primary mathematics1018
初中阶段高阶词汇汇总同步--如何记单词
英语初中阶段高阶词汇同步知识定位词汇是英语学习的基础，是建造语言大厦的建筑材料，词汇的学习，要靠平时多记，积少成多。
中考词汇考查形式通常有单词拼写、单词填空等。
在考查内容上除了考查词的读音（听力测试中）、拼写、词形变化（完形填空、词汇、书面表达等以及对整个试卷的阅读理解都要考查）外，重在考查词的应用，同时还考查词的固定搭配、习惯用法等。
知识梳理英语词汇表单词（Vocabulary）M894. machine n．机器895. mad adj．发疯的；生气的896. magazine n．杂志897. magic n. 魔术；魔法898. mail n．（美）邮件，邮递v.邮寄899. main adj．主要的900. mainly adv．主要地901. major adj./n. 主要的；（大学中的）主修课目902. make (made, made)v．制造，做；使得903. maleadj．男（性）的；雄的n. 男性904. mall n．大型商场905. man (复men）n．成年男人；人；人类906. manage v．管理，经营；处理；设法对付907. manager n．经理908. manners n．方式，态度，举止909. many（more，most）adj．许多的pron．许多人（或物）910. map n．地图911. March n．三月912. mark n．标记，记号；符号；（考试等）分数v．标明；用记号于；给……打分数913. market n．市场；集市914. marry v. （使）成婚，结婚915. match n．比赛，竞赛；火柴v．与……匹配916. material n. 材料，原料917. maths (mathematics) n．（通常做单数用）数学918. matter n．要紧事，麻烦事，事情；问题；物质v．（用于否定句和疑问句）要紧，有关系919. May n．五月920. may（might）aux．v．可以；也许，可能921. maybe adv．可能，大概；也许922. meal n．一餐（饭）923. mean（meant，meant）v．意思是；意味着924. meat n．肉925. medical adj．医学的；医疗的926. medicine n．药927. medium adj．中间的；中等的928. meet（met，met）v．遇见，见到；会见；迎接；满足；会，集会929. meeting n．会；集会；会见930. member n．成员，会员931. memory n．回忆，记忆932. menu n．菜单933. message n．消息；音信934. metal n．金属935. method n．方法，办法936. metre（meter）n．米（公制长度单位）937. middle n. 中间；当中；中部938. mile n．英里939. milk n．牛奶940. mind n．思想；想法v．介意；关心941. minus prep．& adj. 负的，减去的942. minute n．分钟；一会儿，瞬间943. mirror n．镜子944. miss v．1. 失去，错过，缺；2. 惦记945. Miss n．小姐，女士（称呼未婚妇女）946. mistake n．错误v．弄错，误解947. mix v．混合，搅拌948. mixture n．混合物949. model n．模型，范例；模范；模特950. modern adj．现代的951. moment n．片刻；瞬间952. Monday n．星期一953. money n．钱；货币954. monitor n．班长；监视器955. monkey n．猴子956. month n．月；月份957. moon n．月球，月亮958. more（many 或much 的比较级）adj．较多的；更大的；更多的adv．再；更n. 更多的量959. morning n．早晨，上午960. most （many 或much 的最高级）adj．最大的；最多的；大多数的；大部分的adv．最n. 大部分；大多数961. mother n．母亲962. motor n．发动机，引擎963. motorcycle n．摩托车964. mountain n．山，山脉965. mouse (复mice) n．老鼠；（计算机）鼠标966. mouth n．嘴，口967. move v．移动，搬动，搬家968. movie n．电影969. Mr n．先生970. Mrs n．夫人，太太971. much（more，most）adj．（修饰不可数名词）大量的，许多的n. （表示不可数名词）大量，许多adv．（修饰动词）非常；（加强比较级）更加972. mum n. （口语）妈妈973. museum n．博物馆974. mushroom n. 蘑菇975. music n．音乐；乐曲976. must aux．v．必须；需要；应当；必定是，一定N977. name n．名字，姓名，名称v. 命名，名叫978. narrow adj．狭窄的979. nation n. 民族，国家980. national adj．全国性的；国家的；民族的981. nationality n．国籍982. native adj. 本地的；本国的；当地的n. 本地人；本国人983. natural adj. 自然的，天然的984. nature n．自然；本质985. near prep．在……附近adj. 近的，附近的adv. 附近，邻近986. nearby adj．附近的adv．在附近987. nearly adv. 将近，几乎，差不多988. necessary adj．必需的，必要的989. needaux．v．必须v. 需要，必须n. 需要，需求，必要990. neighbour（美neighbor）n．邻居991. neighbourhood（美neighborhood）n．邻近地区，街坊992. neither pron．（两者）都不adj. （两者）都不adv．也不conj．也不993. nervous adj．紧张的994. net n. 网995. network n. 网络；网络系统996. never adv．决不；从来没有997. new adj．新的；新鲜的998. news n．新闻；消息999. newspaper n．报纸1000. next adj．最近的；紧挨着的；隔壁的；下一次的adv．最近地；然后；下一步1001. nice adj．令人偷快的；好的；漂亮的1002. night n．夜；夜间1003. no adv．不，不是，没有，无adj. 没有，无，不1004. nobody pron．没有人，谁也不1005. noise n．声音；噪声；响声1006. noisy adj．嘈杂的；喧闹的1007. none pron．没有任何东西；没有人1008. noodle n．面条1009. nor conj．也不1010. north n．北的；朝北的；北方的adj. 北；北方；北部1011. nose n．鼻子1012. not adv．不，没1013. note n．便条，笔记；钞票1014. nothing pron．没有东西；没有什么1015. notice n. 布告；通知；注意v. 注意；注意到1016. novel n. 小说1017. November n. 十一月1018. now adv. 现在1019. nowadaysadv．当今，现在1020. nowhere adv. 任何地方都不1021. number n．数，数字；号码；数量1022. nurse n．护士；保育员O1023. obeyv．服从；顺从；听从1024. observe v. 观察；观测1025. occupation n．职业1026. ocean n．海洋1027. o’clock n. ……点钟1028. October n．十月1029. of prep．(表示所属、数量、其中等)……的1030. off adv．1．离开；走开2．断掉（电、水等）3．完，光prep．脱离, （走）开；从……离开1031. offerv．& n. 提供1032. office n．办公室1033. officer n．官员，办事员1034. official adj．官方的，政府的；官员的n. （公司、团体或政府的）官员1035. often adv．常常，经常1036. oil n．油；石油1037. OK adj．& adv. 对，好，可以1038. old adj．年老的；古老的；旧的1039. Olympic adj．奥林匹克的n. 奥林匹克（运动会）1040. on prep．在……之上；在……时刻；关于adv．（穿、放……）上（电、水等）接通；进行下去adj．在上映1041. once adv．& conj. & n. 一次；一度；从前1042. one pron．一1043. online adj. & adv. 在线的；在线地1044. only adv．只；只是；仅仅adj. 唯一的；仅有的1045. onto prep. 到……上面1046. open v．打开；张开；开办；开设adj. 开着，开口的；敞开的；营业中的1047. operate v．动手术，开刀；操刀；运转1048. operation n. 手术1049. opinion n．观点，看法1050. opposite adj．& prep．在……对面；相对的；对立的1051. or conj．或者；还是；否则1052. orange n．橘子，橙，橙色adj. 橙色的1053. order n．命令，指示，顺序v. 命令；定购；点（菜、饮料等）1054. ordinary adj．普通的，平常的1055. organization n．组织；团体；机构1056. organize v．组织1057. other adj．其他的；另外的pron．别人；别的东西1058. ought（to）aux. v. 应该，应当1059. out adv．向外，在外，离开1060. outdoor adj．室外的1061. outside prep．在……外adv．在外面1062. oven n．炉子；烤箱1063. over prep．在……上方；越过；遍及；超过adv．结束；越过；翻转1064. own adj．自己的v. 拥有1065. owner n．拥有者；物主P1066. Pacific n．太平洋1067. pack n．包v. 打包1068. page n．页；页码1069. pain n. 疼痛，疼1070. paintv．油漆，粉刷，绘画n. 油漆；涂料；颜料1071. painting n．画，油画，水彩画1072. pair n．一双；一对1073. palace n．宫，宫殿1074. pale adj. 苍白的，灰白的1075. panda n．熊猫1076. paper n．（不可数）纸；（可数）报纸；考卷；文件1077. parent n．父（母），（复数）双亲1078. Paris n．（法国首都）巴黎1079. park n．公园v. 停放（车辆等）1080. part n．部分；角色；部件；零件1081. partner n. 搭档，合作者1082. party n．聚会，晚会；政党；党派1083. pass v．传，递；经过，路过；通过1084. passage n．（文章等的）一节，一段；通道1085. passengern．乘客；旅客1086. passport n．护照1087. past prep．（指时间等）过……；走过某地n. 过去；昔日；往事1088. patient n．病人adj. 耐心的1089. pay（paid，paid）v．付钱，给……报酬n. 工资1090. P.E. 体育1091. peace n．和平1092. pear n．梨1093. pearl n. 珍珠1094. pen n．钢笔；笔1095. pencil n．铅笔1096. penny （复pence）n．（英）便士；美分1097. people n．人；人们；人民1098. per prep．每，每一1099. percent n. 百分比1100. perfect adj. 完美的；极好的1101. performance n．表演，演出；行动1102. perhaps adv．可能；也许1103. person n．人1104. personal adj. 个人的，私人的1105. pet n．宠物1106. phone (telephone) n．电话v. 打电话1107. photo (photograph) n．照片1108. physical adj．身体的；物理的1109. physics n．物理（学）1110. piano n．钢琴1111. pick v．拾起；摘，采集；挑选1112. picnic n．野餐1113. picture n．图片；图画；照片；影片1114. piece n．一块（片，张，件，段……）；碎片；部件1115. pig n．猪1116. pilot n．飞行员1117. pin v．用针别住n. 针1118. pineapple n．菠萝1119. pink n．粉红adj．粉红色的1120. pioneer n. 先锋，开拓者1121. pity n．怜悯，同情；可惜1122. pizza n．比萨饼1123. place n．地方，处所v. 放置1124. plan v．& n．计划，打算1125. plane n．飞机1126. planet n．行星1127. plant n．植物v. 种植；播种1128. plastic adj．塑料的1129. plate n．盘子；碟子；板；牌1130. platform n．讲台；站台1131. play v．玩；打（球）；演奏；扮演；播放n. 玩耍；戏剧；剧本1132. playground n．操场，运动场1133. pleasant adj．令人愉快的，舒适的1134. pleased adj．感到高兴的；感到满意的1135. please v．请1136. pleasure n．高兴；愉快1137. plenty n．丰富，充足；大量1138. P.M.（p.m.）下午1139. plus prep．加，加上1140. pocket n．（衣服的）口袋1141. poem n．诗1142. point v．指；指向n. 点；分数；小数点1143. pole n．杆；极1144. policeman（复policemen）n．警察1145. polite adj. 有礼貌的1146. politely adv．有礼貌地1147. politics n．政治1148. pollute v．污染，弄脏1149. pollution n．污染1150. pool n．水池，水塘1151. poor adj．贫穷的；可怜的；不好的1152. popular（=pop）adj．流行的；大众的；受欢迎的1153. population n. 人口；人数1154. pork n．猪肉1155. possible adj．可能的；也许的1156. possibly adv．可能地；也许1157. post n．邮政，邮寄，邮件v. 投寄；邮寄1158. postcard n．明信片1159. postman（复postmen）n．邮递员1160. pot n．锅，壶，罐1161. potato n．土豆，马铃薯1162. pound n. 磅（重量单位）；英镑1163. power n．电源；能源；动力；电力1164. powerful adj．强大的1165. practice n．练习；实践v. （美practise）练习，实践1166. praise n. & v．赞扬，表扬1167. precious adj. 宝贵的，珍贵的1168. predict v. 预言；预料1169. prefer v．宁愿（选择），更喜欢1170. prepare v．准备，预备1171. present n．礼物；赠品；现在adj. 出现的，出席的1172. president n．总统；主席1173. press v．压，按1174. pretend v. 假装，装作1175. pretty adj．漂亮的1176. prevent v. 防止，预防；阻止，阻挡1177. pricen. 价格1178. primary adj．初等的；初级的1179. print v．打印，印刷1180. prison n．监狱1181. private adj．私人的；私有的1182. prize n．奖赏，奖品1183. probably adv．很可能地，大概，或许1184. problem n．问题，难题，习题1185. process v．处理；加工n. 过程1186. produce v．生产；产生；制造1187. product n. 产品1188. programme (美program) n．节目；项目1189. progress n．进步，进展1190. project n．工程；课程；作业；项目；规划1191. promise n. & v．答应；承诺1192. pronounce v．发音1193. properly adv．适当地，合适地1194. protect v．保护1195. proud adj. 自豪的；骄傲的1196. prove v．证明1197. proverb n. 谚语，格言1198. provide v. 提供1199. public adj．公众的，公共的n. 公众1200. publish v．出版1201. pull v．拉；拖；拔1202. purpose n. 目的；用途1203. push v．& n. 推1204. put（put，put）v．放，摆例题精讲【试题来源】【题目】-- Do you often watch Man and Nature on TV?--Sometimes. It's an interesting program, but I______ Sports News.【选项】A. prefer B. want C. enjoy D. miss【答案】A【解析】词汇辨析。
取遍数字 数学英文
取遍数字数学英文The Language of Mathematics" with a word count exceeding 1000 words, as requested:Numbers are the fundamental building blocks of our understanding of the world around us. They are the universal language that transcends cultures and barriers, allowing us to communicate, quantify, and make sense of the complexities of our existence. From the ancient civilizations of Mesopotamia and Egypt to the modern technological marvels of our time, the power of numbers has been a constant companion in our journey of discovery and progress.At the heart of this numerical odyssey lies the discipline of mathematics, a field that has evolved alongside human civilization, constantly expanding our horizons and challenging our perceptions. Mathematics is not merely a collection of formulas and equations; it is a way of thinking, a language that allows us to explore the intricate patterns and relationships that govern the universe.One of the most captivating aspects of mathematics is the sheer diversity of the numbers themselves. From the simplicity of the natural numbers to the complexity of irrational and imaginarynumbers, each type of number carries its own unique characteristics and applications. The natural numbers, for instance, form the foundation of our numerical system, allowing us to count, quantify, and compare. These familiar digits, from 1 to 9, and the subsequent powers of 10, have become the backbone of our everyday interactions with numbers, from tallying inventory to calculating the cost of our groceries.As we venture deeper into the realm of mathematics, we encounter numbers that defy the traditional boundaries of our understanding. Fractions, for example, represent the division of a whole into equal parts, enabling us to express and manipulate quantities with greater precision. The introduction of negative numbers, once considered a paradox, has revolutionized our ability to represent and solve complex problems, from tracking financial transactions to mapping the movements of celestial bodies.But the true wonders of mathematics lie in the realm of irrational and imaginary numbers. Irrational numbers, such as the famous pi (π), are numbers that cannot be expressed as a simple fraction, their decimal representations continuing on infinitely without repeating. These enigmatic figures have captivated mathematicians and scientists alike, as they reveal the inherent complexity and beauty of the natural world, from the perfect circles of planets to the intricate patterns of snowflakes.Imaginary numbers, on the other hand, represent a whole new dimension of mathematical exploration. These numbers, denoted by the symbol "i," are defined as the square root of -1, a concept that initially defied logical explanation. Yet, these seemingly abstract constructs have become indispensable in fields ranging from quantum mechanics to electrical engineering, allowing us to model and understand phenomena that defy traditional numerical representations.The power of numbers, however, extends far beyond their mathematical applications. In the realm of language and communication, numbers have become an integral part of our daily lives, serving as a universal medium for expressing ideas and conveying information. From the ubiquitous use of numerical codes in our digital world to the symbolic significance of numbers in various cultural and religious traditions, these seemingly simple entities have become woven into the fabric of human expression.In the realm of art and design, numbers have also played a pivotal role, inspiring and shaping the creative process. The golden ratio, a mathematical proportion found in nature and often employed in art and architecture, has long been celebrated for its aesthetic appeal and its ability