随着多元化几何课程理念的提出，数学教科书原有的平面几何内容被消减，逻辑推理形式与深度都随之发生了改变，几何例习题的数量与难度也发生了重大的调整与变化，这些都是几何教科书的热议问题。逻辑推理是数学命题学习的重要手段，是建构数学体系的重要方式，而几何学科在逻辑推理的培养中具有不可替代的作用。新一轮课程改革的关键词数学核心素养赋予逻辑推理新的时代特征，几何教科书该用怎样的推理方式呈现逻辑推理？推理的深度又如何？这些问题再次将几何教科书的研究推到改革的前沿。研究选取中国大陆、中国香港、新加坡、美国、英国和澳大利亚五个国家六个版本的初中数学教科书几何内容，比较年级覆盖初中学段的三个学年。研究首先对比分析不同教科书几何内容章节知识结构，并以具体知识点加以佐证，确认六本教科书几何内容的公共知识结构并分类，进而确定不同教科书几何内容的四个公共知识主题，分别为三角形知识主题、多边形知识主题、圆知识主题和立体图形知识主题。这四个知识主题涵盖不同教科书几何知识点的比重最低约78%，最高达到90%。研究对六本教科书的几何内容分布即几何内容容量问题、几何内容广度即几何知识点数量问题、几何推理方式、几何推理深度以及几何例习题等方面进行比较分析。 几何推理是本研究的重点内容，研究分为几何推理方式与几何推理深度两个维度。对推理方式的分类与推理深度的分层以范希尔几何思维发展水平层级作为分类分层基础，借鉴TIMSS评测研究模式对数学推理层级的划分、几何课程目标体系模型、国内外几何推理分类模型以及数学思维活动经验的维度划分，以各国初中数学课程标准对几何内容呈现的具体要求作为佐证，最终确定几何推理方式分类，为直观辨析、实验演示、类比推理、归纳推理、演绎推理和演绎证明六种，几何推理深度包括层级1到层级4的四个水平层级，通过水平层级确定不同教科书几何推理深度。几何内容分布比较研究指标由几何内容页码数与数学内容页码总数的比值所确定；几何内容广度由四个知识主题知识点数目来确定；几何例习题难度以要求水平、包含知识点数量和背景三个维度确定。研究以几何内容广度、几何推理深度、几何例习题难度确定六本教科书几何内容的整体难度。 主要研究结果如下： 1. 几何内容分布 六个版本教科书几何内容比重的平均值是29.61%，只有中国大陆和香港教科书以40.18%与35.54%比重超过此均值。中国大陆教科书随年级增长而增加几何内容比重；美国教科书随年级增长而减少比重；新加坡教科书重初始两个年级而轻末尾年级比重；英国教科书几何内容平均分配比重；中国香港与澳大利亚教科书都是重首尾轻中间年级比重。中国大陆教科书是立体几何知识容量唯一低于5%的版本。 2. 几何内容广度 几何知识点数量在六本教科书中分布差异很大。排在第一位的教科书知识点个数要比排在末位教科书多出41.28%。中国大陆与香港以109个知识点并列居首，新加坡少了约10%的知识点排在第三位，排在第四位的美国知识点又下降约5%，英国的知识点下降比重很大，约20%，排在第五位。最末位的澳大利亚还要降约14%的知识点。四个知识主题的知识点分布，中国大陆重三角形和圆两个知识主题轻立体图形知识主题；中国香港重立体图形知识主题轻圆知识主题；美国、新加坡、英国和澳大利亚都是注重多边形知识主题而轻圆知识主题内容。 3. 几何推理 几何推理包括几何推理方式、几何推理水平层级以及几何推理深度三个比较领域研究。直观辨析、实验演示、归纳推理和演绎推理具有使用的广泛性，其中直观辨析和实验演示也具有使用比重的突出性。演绎证明只在中国大陆和香港教科书中使用，类比推理使用比重最少，最高比重不到4%。中国大陆教科书多重推理方式使用种类最多但比重低，美国与新加坡教科书重实验演示和归纳推理的二重复合使用。概念和命题的推理深度层级主要分布在层级2、3上，层级2比重最多，层级4只分布在中国大陆和香港两本教科书中。几何推理深度排名依次以中国大陆、美国、中国香港、新加坡、澳大利亚与英国教科书降序排列。六本教科书在几何推理方式与深度层级上呈现两两教科书相似的特征。 4. 几何例习题 美国教科书没有编写例题。中国香港教科书例习题数量突出，澳大利亚教科书例题大题数量最少，中国大陆教科书习题小题数量极少。六本教科书几何例习题编写题型丰富，例题均没有编排开放性题目，习题中开放性题目比重都不高。几何例题难度以中国大陆、中国香港、新加坡、澳大利亚、和英国降序排列。几何习题难度以中国大陆、新加坡、美国、澳大利亚、中国香港和英国降序排列。 5. 几何内容整体难度 教科书几何内容整体难度以三个维度即相对广度、相对深度和习题相对难度确定的难度结果是以中国大陆、中国香港、美国、新加坡、英国和澳大利亚降序排列。在四个维度下即相对广度、相对深度、例题相对难度及习题相对难度确定的难度结果是以中国大陆、中国香港、新加坡、英国和澳大利亚降序排列。两种教科书整体难度排名具有一致性。 通过比较研究得到相关启示：保持现有几何内容容量优势，加强几何例习题的编写；在现有几何推理方式与推理深度优势的基础上坚持教科书几何推理编排的“启发性”功能，适当给予一定的“开放性”和“自由度”；有效达到多样化信息技术与几何内容相融合的目标。

With the pluralism of geometrical curriculum revolution proposed, some geometry content in mathematical textbook was deleted, patterns and depth of geometrical reasoning were changed, and the quantity and difficulties of examples and exercise in geometry were adjusted and changed. All of those are the hot topics in the area of geometrical textbook. Logical reasoning is important method to study mathematical proposition and construct the system of mathematical knowledge. Geometry is irreplaceable for training logical reasoning. The key competency is the key word of new mathematical curriculum revolution which brings new feature of geometrical reasoning. How to choose the proper presence of geometrical knowledge in mathematical textbook? How depth is it? The research of geometrical textbook is pushed to the forward postion. This research is the comparative study of geometrical content in mathematical textbook on the whole middle school level of six different verisons choosed from the mainland of China, Hongkong, Singapore, America, England and Australia. The first work is to confirm four comparative topics of geometrical content which are triangle topic, polygon topic, circle