of geometry transformation in schoolmathematics.So as to provide case for exploring topic learning of middle school mathematics and enlightenment for middle school mathematics related content teaching.