to evoke a sense of harmony and balance. Similarly, the use of numerical patterns and symmetries in music and visualarts has been a source of fascination, as artists explore the interplay between the rational and the emotive.As we delve deeper into the world of numbers, we realize that they are not merely abstract constructs, but rather a reflection of the underlying order and structure of the universe. From the microscopic realm of subatomic particles to the vast expanses of the cosmos, numbers serve as the language through which we can understand and quantify the fundamental principles that govern our existence.In the field of science, numbers have been instrumental in unlocking the mysteries of the natural world. The precise measurement and analysis of physical phenomena, from the speed of light to the half-life of radioactive isotopes, have been made possible through the rigorous application of mathematical principles. These numerical insights have not only expanded our knowledge but have also enabled us to harness the power of nature for the betterment of humanity, from the development of life-saving medical treatments to the harnessing of renewable energy sources.As we continue to push the boundaries of our understanding, the role of numbers in shaping our future becomes ever more apparent. In the realm of artificial intelligence and machine learning, numbers are the foundation upon which complex algorithms and models are built, allowing us to make sense of vast troves of data and uncoverhidden patterns that were once beyond our reach.In the end, the true power of numbers lies not just in their ability to quantify and analyze, but in their capacity to inspire wonder, foster creativity, and unlock the secrets of our universe. As we take a tour of the numerical landscape, we are reminded of the enduring legacy of mathematics, a discipline that has been a constant companion in our quest to understand and shape the world around us. It is a journey of discovery that continues to captivate and challenge us, inviting us to explore the infinite possibilities that lie within the realm of numbers.。
小学数学-英文版Primary maths textbook
Yeap Ban Har, Ph.D. National Institute of Education Nanyang Technological University Singapore
This presentation focuses on selected
44%
Singapore International Average
Research Findings
Students who used the program continued trend internationally.
to
do well in the later grade levels. This is not the
The figure below is made up of 2 identical squares, 4 identical rectangles and 3 identical semi-circles. What is the area of the figure?
x
70 cm
STUDENTS ARE EXPECTED TO HANDLE CHALLENGING SITUATIONS
Opportunities for the majority to reach proficient level …
.. opportunities to engage advanced learners
Provides independent practice
Serves as pre-test which is integrated into instruction
感受数学之美的给孩子看的英文书
感受数学之美的给孩子看的英文书Mathematics is a subject that is often perceived as challenging and intimidating, especially for young learners. However, the beauty of mathematics lies in its elegance, its logical structure, and its ability to unlock the mysteries of the universe. As an educator, it is my mission to help children discover the wonders of this subject and to cultivate a deep appreciation for its beauty.One of the key aspects of mathematics that fascinates me is its inherent patterns and symmetries. From the intricate designs of snowflakes to the spiraling structure of seashells, the natural world is teeming with mathematical principles. By exploring these patterns, children can develop a sense of wonder and curiosity about the world around them. They can begin to see the underlying order that governs the universe and how mathematics is the language that allows us to understand and describe it.Another aspect of mathematics that I find captivating is its ability to solve complex problems. Whether it's calculating the trajectory of a spacecraft, designing a bridge that can withstand the forces ofnature, or predicting the behavior of financial markets, mathematics is the foundation upon which these solutions are built. By engaging children in problem-solving activities, we can help them develop critical thinking skills, logical reasoning, and a sense of accomplishment when they arrive at the correct solution.Moreover, mathematics is a universal language that transcends cultural boundaries. No matter where you are in the world, the principles of mathematics remain the same. This universality allows children to connect with the subject on a global scale and to appreciate the contributions of mathematicians from diverse backgrounds. By exploring the rich history and cultural significance of mathematics, children can develop a deeper understanding and appreciation for the subject.One of the most captivating aspects of mathematics for children is its ability to spark their imagination and creativity. From the intricate patterns of fractals to the mind-bending concepts of infinity, mathematics can open up a world of wonder and imagination. By engaging children in hands-on activities and visual representations of mathematical concepts, we can help them see the beauty and elegance of the subject.For example, the study of symmetry can be a captivating entry point for children. By exploring the symmetrical patterns found in nature,such as the radial symmetry of a sunflower or the bilateral symmetry of a butterfly, children can begin to appreciate the underlying mathematical principles that govern these structures. They can then apply these principles to create their own symmetrical designs, unleashing their creativity and fostering a deeper understanding of the subject.Another way to help children appreciate the beauty of mathematics is through the exploration of mathematical art and architecture. From the breathtaking dome of the Pantheon in Rome to the intricate tessellations of M.C. Escher's artwork, mathematics has long been a source of inspiration for artists and architects. By exposing children to these examples and encouraging them to create their own mathematical art, we can help them see the intersection of mathematics and creativity.Furthermore, the study of mathematics can also cultivate a sense of wonder and awe about the natural world. By exploring the mathematical principles that govern the behavior of celestial bodies, the structure of atoms, or the growth patterns of plants, children can develop a deeper appreciation for the complexity and beauty of the universe. They can begin to see the world through the lens of mathematics and to understand the fundamental laws that govern the physical world.In conclusion, the beauty of mathematics is a gift that we can share with children. By fostering a sense of wonder, curiosity, and appreciation for the subject, we can help them develop a lifelong love of learning and a deeper understanding of the world around them. Whether it's through exploring patterns and symmetries, solving complex problems, or engaging in creative and imaginative activities, there are countless ways to help children discover the beauty of mathematics. As educators, it is our responsibility to nurture this appreciation and to inspire the next generation of mathematicians, scientists, and innovators.。
学科分类号MASS1
学科分类号MASS 1学科分类号MASS(1) 2011年09月24日学科分类与代码 (说明:本表中黑体4号字为一级学科，黑体5号字为二级学科;宋体5号字为三科。
)代码学科名称 110 数学 110.11 数学史 110.14 数理逻辑与数学基础 110.1410 演绎逻辑学(亦称符号逻辑学) 110.1420 证明论(亦称元数学) 110.1430 递归论 110.1440 模型论 110.1450 公理集合论 110.1460 数学基础 110.1499 数理逻辑与数学基础其他学科 110.17 数论 110.1710 初等数论110.1720 解析数论 110.1730 代数数论 110.1740 超越数论 110.1750 丢番图逼近 110.1760 数的几何 110.1770 概率数论 110.1780 计算数论 110.1799 数论其他学科 110.21 代数学 110.2110 线性代数 110.2115 群论 110.2120 域论110.2125 李群 110.2130 李代数 110.2135 Kac-Moody代数 110.2140 环论(包括交换环与交换代数,结合环与结合代数,非结合环与非结合代数等) 110.2145 模论110.2150 格论 110.2155 泛代数理论 110.2160 范畴论 110.2165 同调代数110.2170 代数K理论 110.2175 微分代数 110.2180 代数编码理论 110.2199 代数学其他学科 110.24 代数几何学 110.27 几何学 110.2710 几何学基础110.2715 欧氏几何学 110.2720 非欧几何学(包括黎曼几何学等) 110.2725 球面几何学 110.2730 向量和张量分析 110.2735 仿射几何学 110.2740 射影几何学110.2745 微分几何学 110.2750 分数维几何 110.2755 计算几何学 110.2799 几何学其他学科 110.31 拓扑学 110.3110 点集拓扑学 110.3115 代数拓扑学110.3120 同伦论 110.3125 低维拓扑学 110.3130 同调论 110.3135 维数论110.3140 格上拓扑学 110.3145 纤维丛论 110.3150 几何拓扑学 110.3155 奇点理论 110.3160 微分拓扑学 110.3199 拓扑学其他学科 110.34 数学分析110.3410 微分学 110.3420 积分学 110.3430 级数论 110.3499 数学分析其他学科 110.37 非标准分析 110.41 函数论 110.4110 实变函数论 110.4120 单复变函数论 110.4130 多复变函数论 110.4140 函数逼近论 110.4150 调和分析110.4160 复流形 110.4170 特殊函数论 110.4199 函数论其他学科 110.44 常微分方程 110.4410 定性理论 110.4420 稳定性理论 110.4430 解析理论110.4499 常微分方程其他学科 110.47 偏微分方程 110.4710 椭圆型偏微分方程 110.4720 双曲型偏微分方程 110.4730 抛物型偏微分方程 110.4740 非线性偏微分方程 110.4799 偏微分方程其他学科 110.51 动力系统 110.5110 微分动力系统 110.5120 拓扑动力系统 110.5130 复动力系统 110.5199 动力系统其他学科110.54 积分方程 110.57 泛函分析 110.5710 线性算子理论 110.5715 变分法110.5720 拓扑线性空间 110.5725 希尔伯特空间 110.5730 函数空间 110.5735 巴拿赫空间 110.5740 算子代数 110.5745 测度与积分 110.5750 广义函数论110.5755 非线性泛函分析 110.5799 泛函分析其他学科 110.61 计算数学110.6110 插值法与逼近论 110.6120 常微分方程数值解 110.6130 偏微分方程数值解 110.6140 积分方程数值解 110.6150 数值代数 110.6160 连续问题离散化方法 110.6170 随机数值实验 110.6180 误差分析110.64 概率论 110.6410 几何概率 110.6420 概110.6199 计算数学其他学科率分布 110.6430 极限理论 110.6440 随机过程(包括正态过程与平稳过程、点过程等) 110.6450 马尔可夫过程 110.6460 随机分析 110.6470 鞅论 110.6480 应用概率论 110.6499 概率论其他学科 110.67 数理统计学 110.6710 抽样理论(包括抽样分布、抽样调查等) 110.6715 假设检验 110.6720 非参数统计110.6725 方差分析 110.6730 相关回归分析 110.6735 统计推断 110.6740 贝叶斯统计(包括参数估计等) 110.6745 试验设计 110.6750 多元分析 110.6755 统计判决理论 110.6760 时间序列分析 110.6799 数理统计学其他学科 110.71 应用统计数学 110.7110 统计质量控制 110.7120 可靠性数学 110.7130 保险数学110.7140 统计模拟 110.7199 应用统计数学其他学科 110.74 运筹学 110.7410 线性规划 110.7415 非线性规划 110.7420 动态规划 110.7425 组合最优化110.7430 参数规划 110.7435 整数规划 110.7440 随机规划 110.7445 排队论110.7450 对策论(亦称博奕论) 110.7455 库存论 110.7460 决策论 110.7465 搜索论 110.7470 图论 110.7475 统筹论 110.7480 最优化 110.7499 运筹学其他学科 110.77 组合数学 110.81 离散数学 110.84 模糊数学 110.87 应用数学110.99 数学其他学科 120 信息科学与系统科学 120.10 信息科学与系统科学基础学科 120.1010 信息论 120.1020 控制论 120.1030 系统论 120.1099 信息科学与系统科学基础学科其他学科 120.20 系统学 120.2010 混沌 120.2020 一般系统论120.2030 耗散结构理论 120.2040 协同学 120.2050 突变论 120.2060 超循环论120.2099 系统学其他学科 120.30 控制理论 120.3010 大系统理论120.3020 系统辩识 120.3030 状态估计 120.3040 鲁棒控制 120.3099 控制理论其他学科 120.40 系统评估与可行性分析 120.50 系统工程方法论 120.5010 系统建模 120.5099 系统工程方法论其他学科 120.60 系统工程 120.99 信息科学与系统科学其他学科 130 130.10 基础 130.1010 理论力学力学力学130.1020 理性 130.1030 非线性 130.1040 连续介质 130.1050 力学力学力学摩擦学130.1060 柔性多体力学 130.1070 陀螺 130.1080 飞行力学力学130.1099 基础其他学科 130.15 固体 130.1510 弹性 130.1515 力学力学力学塑性力学(包括弹塑性力学) 130.1520 粘弹性、粘塑性力学 130.1525 蠕变 130.1530 界面力学与表面力学 130.1535 疲劳 130.1540 损伤力学 130.1545 断裂力学 130.1550 散体力学 130.1555 细观力学 130.1560 电磁固体力学 130.1565 结构力学 130.1570 计算固体力学 130.1575 实验固体力学其他学科 130.20 振动与波 130.2010 线性振动 130.1599 固体力学力学130.2020 非线性振动 130.2030 弹性体振动 130.2040 随机振动力学力学力学130.2050 振动控制理论 130.2060 固体中的波 130.2070 流体—固体耦合振动130.2099 振动与波其他学科 130.25 流体力学 130.2511 理论流体力学 130.2514 水动力学 130.2517 气体动力学 130.2521 空气动力学 130.2524 悬浮体力学130.2527 湍流理论 130.2531 粘性流体力学 130.2534 多相流体力学 130.2537 渗流力学 130.2541 物理—化学流体力学 130.2544 等离子体动力学 130.2547 电磁流体力学 130.2551 非牛顿流体力学 130.2554 流体机械流体力学 130.2557 旋转与分层流体力学 130.2561 辐射流体力学 130.2564 计算流体力学 130.2567 实验流体力学 130.2571 环境流体力学 130.2599 流体力学其他学科 130.30 流变学130.35 爆炸力学 130.3510 爆轰与爆燃理论 130.3520 爆炸波、冲击波、应力波130.3530 高速碰撞动力学 130.3599 爆炸力学其他学科 130.40 物理力学130.4010 高压固体物理力学 130.4020 稠密流体物理力学 130.4030 高温气体物理力学 130.4040 多相介质物理力学 130.4050 临界现象与相变 130.4060 原子与分子动力学 130.4099 物理力学其他学科 130.45 统计力学 130.50 应用力学130.99 力学其他学科 140 物理学 140.10 物理学史 140.15 理论物理学140.1510 数学物理 140.1520 电磁场理论 140.1530 经典场论 140.1540 相对论与引力场 140.1550 量子力学 140.1560 统计物理学 140.1599 理论物理学其他学科 140.20 声学 140.2010 物理声学 140.2020 非线性声学 140.2030 量子声学140.2040 超声学 140.2050 水声学 140.2060 