topic and solid figure topic by the comparison and analysis the sharing geometrical knowledge structure in chapters, sections and the exact content of different books. These four topics cover the knowledge points from 78% to 90% in different books. There are five research areas in this paper which are the distribution of geometrical content, the scope of geometrical content, geometrical reasoning, geometrical examples and exercises. On the above basis, this paper compares the degree of difficulty of geometrical content in different books. The most important area in this paper is geometrical reasoning. The new study tool built by this paper is pointed to the research of geometrical reasoning basis on the Van Hiele model of the development of geometric thought, the evaluating modle of TIMSS, the model of the objective sysem of geometrical course in China, the model of geometrical reasoning pattern from Australia and China, dimenstions of experience and activities of mathematical thinking and the mathematical curriculum standard built in different comparative countries and the area. The distribution of geometrical content is decided by the proportion of the numbers of textbook pages of geometrical content and the whole number of textbook pages. The scope of geometrical content is decided by the numbers of points of geometrical knowledge. The difficulty of geometrical examples and exercises is decided by the level of requirement, numbers of points of knowledge in each example and exercise, the background of each example and exercise. At last it determined the difficulty of whole geometrical content of each textbook by the scope of geometrical content, the depth of geometrical reasoning, the difficulty of geometrical examples and exercises. The study has the following conclustions. 1. Distribution of Geometrical Content The percentage of distribution of geometrical content in textbooks of Chinese mainland and Hong Kong are over 29.61% that is the average number of six different textbook versions. The former’s is 40.18% and the latter is 35.54%. From the first to the third grade, Chinese mainland’s textbook is the unique to increase the percentage of geometrical content by the increasing of grades. On the contrary, America decreases, meanwhile its percentages in first two years are higher than the last year’s. Singapore has the similar situation to America. England’s textbook chooses equal division in three grades. The version used in Hong Kong and Australia concentrate on both ends of grades rather than the middle. The percentage of content of 3-dementional geometry in Chinese mailand’s textbook is lower than 5%. 2. Scope of Geometrical Content The situation of distribution of points of geometrical knowledge is very different of six books. The most points is 109 in the textbooks of mainland and Hong Kong of China both. The quantity of points in Singapore’s textbook decreases about 10% . The points of knowledge in textbook of America is very close to the former, just 5 points less. The points in extbook of England decreases sharply by bout 20%. The last position is the textbook of Australia with 14% points of geometrical knowledge less. Obviously, there are 48% points more of the first ranking textbook than the last one. The knowledge arranged in triangle topic is emphasized, but the