应用声学 140.2099 声学其他学科140.25 热学140.2510 热力学 140.2520 热物性学 140.2530 传热学 140.2599 热学其他学科 140.30 光学 140.3010 几何光学 140.3015 物理光学 140.3020 非线性光学140.3025 光谱学 140.3030 量子光学 140.3035 信息光学 140.3040 导波光学140.3045 发光学 140.3050 红外物理 140.3055 激光物理 140.3060 应用光学140.3099 光学其他学科 140.35 电磁学 140.3510 电学 140.3520 静电学140.3530 静磁学 140.3540 电动力学 140.3599 电磁学其他学科 140.40 无线电物理 140.4010 电磁波物理 140.4020 量子无线电物理 140.4030 微波物理学140.4040 超高频无线电物理 140.4050 统计无线电物理 140.4099 无线电物理其他学科 140.45 电子物理学 140.4510 量子电子学 140.4520 电子离子与真空物理140.4530 带电粒子光学 140.4599 电子物理学其他学科 140.50 凝聚态物理学140.5010 凝聚态理论 140.5015 金属物理学 140.5020140.5025 电介质物理学 140.5030 晶体学(包括晶体生长、晶体半导体物理学化学等) 140.5035 非晶态物理学 140.5040 液晶物理学 140.5045 薄膜物理学 140.5050 低维物理 140.5055 表面与界面物理学 140.5060 固体发光140.5065 磁学 140.5070 超导物理学 140.5075 低温物理学 140.5080 高压物理学 140.5099 凝聚态物理学其他学科 140.55 等离子体物理学 140.5510 热核聚变等离子体物理学 140.5520 低温等离子体物理学 140.5530 等离子体光谱学140.5540 凝聚态等离子体物理学 140.5550 非中性等离子体物理学 140.5599 等离子体物理学其他学科 140.60 原子分子物理学 140.6010 原子与分子理论140.6020 原子光谱学 140.6030 分子光谱学 140.6040 波谱学 140.6050 原子与分子碰撞过程 140.6099 原子分子物理学其他学科 140.65 原子核物理学140.6510 核结构 140.6515 核能谱学 140.6520 低能核反应 140.6525 中子物理学 140.6530 裂变物理学 140.6535 聚变物理学 140.6540 轻粒子核物理学140.6545 重离子核物理学 140.6550 中高能核物理学 140.6599 原子核物理学其他学科 140.70 高能物理学 140.7010 基本粒子物理学 140.7020 宇宙线物理学140.7030 粒子加速器物理学 140.7040 高能物理实验 140.7099 高能物理学其他学科 140.75 计算物理学 140.80 应用物理学 140.99 物理学其他学科 150 化学150.10 化学史 150.15 无机化学 150.1510 元素化学 150.1520 配位化学150.1530 同位素化学 150.1540 无机固体化学 150.1550 无机合成化学 150.1560无机分离化学 150.1570 物理无机化学 150.1580 生物无机化学 150.1599 无机化学其他学科 150.20 有机化学 150.2010 元素有机化学(包括金属有机化学等) 150.2020 天然产物有机化学150.2030 有机固体化学 150.2040 有机合成化学 150.2050 有机光化学150.2060 物理有机化学(包括理论有机化学、立体化学等) 150.2070 生物有机化学 150.2099 有机化学其他学科 150.25 分析化学 150.2510 化学分析(包括定性分析、定量分析等) 150.2515 电化学分析 150.2520 光谱分析 150.2525 波谱分析 150.2530 质谱分析 150.2535 热谱分析 150.2540 色谱分析 150.2545 光度分析 150.2550 放射分析 150.2555 状态分析与物相分析 150.2560 分析化学计量学150.2599 分析化学其他学科 150.30 物理化学 150.3010 化学热力学 150.3015 化学动力学(包括分子反应动力学等) 150.3020 结构化学(包括表面化学、结构分析等) 150.3025 量子化学 150.3030 胶体化学与界面化学 150.3035 催化化学150.3040 热化学 150.3045 光化学(包括超分子光化学、光电化学、激光化学、感光化学等)150.3055 磁化学 150.3060 高能化学(包括辐射化学,等离体150.3050 电化学化学) 150.3065 计算化学 150.3099 物理化学其他学科 150.35 化学物理学150.40 高分子物理 150.45 高分子化学 150.4510 无机高分子化学 150.4520 天然高分子化学 150.4530 功能高分子(包括液晶高分子化学) 150.4540 高分子合成化学 150.4550 高分子物理化学 150.4560 高分子光化学 150.4599 高分子化学其他学科 150.50 核化学 150.5010 放射化学 150.5020 核反应化学 150.5030 裂变化学 150.5040 聚变化学 150.5050 重离子核化学 150.5060 核转变化学150.5070 环境放射化学 150.5099 核化学其他学科 150.55 应用化学 150.99 化学其他学科 160 天文学 160.10 天文学史 160.15 天体力学 160.1510 摄动理论160.1520 天体力学定性理论 160.1530 天体形状与自转理论 160.1540 天体力学数值方法 160.1550 天文动力学(包括人造卫星、宇宙飞船动力学等) 160.1560 历书天文学 160.1599 天体力学其他学科 160.20 天体物理学 160.2010 理论天体物理学 160.2020 相对论天体物理学 160.2030 等离子体天体物理学 160.2040 高能天体物理学(包括天体核物理学) 160.2050 实测天体物理学 160.2099 天体物理学其他学科 160.25 天体化学 160.30 天体测量学 160.3010 天文地球动力学160.3020 基本天体测量学 160.3030 照相天体测量学 160.3040 射电天体测量学160.3050 空间天体测量学 160.3060 方位天文学 160.3070 实用天文学 160.3099 天体测量学其他学科 160.35 射电天文学 160.3510 射电天体物理学 160.3520 射电天文方法 160.3599 射电天文学其他学科 160.40 空间天文学 160.4010 红外天文学 160.4020 紫外天文学 160.4030 X射线天文学 160.4040 r射线天文学160.4050 中微子天文学160.4099 空间天文学其他学科 160.45 天体演化学(各层次天体形成与演化入各学科) 160.50 星系与宇宙学 160.5010 星系动力学 160.5020 星系天文学160.5030 运动宇宙学 160.5040 星系际物质 160.5050 大爆炸宇宙论 160.5060 星系形成与演化 160.5070 宇宙大尺度结构起源与演化 160.5099 星系与宇宙学其他学科 160.55 恒星与银河系 160.5510 恒星物理学 160.5520 恒星天文学160.5530 恒星形成与演化 160.5540 星际物质物理学 160.5550 银河系结构与运动 160.5599 恒星与银河系其他学科 160.60 太阳与太阳系 160.6010 太阳物理学160.6020 太阳系物理学 160.6030 太阳系形成与演化 160.6040 行星物理学160.6050 行星际物理学 160.6060 陨星学 160.6099 太阳与太阳系其他学科160.65 天体生物学 160.99 天文学其他学科 170 地球科学 170.10 地球科学史170.15 大气科学 170.1510 大气物理学(包括大气光学、大气声学、大气电学、云雾物理学、边界层物理学、中层物理学等) 170.1515170.1520 大气探测(包括大气遥感) 170.1525 动力气象学(包括数值大气化学天气预报与数值模拟等) 170.1530 天气学 170.1535 气候学 170.1540 云与降水物理学 170.1545 应用气象学 170.1599 大气科学其他学科 170.20 固体地球物理学 170.2010 地球动力学 170.2015 地球重力学 170.2020 地球流体力学170.2025 地壳与地形变 170.2030 地球内部物理学 170.2035 地声学 170.2040 地热学 170.2045 地电学 170.2050 地磁学 170.2055 放射性地球物理学170.2060 地震学 170.2065 勘探地球物理学 170.2070 计算地球物理学 170.2075 实验地球物理学 170.2099 固体地球物理学其他学科 170.25 空间物理学170.2510 电离层物理学 170.2520 高层大气物理学 170.2530 磁层物理学170.2540 空间物理探测 170.2550 空间环境学 170.2599 空间物理学其他学科170.30 地球化学 170.3010 元素地球化学 170.3015 有机地球化学 170.3020 放射性地球化学 170.3025 同位素地球化学 170.3030 生物地球化学 170.3035 地球内部化学 170.3040 同位素地质年代学 170.3045 成矿地球化学 170.3050 勘探地球化学 170.3055 实验地球化学 170.3099 地球化学其他学科 170.35 大地测量学170.3510 地球形状学 170.3520 几何大地测量学 170.3530 物理大地测量学170.3540 动力大地测量学 170.3550 空间大地测量学 170.3560 行星大地测量学170.3599 大地测量学其他学科 170.40 地图学 170.45 地理学 170.4510 自然地理学(包括化学地理学、生态地理学、地貌学、冰川学、冻土学、沙漠学、岩溶学等) 170.4520 人文地理学(包括区域地理、旅游地理, 其他入有关学科) 170.4599 地理学其他学科 170.50 地质学170.5011 数学地质学 170.5014 地质力学 170.5017 动力地质学 170.5021矿物学(包括放射性矿物学) 170.5024 矿床学与矿相学(包括放射性矿床学，不包括石油、天然气和煤。
学术论文指导 04讲 二手数据研究方法(物流班)
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（二）基于微观企业数据库的研究
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•全国29个省市自治区（除西藏重庆）的上述数据
•上述三组数据在《中国统计年鉴》中全部有呈现，都是 国家统计局逐年统计发布的数据，此外，统计年鉴还会 公布许多别的统计项目，非常齐全，感兴趣的同学可以 到学院图书馆查阅。可以做出非常有趣的研究出来。
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primary mathematics
Student participation in elementary mathematicsclassrooms:the missing link between teacher practicesand student achievement?Marsha Ing 1&Noreen M.Webb 2&Megan L.Franke 2&Angela C.Turrou 2&Jacqueline Wong 2&Nami Shin 2&Cecilia H.Fernandez 2Published online:5August 2015#Springer Science+Business Media Dordrecht (outside the USA)2015Abstract Engaging students as active participants in mathematics classroom discussions has great potential to promote student learning.Less well understood is how teachers can promote beneficial student participation,and how teacher-student interaction relates to student achieve-ment.This study examined how the kinds of teacher practices that may encourage beneficial student participation relate to student achievement in elementary school mathematics ing videotaped recordings,we examined the extent to which students explained their own ideas and engaged with others ’ideas and how teachers supported these kinds of student participation.Linking teacher practices,student participation,and achievement all at the individual student level,we found that student achievement was best predicted by the combination of teacher practices and student participation.The results show that taking intoEduc Stud Math (2015)90:341–356DOI 10.1007/s10649-015-9625-z*Marsha Ing marsha.ing@Noreen M.Webbwebb@Megan L.Frankemfranke@Angela C.Turrouachan@Jacqueline Wongwritejackie@Nami Shinnami0623@Cecilia H.Fernandezceci.henriquez@1University of California,Riverside,1207Sproul Hall,Riverside,CA 92521,USA 2University of California,Los Angeles,Los Angeles,CA,USA342M.Ing et al. account student participation is necessary for understanding how teaching practices relate to student mathematics learning.Keywords Participation.Instruction.Achievement.MathematicsAs researchers,policy makers,and practitioners work toward improving student learning in mathematics,they increasingly recognize the importance of engaging students as active participants in the learning process.A central component of students’participation is their engagement in classroom mathematics conversations.For example,in the USA,the Common Core State Standards Initiative for Mathematical Practice explicitly addresses students’com-munication competence,including the ability to construct viable arguments and critique the reasoning of others(National Governors Association Center for Best Practices&Council of Chief State School Officers,2010).Teachers are expected to work to enable students to justify their conclusions,communicate conclusions to others,listen to arguments of others,decide whether those arguments make sense,and ask useful questions to clarify or improve the arguments.1This study examines the kinds of teacher practices that may encourage beneficial student participation in discussions around important mathematical content and how teacher practices and student participation relate to student achievement.It extends previous research in two ways.First,we code the participation of individual students,the teacher practices experienced by each student,and the achievement of individual students to help us understand the encounters that matter for students.Second,we present and analyze a single,unified model that links student participation,teacher practices,and student achievement in order to better understand the relationships among them.We focus on two fundamental dimensions of student participation that underlie the recom-mendations for students to develop competency in constructing arguments and critiquing the reasoning of others:explaining one’s own ideas and engaging in the ideas of others.The power of explaining one’s ideas to others and engaging with others’ideas for promoting mathematics learning has a broad range of theoretical support.One perspective is cognitive.During the processes of formulating ideas and communicating with others,students offering explanations monitor and revise their own thinking and may recognize their own misconceptions,or contradictions or incompleteness in their ideas(e.g.,Hatano,1993).To resolve these issues, explainers may bring concepts or details together in ways they had not thought of previously (Roscoe&Chi,2008).Students may also see other ways to revise their ideas,such as making their arguments more compact,their ideas clearer or more elaborated,or their problem-solving strategies more paring one’s ideas with others’ideas may help reveal errors and gaps in students’understanding,as well as differences in perspectives.Students can use each other’s ideas to re-examine and question their own thinking,to correct misconceptions and make their understanding more complete,and to extend their own thinking in the building of more elaborated mental models(Chi,2000).Another perspective is sociocultural.As Sfard,Nesher,Streefland,Cobb,and Mason (1998)describe,mathematical ideas being discussed shift as students compare,challenge, criticize,refute,complete,reject,and amplify each other’s suggestions.Participating in these 1Similar recommendations appear in curriculum standards in other countries,such as the Australian Association of Mathematics Teachers Standards for Excellence in Teaching Mathematics in Australian Schools(2006); Finnish National Core Curriculum for Basic Education(2012),and New Zealand Curriculum(2008).Student participation in elementary mathematics classrooms343 B reflective shifts in discourse^gives students opportunities to reflect on and reorganize mathematical activity.As mathematical conversations unfold,students’B reflection is enabled by their participation in the discourse,and by reflecting they contribute to the shift in discourse^(p.48).Participating in discussions,especially explaining one’s own ideas and engaging with others’ideas,helps students understand each other’s thinking as well as the changes in ideas that develop.As discussed by Sfard and Kieran(2001),interaction that is mathematically productive for students depends on students making their emergent thinking available for their conversation partners.In addition to theoretical support,there is empirical support for the relationship between explaining one’s ideas to others and engaging with others’ideas and students’learning.In a recent study,we found that the extent of detail in elementary school students’explanations of their mathematics problem-solving strategies and the degree to which students added further detail to others’ideas,or described how and why they disagreed with others,or suggested alternatives to others’approaches significantly predicted student achievement(Webb et al., 2014).Along similar lines,other researchers have demonstrated that giving complex expla-nations(e.g.,integrating multiple concepts)and engaging with others’ideas at a high level (e.g.,acknowledging,repeating,challenging,elaborating)are related to students’learning outcomes in mathematics(e.g.,Veenman,Denessen,van den Akker,&van der Rijt,2005). Warner(2008)found that student questioning,explaining,re-explaining,and using their own or others’ideas were associated with the growth of students’mathematical ideas.Other researchers suggest that the frequency of students’substantive mathematical explanations may be a factor in achievement differences between countries(e.g.,Perry,2000).Considering the positive effects of explaining one’s own thinking and engaging with the ideas of others,an important question is how teachers can interact with students during classroom dialogue to facilitate and support these productive student interactions.In this paper, we focus on teacher moves that invite students to explain their own and others’thinking,that probe specific details in students’explanations to elicit further student elaboration,and that encourage students to compare and connect their ideas with others’ideas.Our previous research(e.g.,Franke et al.,2009)has given examples of how these moves may be productive for student participation.Other researchers have examined similar types of teacher moves that may promote productive student participation,such as asking students probing and clarifying questions to encourage them to elaborate on their ideas and problem-solving strategies(e.g., Kazemi&Stipek,2001)and asking students to add to the ideas presented by others(e.g., O’Connor,Michaels,&Chapin,2015).Few studies have examined whether teacher support for student participation translates into improved student learning outcomes.One exception is a series of studies by Gillies and colleagues that compared classrooms of teachers trained to carry out specific support moves (or who were observed to carry out the support moves frequently)to classrooms of teachers not trained to do so(or who were observed to carry out the support moves infrequently). Students in classrooms whose teachers frequently paraphrased students’ideas,prompted students to explain their ideas,and used questions to challenge and scaffold children’s learning (1)exhibited more elaboration of ideas and(2)obtained higher scores on reasoning and problem-solving tasks than students in classrooms where teachers carried out these behaviors less frequently(e.g.,Gillies&Boyle,2008;Gillies&Haynes,2011;Gillies&Khan,2009).While these studies suggest important links between teacher practices,student participation, and student learning,unanswered questions remain about the extent to which the effects of teacher practices on student learning can be explained by the mediating role of student344M.Ing et al. participation.The purpose of this study is to examine this question.To do so,we examine the relationships among teacher practices,student participation,and student learning outcomes in a single unified model.To comprehensively analyze this model,we examine the relationships among all variables at the individual student level,that is,the links among teacher practices that are focused on individual students,the participation of the students experiencing those teacher moves,and those students’learning outcomes.The intent of this paper,then,is to clarify these relationships in a way that attends to the experiences of each individual student.1Method1.1SampleThe teachers and students analyzed here are largely the same sample as that reported in previous studies(Franke et al.,in press;Webb et al.,2014).The data were collected during the2008–2009academic year at an elementary school affiliated with a major university in California.A goal of the school is for teachers to teach for mathematical proficiency(as defined by Adding it Up,National Research Council,2001)by asking students about their mathematical thinking,supporting mathematical discussion,and focusing on problem solving. While the school is organized into multiage groups,ranging from early childhood level(4to 6year olds)to upper level(10to12year olds),we focus on the intermediate classrooms(8to 10year olds)because these classrooms covered similar mathematical content and included standardized test scores from the previous school year that could be used as a covariate in the analyses.The sample with complete video and achievement test data consisted of71students from six classrooms(the total enrollment in these six classrooms was113).To ensure an ethnically and socioeconomically diverse student population,the school is not designed to serve the children of the university community,and students apply for admission (stratified by ethnicity and income).The school demographics include41%white,20% Latino,4%Asian,3%African-American,and33%multiethnic;23%of families had annual incomes below$50,000(13%below$35,000),and18%of students qualified for free or reduced meals.The specific demographics for the sample of students analyzed in this study are not reported to maintain confidentiality.However,the ethnic and socioeconomic diversity is similar to that of the school as a whole.The previous studies reported quantitative information about the relationship between student participation and achievement scores and qualitative information about instructional practices that may have encouraged students to attend to and engage with each other’s thinking.The current study extends the previous studies by(a)systematically coding and analyzing instructional practices that were experienced by students individually(for all students in the sample);(b)coding instructional practices in ways suitable for quantitative analysis;and(c)linking information about instructional practices,student participation,and student achievement in a single unified analysis(as described in the following sections).1.2Observation:recording and coding proceduresRecording procedures Project members spent6months prior to formal data collection taking field notes,talking with teachers and practicing videotaping to become familiar with the classrooms and teachers’customary practices.All observation data were collected during aStudent participation in elementary mathematics classrooms345 regular mathematics lesson(approximately one hour).Teachers were asked to teach what they would consider a typical mathematics lesson.The mathematical content of the lesson included whole number operations,fractions,and decimals.Teachers were asked to teach what they would consider a typical mathematics lesson.The mathematical content of the lessons varied across classrooms,ranging from whole number operations to fractions and decimals.All lessons included both problem solving(with attention to conceptual ideas and procedures) and work on understanding number relationships.Teachers were aware that the focus of the data collection was around engagement of students’thinking around the mathematics and student participation,but they were encouraged to interact with students as they would in a typical mathematics lesson.Since a goal of the video and audio observation recording procedure was to capture all discussions in the whole-class and small-group settings,multiple video and audio recordings were captured from at least12students per classroom.For video recording,we used one stationary video camera with two flat microphones to capture the ongoing flow of the classroom and the discussion of up to two groups of students;in addition, we used four mobile cameras to capture the discussion of the remaining groups of students.We supplemented the video recording with six digital audio recorders that were placed on students’desks to capture the audio for student-to-student discussion that may not have been captured clearly by the audio on the video camera.This combination of multiple video and audio cameras made it possible to identify which student was speaking even during simulta-neous small group conversations.The number of math problem-solving discussions recorded per student(number of prob-lems discussed during either whole-class discussion or small-group discussion)ranged from4 to13across the classrooms.Multiple video and audio recordings for a classroom were merged into a single classroom movie using Final Cut Pro.We carried out a sequence of steps to create the single file for each class.First,we created a movie for the whole-class setting(that followed the teacher during the entire class period and captured all interactions of students with the teacher)and a separate movie for each small-group discussion(that captured every conversation on every problem).To create each of these movies,we selected the best video and sound sources at each point in time and merged them into a single file.Second,we merged these whole-class and small-group movies into a single file.As described below,we carried out the coding using the omnibus file(movie)for each classroom.Coding procedures We coded each classroom movie using video analysis software that allows for moment-to-moment coding of all interaction occurring in the movie(Studiocode Business Group,1997–2013).The video analysis software creates B timelines^that are linked to a video and provide chronologically accurate representations of the classroom lesson.These timelines make it possible to create instances of codes that simultaneously capture time length (e.g.,5-minute warm-up activity,30-second student explanation)and code descriptors(e.g., fully detailed student explanation).Multiple codes can be applied simultaneously.We included codes for multiple types of student participation and teacher support of student participation (described in detail below),as well as codes for the classroom context(e.g.,whole-class discussion,small-group discussion,individual seatwork),the type of mathematical activity (e.g.,question-and-answer warm-up exercise,extended problem),and the mathematical con-tent of the activity or problem(necessary for coding decisions such as what constituted fully detailed student explanations;see below).The spatial organization of the timeline greatly facilitated coding decisions by presenting the full context of each interaction episode between teacher and students and among students(e.g.,the ideas that had already been shared,and by346M.Ing et al. whom,prior to an interaction episode;the full sequence of the interaction episode as it unfolded).A team of raters participated in discussions about the development of the coding procedures and refinement of the variables used in the analyses.Each rater was initially responsible for coding a single classroom.Raters then brought questions to the rater team for discussion, which led to further refinement of the coding scheme,and further review of coding decisions. After the final version of the codes were established,a second rater double-coded excerpts of the timeline for each classroom.The percent of exact agreement between the first and second raters for the variables used to create the student participation composite and the teacher support of student participation composite measures used in the analysis(described in detail below)was high(ranging from82to100%).Generalizability analyses(Shavelson&Webb, 1991)also showed high consistency among raters(generalizability coefficients for a single rater were.89for the student participation composite and.95for the teacher support of student participation composite).1.3Coding of student participationThe student participation variable used in the analyses is a composite(sum)of three variables that were coded and analyzed as separate variables in Webb et al.