weakness of solid figure topic in the textbook of the mainland of China. Hong Kong’s textbook is emphasized on the solid figure topic rather than the circle topic. The polygon topic is emphasized in other four textbooks, but the circle topic is weak. 3. Geometrical Reasoning The comparative study of geometrical reasoning is composed by the reasoning patterns, the reasoning levels and the depth of reasoning. The reasoning patterns has six types: visual discrimination, experimental demonstration, induction reasoning, analogic reasoning, deductive reasoning, deductive proof. The reasoning levels has level 1, level 2, level 3 and level 4. It is found that visual discrimination, experimental demonstration, induction reasoning and deductive reasoning have comprehensive using in six textbooks, but the only first two of them have highlighting in using percentage. Deductive proof was used in Chinese mainland version and Hong Kong version mostly. The most weakness is analogic reasoning using in whole comparative textbooks because the highest percentage is near to 4% in the textbook of the mainland from China. The depth levels of six textbooks are focus on level 2 and level 3, and the former percentage is more. Chinese mainland version and Hong Kong version are only two books which have level 4. The decreasing ranking of reasoning depth is composed by the mainland of China, America, Hongkong, Singapore, Australia and England. 4. Examples and Exercises of Geometrical Content American textbook has no geometrical examples. Hong Kong textbook has the most quantities of geometrical examples and questions in the exercise, and the quantity of them is more large than the other books. On the contrary, Austrilian textbook organizes the least quantity. In Chinese two verstions, it is not good at the organization of sub-questions. In general, all questions of different versions are in eight types. The proportion of calculation is more than answer questions. There is no examples in open topic, and the proportion of this type in exercise is very low. The ranking of the degree of difficulty of examples is descended by the mainland of China, Hongkong, Singapore, Australia and England. Descending order of difficulty of exercise is the mainland of China, Singapore, America, Australia, Hongkong and England. 5. Degree of Difficulty of Geometrical Content The ranking of difficulty degree of geometrical content in six textbooks decided by three dimensions in descending order is the mainland of China, Hong Kong, America, Singapore, England and Australia. The ranking in five verstions without America decided by four dimensions in descending order is the mainland of China, Hongkong, Singapore, England and Australia. There is almost no difference of the rankings between two books above. The results give the following implications. Firstly, the mainland of China pay more attension to geometry with the most capacity, but enhancing examples and exercises is necessary in the future. Secondly, geometrical reasoning is the advantage of the book of Chinese mainland. It will bring positive results if giving more degree of freedom and opening with the retaining of illuminating by the content of geometrical reasoning. The last point is to achieve the mix of modern information technology and geometrical content by different methods.