(2014):a student giving fully detailed explanations,a student engaging with other students’ideas,and a student having other students engage with his or her ideas.We used a single composite participation variable here instead of the separate variables because(1)conceptually,they are linked dimensions of student participation,(2)the variables were substantially intercorrelated(ranging from r=.31,p<.05to r=.59,p<.01),and(3)using a single student participation variable greatly simplified the unified analyses predicting student achievement from teacher practice and student participation.Student explanations The coding of student explanations focused on the extent to which students provided fully detailed explanations of a valid approach for solving the problem.This decision was based on our previous work showing strong positive correlations between giving correct and complete(or fully detailed)explanations and achievement,but nonsignificant or negative correlations between giving explanations that were ambiguous,incorrect,incomplete, or not detailed and achievement(Webb et al.,2008,2009,2014).We considered an explana-tion to be fully detailed if the student verbalized every step involved in solving the problem and the student’s strategy was correct and valid.The explanation variable reflected the number of fully detailed explanations given by a student across the problem-solving discussions(0= none,1=1,2=2or more).Because few students gave more than two fully detailed explana-tions,they were coded in the same way as students who gave two fully detailed explanations. Student engagement with each other’s ideas The coding of student engagement with each other’s ideas focused on the extent to which students engaged with other students around the mathematics.As in our previous study(Webb et al.,2014),we created two variables representing the level of student engagement with each other’s ideas.One variable represented the level at which a student engaged with other students’ideas;the other variable represented the level at which other students engaged with a student’s ideas.In our previous study,each of these variables was significantly correlated with achievement.That is,higher levels ofStudent participation in elementary mathematics classrooms347 engaging with other students’ideas and others engaging with a students’idea were related to higher student achievement.Consistent with the coding in the previous study(Webb et al.,2014),we used three levels of student engagement for both engaging in another student’s ideas and having other students engage with a student’s ideas:low,medium,and high.Low-level engagement with another student’s idea consisted of referencing or acknowledging another student’s idea in a general way without providing details(e.g.,B I get it,^B I agree^).Medium-level engagement with another student’s idea consisted of explicitly referencing the details of the other student’s idea for how to solve the problem but not adding any further detail to the explanation originally offered(e.g.,repeating what another student said or describing what he or she did,asking questions about the details of another student’s strategy,or disagreeing with details in another student’s strategy without suggesting any alternative).High-level engagement with another student’s idea consisted of adding to another student’s suggested strategy by(a)disagreeing with what had been being shared and suggesting an alternative,(b)adding further detail to the approach articulated by another student,or(c)suggesting an alternative approach that explic-itly referenced the idea already posed(e.g.,stating that the alternative approach is better than or different from the original suggestion).The engagement variables reflected the maximum level of engagement with another student’s idea that a student showed during the course of the discussions and the maximum level that other students engaged with a student’s idea(0=low,1=medium,2=high).Students who never showed evidence of engagement with other students’ideas were included in the low engagement level of engaging with other students’ideas.Similarly,students who never received any engagement with their ideas(or received ambiguous statements)were included in the low engagement level for receiving engagement from others.The three student participation variables(giving fully detailed explanations,engaging with other students’ideas,having other students engage with a student’s ideas)were then combined to form a single composite student participation variable.For the composite,we summed the scores on the separate variables,resulting in a scale from0to6.1.4Coding of teacher support of student participationThe teacher support of student participation(TSSP)variable used in the analyses is a composite (sum)of two variables:teacher eliciting of a student’s thinking and teacher support for a student’s engaging with other students’ideas.Each component in the composite was coded for each student and is described in detail below.Given the focus of this work to analyze the instructional practices that individual students experienced,we coded teacher support provided to individual students but not general support offered to the class as a whole,such as asking instructions for students to explain their thinking to each other when working in small groups.Because general support to the whole class was similar across all students,it did not provide information about the student-specific instructional practices experienced by any particular student.Teacher eliciting of student thinking The coding of teacher eliciting of student thinking focused on the extent to which the teacher encouraged each student to share his or her thinking. First,based on our previous work,we included the teacher asking a student probing questions about specific details that a student had offered(B I see you drew some lines there;can you tell us what those lines mean?^)because such questions often elicited further elaboration from348M.Ing et al. students(Franke et al.,2009;Webb et al.,2008).Second,extending our previous work,we broadened the concept of eliciting student thinking to include the teacher’s initial invitation for a student to explain his or her thinking(B Can you explain your strategy?^)and instances of teacher repetition or revoicing of particular details of a student’s explanation that signaled (typically through tone of voice)that the teacher was requesting further elaboration or clarification of those details.Each student’s code for teacher eliciting of student thinking reflected the number of teacher moves that the student experienced during the course of the discussions(0=none,1=1,2=2or more).Because few students experienced more than two eliciting student thinking moves,those students were assigned the same code as students who experienced two eliciting moves.Teacher support for student engagement with others’ideas The coding of teacher support for student engagement with others’ideas focused on the extent to which the teacher encouraged a student to engage with other students’ideas.We observed teachers making six moves to foster student engagement with each other’s ideas:asking a student to explain someone else’s strategy,asking students to discuss differences between multiple ideas already shared,asking one student to make a suggestion to another student based on the other student’s work,asking students to connect their own ideas to the ideas of another student,asking students to work together to jointly create a solution,and asking one student to use another student’s strategy.The teacher support for student engagement variable for each student reflected the number of moves that a student experienced in which the teacher encouraged the student to engage in others’ideas(0=none,1=1,2=2,or more).Because few students experienced more than two teacher moves to support their engagement with others’ideas,they were assigned the same code as students who experienced two moves.The correlation between the two teacher practice variables(teacher eliciting of student thinking and teacher support for student engagement with others’ideas)was positive and significant,r(71)=0.32,p<.01.The two teacher practice variables were then combined(sum of the scores on the two variables)into a single teacher practice composite(TSSP)with a scale of0to4.Similar to the rationale provided for the use of a single student participation composite variable,this single composite teacher practice variable was used in the analyses instead of the separate variables.As is the case for the student participation composite,using a teacher practice composite greatly simplified the unified analyses predicting student achieve-ment from teacher practices and student participation.1.5Student achievement measuresPosttest achievement Student achievement was measured using a researcher-designed, written assessment of students’mathematical thinking that focused on aspects of whole number addition,subtraction,multiplication,and division,as well as place value,and fair sharing using four problems situated in a story context(e.g.,B There are12pieces of gum in each package.Megan had5packages of gum.She gave15pieces of gum to her friends.How many pieces of gum does she have left?^).2The problems are parallel to those used in previous 2An additional standardized mathematics achievement measure was initially considered as a posttest measure. However,the high,positive correlation between the prior achievement measure and standardized achievement measure,r(71)=.69,p<.01,indicated that none of the other variables would significantly relate to posttest achievement after partialling out the influence of prior achievement.。
数学本科和研究生初等教材
9月1日数学研究生基础课程参考书目*这个计划是按照美国的体系制订的，美国一年级的研究生课程大概相当于我国重点大学数学本科大三、大四的水平第一学年秋季学期春季学期几何与拓扑I 几何与拓扑II1、James R. Munkres, Topology较新的拓扑学的教材适用于本科高年级或研究生一年级2、Basic Topology by Armstrong本科生拓扑学教材3、Kelley, General Topology一般拓扑学的经典教材，不过观点较老4、Willard, General Topology一般拓扑学新的经典教材5、Glen Bredon, Topology and geometry研究生一年级的拓扑、几何教材6、Introduction to Topological Manifolds by John M. Lee研究生一年级的拓扑、几何教材，是一本新书7、From calculus to cohomology by Madsen很好的本科生代数拓扑、微分流形教材代数I 代数II1、Abstract Algebra Dummit最好的本科代数学参考书，标准的研究生一年级代数教材2、Algebra Lang标准的研究生一、二年级代数教材，难度很高，适合作参考书3、Algebra Hungerford标准的研究生一年级代数教材，适合作参考书4、Algebra M,Artin标准的本科生代数教材5、Advanced Modern Algebra by Rotman较新的研究生代数教材，很全面6、Algebra：a graduate course by Isaacs较新的研究生代数教材7、Basic algebra V ol I&II by Jacobson经典的代数学全面参考书，适合研究生参考分析基础复分析I实分析I1、Walter Rudin, Principles of mathematical analysis本科数学分析的标准参考书2、Walter Rudin, Real and complex analysis标准的研究生一年级分析教材3、Lars V. Ahlfors, Complex analysis本科高年级和研究生一年级经典的复分析教材4、Functions of One Complex Variable I，J.B.Conway研究生级别的单变量复分析经典5、Lang, Complex analysis研究生级别的单变量复分析参考书6、Complex Analysis by Elias M. Stein较新的研究生级别的单变量复分析教材7、Lang, Real and Functional analysis研究生级别的分析参考书8、Royden, Real analysis标准的研究生一年级实分析教材9、Folland, Real analysis标准的研究生一年级实分析教材第二学年秋季学期春季学期代数III 代数IV1、Commutative ring theory, by H. Matsumura较新的研究生交换代数标准教材2、Commutative Algebra I&II by Oscar Zariski , Pierre Samuel经典的交换代数参考书3、An introduction to Commutative Algebra by Atiyah标准的交换代数入门教材4、An introduction to homological algebra ,by weibel较新的研究生二年级同调代数教材5、A Course in Homological Algebra by P.J.Hilton,U.Stammbach经典全面的同调代数参考书6、Homological Algebra by Cartan经典的同调代数参考书7、Methods of Homological Algebra by Sergei I. Gelfand, Yuri I. Manin高级、经典的同调代数参考书8、Homology by Saunders Mac Lane经典的同调代数系统介绍9、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud 高级的代数几何、交换代数的参考书，最新的交换代数全面参考代数拓扑I 代数拓扑II1、Algebraic Topology, A. Hatcher最新的研究生代数拓扑标准教材2、Spaniers "Algebraic Topology"经典的代数拓扑参考书3、Differential forms in algebraic topology, by Raoul Bott and Loring W. Tu 研究生代数拓扑标准教材4、Massey, A basic course in Algebraic topology经典的研究生代数拓扑教材5、Fulton , Algebraic topology：a first course很好本科生高年级和研究生一年级的代数拓扑参考书6、Glen Bredon, Topology and geometry标准的研究生代数拓扑教材，有相当篇幅讲述光滑流形7、Algebraic Topology Homology and Homotopy高级、经典的代数拓扑参考书8、A Concise Course in Algebraic Topology by J.P.May研究生代数拓扑的入门教材，覆盖范围较广9、Elements of Homotopy Theory by G.W. Whitehead高级、经典的代数拓扑参考书实分析II 泛函分析1、Royden, Real analysis标准研究生分析教材2、Walter Rudin, Real and complex analysis标准研究生分析教材3、Halmos，"Measure Theory"经典的研究生实分析教材，适合作参考书4、Walter Rudin, Functional analysis标准的研究生泛函分析教材5、Conway,A course of Functional analysis标准的研究生泛函分析教材6、Folland, Real analysis标准研究生实分析教材7、Functional Analysis by Lax高级的研究生泛函分析教材8、Functional Analysis by Yoshida高级的研究生泛函分析参考书9、Measure Theory, Donald L. Cohn经典的测度论参考书微分拓扑李群、李代数1、Hirsch, Differential topology标准的研究生微分拓扑教材，有相当难度2、Lang, Differential and Riemannian manifolds研究生微分流形的参考书，难度较高3、Warner,Foundations of Differentiable manifolds and Lie groups标准的研究生微分流形教材，有相当的篇幅讲述李群4、Representation theory: a first course, by W. Fulton and J. Harris李群及其表示论的标准教材5、Lie groups and algebraic groups, by A. L. Onishchik, E. B. Vinberg李群的参考书6、Lectures on Lie Groups W.Y.Hsiang李群的参考书7、Introduction to Smooth Manifolds by John M. Lee较新的关于光滑流形的标准教材8、Lie Groups, Lie Algebras, and Their Representation by V.S. Varadarajan最重要的李群、李代数参考书9、Humphreys, Introduction to Lie Algebras and Representation Theory , Springer-Verlag, GTM-9标准的李代数入门教材第三学年秋季学期春季学期微分几何I 微分几何II1、Peter Petersen, Riemannian Geometry标准的黎曼几何教材2、Riemannian Manifolds: An Introduction to Curvature by John M. Lee最新的黎曼几何教材3、doCarmo, Riemannian Geometry.标准的黎曼几何教材4、M. Spivak, A Comprehensive Introduction to Differential Geometry I—V全面的微分几何经典，适合作参考书5、Helgason , Differential Geometry,Lie groups,and symmetric spaces标准的微分几何教材6、Lang, Fundamentals of Differential Geometry最新的微分几何教材，很适合作参考书7、kobayashi/nomizu, Foundations of Differential Geometry经典的微分几何参考书8、Boothby,Introduction to Differentiable manifolds and Riemannian Geometry标准的微分几何入门教材，主要讲述微分流形9、Riemannian Geometry I.Chavel经典的黎曼几何参考书10、Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”V ol 1—3经典的现代几何学参考书代数几何I 代数几何II1、Harris,Algebraic Geometry: a first course代数几何的入门教材2、Algebraic Geometry Robin Hartshorne经典的代数几何教材，难度很高3、Basic Algebraic Geometry 1&2 2nd ed. I.R.Shafarevich.非常好的代数几何入门教材4、Principles of Algebraic Geometry by giffiths/harris全面、经典的代数几何参考书，偏复代数几何5、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud 高级的代数几何、交换代数的参考书，最新的交换代数全面参考6、The Geometry of Schemes by Eisenbud很好的研究生代数几何入门教材7、The Red Book of Varieties and Schemes by Mumford标准的研究生代数几何入门教材8、Algebraic Geometry I : Complex Projective Varieties by David Mumford复代数几何的经典调和分析偏微分方程1、An Introduction to Harmonic Analysis,Third Edition Yitzhak Katznelson调和分析的标准教材，很经典2、Evans, Partial differential equations偏微分方程的经典教材3、Aleksei.A.Dezin，Partial differential equations，Springer-Verlag偏微分方程的参考书4、L. Hormander "Linear Partial Differential Operators, " I&II偏微分方程的经典参考书5、A Course in Abstract Harmonic Analysis by Folland高级的研究生调和分析教材6、Abstract Harmonic Analysis by Ross Hewitt抽象调和分析的经典参考书7、Harmonic Analysis by Elias M. Stein标准的研究生调和分析教材8、Elliptic Partial Differential Equations of Second Order by David Gilbarg偏微分方程的经典参考书9、Partial Differential Equations ，by Jeffrey Rauch标准的研究生偏微分方程教材复分析II 多复分析导论1、Functions of One Complex Variable II，J.B.Conway单复变的经典教材，第二卷较深入2、Lectures on Riemann Surfaces O.Forster黎曼曲面的参考书3、Compact riemann surfaces Jost黎曼曲面的参考书4、Compact riemann surfaces Narasimhan黎曼曲面的参考书5、Hormander " An introduction to Complex Analysis in Several Variables" 多复变的标准入门教材6、Riemann surfaces , Lang黎曼曲面的参考书7、Riemann Surfaces by Hershel M. Farkas标准的研究生黎曼曲面教材8、Function Theory of Several Complex Variables by Steven G. Krantz高级的研究生多复变参考书9、Complex Analysis: The Geometric Viewpoint by Steven G. Krantz高级的研究生复分析参考书专业方向选修课：1、多复分析2、复几何3、几何分析4、抽象调和分析5、代数几何6、代数数论7、微分几何8、代数群、李代数与量子群9、泛函分析与算子代数10、数学物理11、概率理论12、动力系统与遍历理论13、泛代数*数学基础：1、halmos ,native set theory2、fraenkel ,abstract set theory3、ebbinghaus ,mathematical logic4、enderton ,a mathematical introduction to logic5、landau, foundations of analysis6、maclane ,categories for working mathematican应该在核心课程学习的过程中穿插选修假设本科应有的水平分析Walter Rudin, Principles of mathematical analysisApostol , mathematical analysisM.spivak , calculus on manifoldsMunknes ,analysis on manifoldsKolmogorov/fomin , introductory real analysisArnold ,ordinary differential equations代数：linear algebra by Stephen H. Friedberglinear algebra by hoffmanlinear algebra done right by Axleradvanced linear algebra by Romanalgebra ,artina first course in abstract algebra by rotman几何：do carmo, differential geometry of curves and surfacesDifferential topology by PollackHilbert ,foundations of geometryJames R. Munkres, Topology12:54 | 阅读评论(2) | 固定链接 | Mathematics数学分析的一些课本一、中文的：最难的5套书：1、《数学分析新讲》（1、2、3册），张筑生，北大版2、《数学分析》（1、2、3册），方企勤，高等教育版3、《数学分析教程》（上、下册）常庚哲等，高等教育版4、《数学分析》（上、下册）黄玉民等，科学出版社5、《简明数学分析》王昆扬，高等教育版最抽象的教材：《数学分析》（上、下册），邹应，武汉大学数学基地班教材（个人认为是目前国内观点最高，最抽象的书）二、国外的书好书太多，菲赫金哥茨的《数学分析原理》太老了，他的那套《微积分学教程》3卷（共8本）才是他的成名作，不过也太老了。
Mathematica软件使用入门
Mathematica软件使用入门目录第一章基本知识与基本操作 (3)1.1 Mathematica的基本语法特征 (3)1.2 Mathematica的启动、基本操作 (4)1.3 操作小技巧 (7)1.4 数值计算 (8)1.5 赋值与替换 (9)1.6 自定义函数 (10)1.7 方程与方程组解 (11)1.8 解不等式与不等式组 (12)1.9 由递推式求数列的通项公式 (13)1.10 作函数图像 (14)第二章运用Mathematica实现高等数学中的基本运算 (16)2.1 求极限运算 (16)2.2 求导数与微分 (18)2.3 求不定积分 (25)2.4 求定积分 (25)第三章实验练习题 (28)Mathematica是当今世界上最为流行的计算机代数系统之一．Mathematica系统是美国物理学家Stephen.Wolfram领导的一个小组开发的,后来他们成立了Wolfram研究公司．1987年推出了系统的1.0版；现在的最新版本是8.0版．Mathematica可以做：●符号计算和数值计算问题,如：能做多项式的计算、因式分解和展开等；●做各种有理式计算，求多项式、有理式方程和超越方程的精确解和近似解；●做向量、矩阵的各种计算；●求极限、导数、积分，做幂级数展开，求解某些微分方程等；●做任意位数的整数或分子分母为任意大整数的有理数的精确计算，做具有任意位精度的数值（实、复数值）的计算．●可以很方便地画出用各种方式表示的一元和二元函数的图形,通过图形,可以立即形象地掌握函数的某些特性，而这些特性一般是很难从函数的符号表达式中看清楚．第一章 基本知识与基本操作1.1 Mathematica 的基本语法特征使用Mathematica ，一定要牢牢记住：● Mathematica 中大写小写是有区别的，如Name 、name 、NAME 等是不同的变量名或函数名；● 系统所提供的功能大部分以系统函数的形式给出， 内部函数一般写全称， 而且一定是以大写英文字母开头， 如Sin[x], Cos[z]等；● 乘法即可以用*，又可以用空格表示，如2 3＝2*3＝6 , 2 Sin[x]＝2* Sin[x] ● 乘幂可以用“^”表示，如x^0.5 表示： Tan[x]^y 表示： ● 自定义的变量可以取几乎任意的名称，长度不限，但不可以数字开头． ● 当你赋予变量任何一个值，除非你：明显地改变该值或 使用Clear[变量名] 或 使用“变量名=.”取消该值，否则它将始终保持原值不变．● 一定要注意四种括号的用法：0.5x yTan[x]( )：表示项的结合顺序，如： (x+(y^x+1/(2x)))；[ ]：表示函数，如：Log[x], Sin[x]；{ }：表示一个“表”(即是一组数字、或任意表达式、或函数等的一个有序集合)，如：{2x,Sin[12 Pi]，A，1}, {1+A,y*x，1，2}；[[ ]]：双方括号表示“表”或“表达式”的下标，如：a； {3,5,7}[[2]]=5．a[[2,3]]表示：23●Mathematica的语句书写十分方便，一个语句可以分为多行写，同一行可以写多个语句（但要以分号间隔）．●当语句以分号结束时，语句计算后不做输出（输出语句除外），否则将输出计算的结果．●Mathematica命令中的标点符号必须是英文的．1.2 Mathematica的启动、基本操作1.2.1 启动“Mathematica”：在windows操作系统中安装了Mathematica后，与其他的常用软件一样，可从“开始”→“程序”→“Mathematica5” Mathematica的主窗口并出现第一个notebook窗口（Untitled-1）:1.2.2 简单使用：例1.1 计算＋33的值①在“Ｕntitled－１”窗口中输入：329/412+3^3②按下“Shift＋Enter”（或数字键盘上的Enter键）,就得到计算结果：其中“In[1]:=”是Mathematica自动加上的，表示第一个输入；“Out[1]:=”表示第一个输出．一般地：In[n]:= 表示第n个输入Out[n]:=表示第n个输出．注意：“In[n]:=”自动加上的，不能人工输入!1.2.3 保存结果：保存方法同一般的Windows软件：“文件”→“保存”⇒“另存为”窗口→在“查找范围”内找到目标文件夹→输入文件名（比如输入“1”）→“”．Mathematica 4或Mathematica 5的文件的后缀是“nb”，当输入“1”时，即产生文件“1.nb”．1.2.4打开文件1.nb启动Mathematica →“文件”→“打开”⇒打开”窗口：→在“查找范围”内找到文件“1.nb”→“”即可．1.2.5 退出Mathematica：与一般应用软件一样，单击右上方的“”按钮（或用菜单：“文件”→“退出”）．1.3 操作小技巧1.3.1Ctrl+K的用途如果只知道命令的首写字母，可在输入该首写字母（要大写），再按下“Ctrl+K”组合键，则所有以该字母为首的命令都列出来，只要用鼠标双1.3.2使用前面已有的结果举例如下：例1.2 做如下操作：①输入：Integrate[x^2*(11-Sin[x]),{x,-1,1}]按：“Shift＋Enter”；②输入：%＋１，按：“Shift＋Enter”；③输入：%＋１，按：“Shift＋Enter”；④输入：%1＋1，按：“Shift＋Enter”；⑤输入：%3＋1，按：“Shift＋Enter”，计算结果如下：可见，“％”表示前一个计算结果；“％n ”表示第n 个计算结果. 1.3.3 删除行：见下图示1.4 数值计算请看下例：1.5 赋值与替换X=. 或Clear[x] 清除赋给x 的值expr/.{x->xval,y->yval} 用xval 、yval 分别替换expr 中的x 、y ． 例1.3 输入：x=3；y=4；w=x+y 输入：Clear[x,y]； 计算输入：z=(x+y)^2 计算输入：z/.x->5 计算输入：Clear[x,y]； 计算输入：u=x+y 计算输入：u/.{x->5,y->6} 计算 计算结果如下：1.6 自定义函数用户可以自行定义函数，一个函数一旦被定义好之后就可以象系的内部函数一样使用．例1.4 如要定义函数f(x)=x 2＋3x-2只要键入：即可．又如要定义分段函数2+1 < 0()= 2sin 0x x g x x x ⎧⎨≥⎩可键入：g[x_]:= Which[x<0,x^2+1,x>=0,2Sin[x]]或g[x_]:=If[x<0,x^2+1,2Sin[x]]请见以下计算结果：1.7 方程与方程组解例1.5 ① 解方程:0652=+-x x输入：Solve[x^2-5x+6==0,x]即可．② 解方程组 输入：即可（结果见下图）．22131x y x y +=⎧⎨-=⎩1.8例1.6 ① ⎪⎩⎪⎨⎧>---01222x x x 输入即可．② 解不等式)3(12>--x x 输入：即可（结果见下图）注： Mathematica系统有内部函数.还有一些系统扩展的功能但不是作为内部函数的、以文件的形式存储在磁盘上的文件，要使用它们，必须用一定的方式来调用这些文件，这些文件我们称之为程序包. 调用方式之一如上所述：或用：Needs["Algebra`InequalitySolve`"] 1.9 由递推式求数列的通项公式例1.7 设求数列的通项公式只要输入：11,1,n na na a-==1.10 作函数图像例1.8在同一坐标系中作出2-1y x 和y=sinx在[－2,2]内的图像．输入： Plot[{x^2-1,Sin[x]},{x,-2,2}] 结果见下图例1.9作出sinxcosy的三维图形输入：Plot3D[Sin[x]*Cos[y],{x,－2Pi,2Pi},{y,－即可（结果见下图）第二章 运用Mathematica 实现高等数学中的基本运算极限、导数和积分是高等数学中的主要概念和基本运算，如果你在科研中遇到较复杂的求极限、求导数或求积分问题，Mathematica 可以帮你快速解决这些问题。
普林斯顿数学指南(英文版)
普林斯顿数学指南(英文版)The Princeton Companion to Mathematics is a comprehensive guide that explores the vast and intricate world of mathematics. This encyclopedic reference work is an indispensable resource for students, educators, and anyonewith an interest in the subject. It covers a wide range of topics, from the fundamentals of arithmetic and geometry tothe more advanced concepts of calculus, number theory, and topology.The book is organized into several sections, eachfocusing on a specific area of mathematics. The first section, "The Foundations of Mathematics," provides an overview of the basic principles and concepts that underlie all mathematical disciplines. This includes an introduction to the history of mathematics, its major branches, and the key figures who have shaped its development over the centuries.The second section, "Algebra and Number Theory," delves into the study of abstract structures, such as groups, rings, and fields, as well as the properties and relationships of numbers. This section also covers the foundations of algebraic geometry, which is concerned with the study of geometric objects defined by polynomial equations.The third section, "Analysis and Calculus," explores the study of continuity, change, and limits, as well as the techniques used to solve problems involving rates of change, such as differentiation and integration. This section also covers the theory of complex analysis, which extends the ideas of real analysis to complex numbers.The fourth section, "Geometry and Topology," focuses on the study of shape, size, and spatial relationships, as well as the properties of abstract spaces that are not necessarily Euclidean. This section includes discussions of classical geometry, such as Euclidean and non-Euclidean geometries, aswell as more modern areas like topology and differential geometry.The fifth section, "Probability and Statistics," deals with the study of randomness and uncertainty, as well as the collection, analysis, and interpretation of data. This section covers the basic principles of probability theory, statistical inference, and statistical modeling.The final section, "Applied Mathematics," highlights the many ways in which mathematics is used to solve real-world problems in fields such as physics, engineering, economics, and computer science. This section includes discussions of optimization, game theory, cryptography, and other areas where mathematical techniques are essential for solving practical problems.Throughout the book, readers will find numerous examples, exercises, and applications that illustrate the concepts and techniques discussed in each section. These include bothhistorical examples, such as the development of calculus or the proof of Fermat's Last Theorem, and contemporary applications, such as the use of chaos theory in meteorology or the role of Fourier analysis in image processing.In addition to its comprehensive coverage of mathematical topics, The Princeton Companion to Mathematics also features biographical sketches of many of the most influential mathematicians in history, from ancient Greek geometers like Euclid and Archimedes to modern giants like Isaac Newton, Carl Friedrich Gauss, and Emmy Noether. These profiles provide insights into the lives and achievements of these remarkable individuals, as well as their contributions to the development of mathematics.Furthermore, the book includes a detailed glossary of mathematical terms and symbols, which can be especially helpful for readers who are new to the subject or encounter unfamiliar concepts. The glossary defines key terms andprovides examples that illustrate their meanings and uses in various contexts.As a reference work, The Princeton Companion to Mathematics is designed to be accessible to a wide range of readers, from beginners to advanced students and professionals. Its clear explanations, extensive examples, and engaging historical narratives make it an ideal resource for anyone seeking to deepen their understanding of mathematics or explore new areas of the subject.In conclusion, The Princeton Companion to Mathematics is a valuable resource for anyone interested in learning more about the fascinating world of mathematics. Its comprehensive coverage of topics, engaging historical narratives, and clear explanations make it an invaluable tool for anyone seeking to enhance their knowledge and appreciation of this vital field of study. Whether you are a student, educator, or simply acurious mind looking to explore the beauty and elegance of mathematics, this guide is an excellent place to start.。
如何使小学数学课堂教学更高效
当今社会对学生的学习能力、思维能力、数学能力等要求日益提高，小学数学教学的重要性不断凸显，要求数学教学的效率和质量持续提升，因此构建高校数学课堂成为教育者面临的重要课题。
使教学课堂更高效，即在有限的课堂时间内，完成更高的教学目标，取得更好的教学效果，这就需要教师全面研究教学内容、方法以及学生特点，然后坚持以学生为中心，运用科学的教学方法和模式，全面培养小学生的数学能力。
因此，数学教师需要在正确教育理念的指导下，充分激发学生在数学课堂上的主体意、和学习兴趣以及探究能力，将学生的学习潜能进行有效的激发，进而实现整个教学过程的高效性。
具体来说，可以从以下几个方面来提升小学数学课堂教学效率：一、设计课堂导入，激发学习兴趣小学数学学习是引导学生从具象思维向抽象思维转换的过程，小学生受到年龄和心理发展的限制，对抽象的数学知识往往理解困难，由此导致学生课堂注意力不集中，对数学学习兴趣不高，不利于课堂教学效率的提升，因此激发学生学习兴趣对构建高效课堂来说是必不可少的。
课堂导入是课堂教学的核心环节之一，设计科学、内容丰富、形式活泼的课堂导入可以很好的吸引学生的注意力，激发他们的学习兴趣和求知欲，使学生在接下来的课堂学习中集中精神，保证课堂教学高效进行，因此教师要重视课堂导入环节的研究和设计。
一方面，教师可以在课堂导入环节中设计小学生喜爱的各种活动，如讲故事、做游戏、讲解百科知识等，并将教学内容融入其中，提升学生的学习兴趣，必要时可以将其制作出微视频，以微课的形式导入课堂，其教学促进效果也是显而易见的；另一方面，教师可以运用小学生熟悉的生活物品或元素导入数学课堂，这样可以更加有效的集中学生的学习注意力，让学生消除数学学习的思维障碍，尽快的投入到学习活动中来。
例如，在学习“轴对称图形”一节时，教师在课堂导入环节可以向学生展示一些轴对称的物品，如对称剪纸、玩具飞机、蝴蝶、城堡等，然后让学生观察并思考他们的共同特点，激起学生的兴趣和探索欲望后引入轴对称的数学概念和相关知识，可以有效提升课堂教学效率。
maths-primary-2013
Learning Mathematics
A 21st Century Necessity Learning mathematics is a key fundamental in every education system that aims to prepare its citizens for a productive life in the 21st century. As a nation, the development of a highly-skilled and well-educated manpower is critical to support an innovation- and technology-driven economy. A strong grounding in mathematics and a talent pool in mathematics are essential to support the wide range of value-added economic activities and innovations. Many countries are paying attention to the quality of their mathematics education. The growing interest in TIMSS and PISA speaks of the global interest and importance placed on mathematics education. At the individual level, mathematics underpins many aspects of our everyday activities, from making sense of information in the newspaper to making informed decisions about personal finances. It supports learning in many fields of study, whether it is in the sciences or in business. A good understanding of basic mathematics is essential wherever calculations, measurements, graphical interpretations and statistical analysis are necessary. The learning of mathematics also provides an excellent vehicle to train the mind, and to develop the capacity to think logically, abstractly, critically and creatively. These are important 21st century competencies that we must imbue in our students, so that they can lead a productive life and be life-long learners. Students have different starting points. Not all will have the same interests and natural abilities to learn mathematics. Some will find it enjoyable; others will find it challenging. Some will find the theorems and results intriguing; others will find the formulae and rules bewildering. It is therefore important for the mathematics curriculum to provide differentiated pathways and choices to support every learner in order to maximise their potential. The curriculum must engage the 21st century learners, who are digital natives comfortable with the use of technologies and who work and think differently. The learning of mathematics must take into cognisance the new generation of learners, the innovations in pedagogies as well as the affordances of technologies. It is the goal of the national mathematics curriculum to ensure that all students will achieve a level of mastery of mathematics that will serve them well in their lives, and for those who have the interest and ability, to pursue mathematics at the highest possible level. Mathematics is an important subject in our national curriculum. Students begin to learn mathematics from the day they start formal schooling, and minimally up to the end of secondary education. This gives every child at least 10 years of meaningful mathematics education.
一年级数学教学指导书说明书
Everyday Mathematics Teacher's Lesson Guide FIRST GRADE Marilyn Burns Classroom MathLibrary*Marilyn Burns Classroom Math LibraryInstructional ContentUnit 1 Establishing Routines1.1 Daily Routines From One to One Hundred, pp. 20-21; One, Two, Skip a Few!, pp. 36-37Students count by ones and tens. Students recite and repeat rhymes that promote counting.1.2 Investigating the Number Line One, Two, Skip a Few!, pp. 36-37 Students use tally marks to interpret results.Students recite and repeat rhymes thatpromote counting.1.3 Tools for Doing Mathematics One, Two, Skip a Few!, pp. 36-37;Sea Shapes, pp. 38-39; The Shape ofThings, pp. 42-43; When a LineBends … A Shape Begins, pp. 60-61; Students identify shapes and relate them to their surroundings. Students draw shapes in patterns. Students draw different shapes. Students recite and repeat rhymes that promote counting.1.4 Number-Writing Practice Cats Add Up!, pp. 14-15; MouseCount, pp. 28-29; One MondayMorning, pp. 32-33; Splash!, pp. 46-47; Ten Red Apples, pp. 52-53Students count objects.1.5 One More, One Less Just a Little Bit, pp. 26-27; MouseCount, pp. 28-29; One MondayMorning, pp. 32-33; Ten Sly Piranhaspp. 54-55 Students predict outcomes and add or subtract numbers that are one more than or less than a given number.1.6 Comparing Numbers1.7 Recording Tally Counts One, Two, Skip a Few!, pp. 36-37 Students use tally marks to represent data.1.8 Investigating Equally LikelyOutcomes1.9 The Calendar1.10 Working in Small Groups Jelly Beans for Sale, pp. 24-25; One,Two, Skip a Few!, pp. 36-37; ShapeSpace, pp. 44-45; Ten Sly Piranhas,pp. 54-55; When a Line Bends … AShape Begins, pp. 60-61Multiple copies of these books are included for small group instruction.Everyday Mathematics Teacher's Lesson Guide FIRST GRADE Marilyn Burns Classroom MathLibrary*Marilyn Burns Classroom Math LibraryInstructional Content1.11 Explorations: Exploring Math Materials Cats Add Up!, pp. 14-15; Fat Frogson a Skinny Log, pp. 16-17; JellyBeans for Sale, pp. 24-25; Just aLittle Bit, pp. 26-27; Mouse Count,pp. 28-29; … 98, 99, 100! Ready orNot, Here I Come!, pp. 30-31; OneMonday Morning, pp. 32-33; OneMore Bunny, pp. 34-35; Seven BlindMice, pp. 40-41; The Shape ofThings, pp. 42-43; Shape Space, pp.44-45; Splash!, pp. 46-47; Ten BlackDots, pp. 48-49; Ten Red Apples, pp.52-53; When a Line Bends ... AShape Begins, pp. 60-61These books provide opportunities forexploration with manipulatives.1.12 Weather and TemperatureRoutinesOne, Two, Skip a Few!, pp. 36-37 Students use tally marks to represent data.1.13 Number Stories Quack and Count, pp. 10-13; CatsAdd Up!, pp. 14-15; Fat Frogs on aSkinny Log, pp. 16-17; Jelly Beansfor Sale, pp. 24-25; Just a Little Bit,pp. 26-27; Mouse Count, pp. 28-29;One Monday Morning, pp. 32-33;Splash!, pp. 46-47; Ten Black Dots,pp. 48-49; Ten Red Apples, pp. 52-53; Ten Sly Piranhas, pp. 54-55;Trick-or-Treat!, pp. 56-57 Students solve problems by retelling and/or reenacting stories using number sentences and/or manipulatives to represent the actions of the stories.1.14 Unit 1 Review andAssessmentUnit 2 Everyday Uses of Numbers2.1 Number Grids2.2 Numbers All Around2.3 Complements of 10Fat Frogs on a Skinny Log, pp. 16-17Students use manipulatives and write numbersentences to represent the actions of thestory which involve addition facts of 10.2.4 Unit Labels for Numbers… 98, 99, 100! Ready or Not, Here ICome!, 30-32Students skip count by twos, fives and tens.2.5 Analog Clocks10 Minutes till Bedtime, 50-51 Students estimate elapsed time.Everyday Mathematics Teacher's Lesson Guide FIRST GRADE Marilyn Burns Classroom MathLibrary*Marilyn Burns Classroom Math LibraryInstructional Content 2.6 Telling Time to the Hour10 Minutes till Bedtime, 50-51Students estimate duration of one minute.2.7 Explorations: ExploringLengths, Straightedges, andDominoes2.8 Pennies Jelly Beans for Sale, pp. 24-25Students recognize coins and determine theirvalue.2.9 Nickels Jelly Beans for Sale, pp. 24-25Students recognize coins and determine theirvalue.2.10 Counting Nickels and Pennies Jelly Beans for Sale, pp. 24-25Students recognize coins and determine theirvalue.2.11 Number Models Cats Add Up!, pp. 14-15; MouseCount, pp. 28-29; Splash!, pp. 46-47Students add objects.2.12 Subtraction Number Models Cats Add Up!, pp. 14-15; MouseCount, pp. 28-29;Students subtract objects.2.13 Number Stories Jelly Beans for Sale, pp. 24-25; Justa Little Bit, pp. 26-27; Ten RedApples, pp. 52-53; Trick-or-Treat!,pp. 56-57 Students count money. Students create and solve number stories.2.14 Unit 2 Review andAssessmentUnit 3 Visual Patterns, NumberPatterns, and Counting3.1 Visual Patterns The Shape of Things, pp. 42-43Students investigate shape patterns.3.2 Even and Odd NumberPatternsOne More Bunny, pp. 34-35Students investigate even and odd numbers.3.3 Number-Grid Patterns Mouse Count, pp. 28-29; … 98, 99,100! Ready or Not, Here I Come!, pp.30-31; Trick-or-Treat!, pp. 56-57Students skip count.3.4 Explorations: Exploring Even & Odd Numbers, Covering Shapes, and Patterns One More Bunny, pp. 34-35; TheShape of Things, pp. 42-42;Students investigate even and odd numbers.Students investigate shape patterns.3.5 Counting on the Number Line From One to One Hundred, pp. 20-21; ... 98, 99, 100! Ready or Not,Here I Come!, pp. 30-31; Trick-or-Treat, pp. 56-57Students explore counting patterns.3.6 Adding and Subtracting on theNumber Line3.7 Telling Time to the Half-HourEveryday Mathematics Teacher's Lesson Guide FIRST GRADE Marilyn Burns Classroom MathLibrary*Marilyn Burns Classroom Math LibraryInstructional Content 3.8 Introduction to the Frames-and-Arrows Routine3.9 More Frames-and-ArrowsProblems3.10 Counting with a Calculator3.11 Dimes Jelly Beans for Sale, pp 24-25; FromOne to One Hundred, pp. 20-21;…98,99,100! Ready or Not, Here ICome!, pp. 30-31 Students recognize coins and determine their value for exchanging. Students count by tens.3.12 Counting Dimes, Nickels, and Pennies Jelly Beans for Sale, pp. 24-25Students recognize coins and determine theirvalue.3.13 Data Day One, Two, Skip a Few!, pp. 36-37 Students use tally marks to represent data.3.14 Domino Addition One More Bunny, pp 34-35 Students explore patterns, odd and evennumbers, and addition of numbers.3.15 Unit 3 Review andAssessmentUnit 4 Measurement and BasicFacts4.1 Math Message and Reading aThermometer4.2 Nonstandard Linear Measure4.3 Personal “Foot” and StandardFoot4.4 The Inch4.5 The 6-Inch Ruler4.6 Measuring with a TapeMeasure4.7 Explorations: Exploring Data, Shapes, and Base-10 Blocks One, Two, Skip a Few!, pp. 36-37;Two Eyes, a Nose, and a Mouth, pp.58-59Students collect data to make a graph.4.8 Telling Time on the Quarter-Hour4.9 Timelines4.10 Number Scrolls4.11 Introducing Fact Power One More Bunny, pp. 34-35Students explore addition facts and theirsums.4.12 Good Fact Habits Quack and Count, pp.10-11 ; Cat'sAdd Up, pp.14-15Students practice addition factsEveryday Mathematics Teacher's Lesson Guide FIRST GRADE Marilyn Burns Classroom MathLibrary*Marilyn Burns Classroom Math LibraryInstructional Content 4.13 Unit 4 Review andAssessmentUnit 5 Place Value, NumberStories, and Basic Facts5.1 Place Value: Tens and Ones From One to One Hundred, pp. 20-21Students explore place value concepts for tens and ones.5.2 Place Value with Calculators From One to One Hundred, pp. 20-21Teachers can include calculators with these Marilyn Burns activities.5.3 Relations: Greater Than, Less Than, and Equal To Mouse Count, pp. 28-29; Ten SlyPiranhas, pp. 54-55Students learn that the equal sign showsequivalence.5.4 Explorations: Exploring Area, Weight, and Counting Just a Little Bit, pp. 26-27 Students compare weights using a balancescale.5.5 Animal Weights5.6 More Than and Less Than Number Stories5.7 Comparison Number Stories Cats Add Up!, pp. 14-15; Fat Frogson a Skinny Log, pp. 16-17; MouseCount, pp. 28-29; Ten Red Apples,pp. 52-53; Ten Sly Piranhas, pp. 54-55; Trick-or-Treat, pp. 56-57Students write subtraction equations to correspond to stories.5.8 Solving Number Stories Cats Add Up!, pp. 14-15; Fat Frogson a Skinny Log, pp. 16-17; MouseCount, pp. 28-29; Ten Red Apples,pp. 52-53; Trick-or-Treat, pp. 56-57Students write addition and subtraction equations to correspond to stories.5.9 Dice Sums5.10 Turn-Around Facts Cats Add Up!, pp. 14-15Students write number sentences, whichinclude "turn-around facts", to represent thestory.5.11 “Easy” Facts Fat Frogs on a Skinny Log, pp. 16-17; One, Two, Skip a Few!, pp. 36-37 Students write number equations to represent the addition facts of 10. Students explore addition facts of doubles.5.12 “What’s My Rule?”5.13 Applying Rules5.14 Unit 5 Review and AssessmentUnit 6 Developing Fact Power6.1 The Addition/Subtraction Facts Table Cats Add Up!, pp. 14-15Students write number sentences showingaddition fact combinations.Everyday Mathematics Teacher's Lesson Guide FIRST GRADE Marilyn Burns Classroom MathLibrary*Marilyn Burns Classroom Math LibraryInstructional Content6.2 Equivalent Names Cats Add Up!, pp. 14-15; One MoreBunny, pp. 34-35Students write equivalent number sentences showing different combinations.6.3 Fact Families Quack and Count, pp.10-11 ; Cat'sAdd Up, pp.14-15; Fat Frogs on aSkinny Log, pp. 16-17Students explore fact families through addtion and subtraction.6.4 Fact Triangles6.5 Using the Addition/Subtraction Facts Table for Subtraction6.6 The Centimeter6.7 Explorations: Exploring Pattern Blocks, Addition Facts, and Triangles Shape Space, pp. 44-45; Students explore and compare variousshapes.6.8 Addition Facts Practice with“What’s My Rule?”6.9 Quarters Jelly Beans for Sale, pp. 24-25Students recognize coins and determine theirvalue.6.10 Digital Clocks6.11 Timing in Seconds6.12 Data Landmarks One, Two, Skip a Few!, pp. 36-37;Two Eyes, a Nose, and a Mouth, pp.58-59Students collect data to make a graph.6.13 Unit 6 Review andAssessmentUnit 7 Geometry and Attributes7.1 Attribute Rules Feast for 10, pp. 18-19; From One toOne Hundred, pp. 20-21Students classify and sort objects.7.2 Explorations: Exploring Attributes, Designs, and Fact Platters Feast for 10, pp. 18-19; From One toOne Hundred, pp. 20-21Students classify and sort objects.7.3 Pattern-Block and Template Shapes Sea Shapes, pp. 38-39; The Shape ofThings, pp. 42-42; Shape Space, pp.44-45; When a Line Bends … AShape Begins, pp. 60-61Students explore and identify various shapes.7.4 Making Polygons Sea Shapes, pp. 38-39; The Shape ofThings, pp. 42-42; Shape Space, pp.44-45; When a Line Bends … AShape Begins, pp. 60-61Students explore and identify various shapes.Everyday Mathematics Teacher's Lesson Guide FIRST GRADE Marilyn Burns Classroom MathLibrary*Marilyn Burns Classroom Math LibraryInstructional Content 7.5 Spheres, Cylinders, andRectangular Prisms7.6 Pyramids, Cones, and Cubes7.7 Symmetry7.8 Unit 7 Review and AssessmentUnit 8 Mental Arithmetic, Money,and Fractions8.1 Review: Money Jelly Beans for Sale, pp. 24-25Students recognize coins and determine theirvalue for exchanging.8.2 Dollars8.3 Place Value: Hundreds, Tens, and Ones From One to One Hundred, pp. 20-21Students explore place value concepts fortens and ones.8.4 Application: Shopping at theSchool Store8.5 Making Change Quack and Count, pp. 10-13; MouseCount, pp. 28-29Students count on from a given number.8.6 Equal Shares8.7 Fractions8.8 Sharing Pennies8.9 Explorations: ExploringFractional Parts and Addition Facts8.10 Unit 8 Review andAssessmentUnit 9 Place Value and Fractions9.1 Tens and Ones Patterns on the Number Grid From One to One Hundred, pp. 20-21; …98, 99, 100! Ready or Not, HereI Come!, pp. 30-31Students explore place value. Students skipcount by tens.9.2 Adding and Subtracting Tens From One to One Hundred, pp. 20-21Teachers can use this title to support lesson.9.3 Number-Grid Puzzles9.4 Adding and Subtracting 2-DigitNumbers9.5 Explorations: ExploringCapacity, Symmetry, and HeightsEveryday Mathematics Teacher's Lesson Guide FIRST GRADE Marilyn Burns Classroom MathLibrary*Marilyn Burns Classroom Math LibraryInstructional Content 9.6 Fractional Parts of the Whole9.7 Comparing Fractions9.8 Many Names for FractionalParts9.9 Unit 9 Review and AssessmentUnit 10 Year-End Review andAssessment10.1 Data Day: End-of-YearHeights10.2 Review: Telling Time10.3 Mental Arithmetic Using a Vending Machine Poster Jelly Beans for Sale, pp. 24-25Students recognize coins and determine theirvalue to show different amounts.10.4 Mental Arithmetic(Continued)10.5 Year-End Geometry Review Sea Shapes, pp. 38-39; The Shape ofThings, pp. 42-42; Shape Space, pp.44-45; When a Line Bends … AShape Begins, pp. 60-61Students explore and identify various shapes.10.6 Review: Thermometers andTemperature10.7 Review: Place Value, Scrolls,and Number Grids。
五下数学第一单元作文英语
五下数学第一单元作文英语Reflections on Unit 1 of Grade 5 Mathematics":A Mathematical Journey: Reflections on Unit 1 of Grade 5 MathematicsMathematics has always been a subject that captivates and challenges me. As I embark on the first unit of my fifth-grade mathematics curriculum, I am filled with a sense of excitement and anticipation. This unit promises to be a transformative experience that will not only deepen my understanding of mathematical concepts but also inspire me to think critically and creatively.At the heart of this unit lies the exploration of place value. This foundational concept serves as the cornerstone upon which we build our understanding of numbers and their relationships. As I delve into the intricacies of place value, I am struck by the elegance and simplicity of this system. The way in which digits are positioned to represent the value of a number is a testament to the ingenuity of human thought.One of the key components of this unit is the comparison ofnumbers. I find myself fascinated by the various strategies and techniques employed to determine the relative magnitude of different numbers. Whether it is comparing two-digit numbers, three-digit numbers, or even numbers with different place values, the process of identifying the greater or lesser value is a captivating exercise in logical reasoning.As I progress through the unit, I am introduced to the concept of rounding numbers. This skill, which may seem straightforward at first glance, actually requires a deep understanding of place value and the ability to make strategic decisions. Rounding numbers to the nearest ten or hundred can have significant implications in real-world applications, and I am eager to master this technique.Another fascinating aspect of this unit is the exploration of patterns and relationships within the number system. I find myself delving into the intricacies of odd and even numbers, as well as the various ways in which numbers can be decomposed and recomposed. The ability to recognize and analyze these patterns not only strengthens my mathematical foundation but also fosters a sense of wonder and appreciation for the underlying structures of the numerical world.One of the most rewarding aspects of this unit is the opportunity to apply my newfound knowledge to real-world problems. Whether it is calculating the total cost of a shopping trip, estimating thepopulation of a city, or determining the appropriate amount of supplies for a classroom project, the practical applications of place value, comparison, and rounding skills are endless. I relish the challenge of translating abstract mathematical concepts into tangible, meaningful solutions.As I navigate through the various lessons and activities within this unit, I am struck by the interconnectedness of the mathematical concepts. Each topic builds upon the previous one, creating a tapestry of knowledge that is both comprehensive and cohesive. This holistic approach to learning mathematics not only deepens my understanding but also helps me to see the subject as a unified whole, rather than a collection of disparate parts.One of the most valuable lessons I have learned in this unit is the importance of perseverance and a growth mindset. Mathematics, by its very nature, can be challenging and at times, frustrating. However, I have come to understand that the struggle is an integral part of the learning process. When I encounter a problem that seems insurmountable, I am learning to approach it with a sense of curiosity and determination, rather than giving up. This mindset not only helps me to overcome obstacles but also fosters a deep appreciation for the process of discovery and problem-solving.As I reflect on my journey through this first unit of fifth-grademathematics, I am filled with a sense of pride and accomplishment. The skills and knowledge I have gained have not only strengthened my mathematical foundation but have also instilled in me a deeper love and appreciation for the subject. I am excited to continue exploring the wonders of mathematics, knowing that each new challenge will bring with it an opportunity for growth and learning.In conclusion, this first unit of fifth-grade mathematics has been a transformative experience. Through the exploration of place value, number comparison, rounding, and patterns, I have developed a more profound understanding of the underlying principles of mathematics. This journey has not only challenged me intellectually but has also ignited a passion within me to continue exploring the limitless possibilities of this fascinating subject. As I look ahead to the rest of my fifth-grade mathematics curriculum, I am filled with a sense of anticipation and a deep desire to continue expanding my mathematical horizons.。
英国数学竞赛教材
英国数学竞赛教材
针对英国数学竞赛，推荐以下教材：
《First Steps For Math Olympians》：这本书是特别针对AMC全球数学竞赛考试所作的，该书籍把AMC考试中的考点和知识点一一进行说明分析，讲解了解决大多数数学竞赛的基本思路，并提供了过去考试中的真题和练习题来说明这些概念。
《 Chung's AMC10,12》：这本书由John Chung 博士撰写。
本书创建了一套AMC竞赛系统，能够帮助学生轻松掌握AMC概念，除了知识点讲解，这本书还有AMC的历年真题和难题，让学生从基础到强化，层层进阶，直至掌握高等数学的技巧。
此外，还有《高中数学竞赛教程》和《高中数学习题册》也是不错的选择，前者主要适用于对数学有特别兴趣并希望挑战更难题的学生，后者是一本练习题型比较全面的教材。
这些教材都是针对数学竞赛的专门教材，能够帮助学生深入理解数学知识，提高解题能力，为参加数学竞赛做好准备。
以上推荐仅供参考，建议咨询专业人士获取更全面准确的信息。
核心素养背景下小学数学课堂作业设计
核心素养的培育是当今教育发展的根本趋势，是我国推进实施新时期教育的重要内容和战略目标。
对小学核心素养特别是数学核心素养进行有针对性的研究，对其内涵进行精准的把握，对其要素构成与价值体现进行剖析，同样具有积极的意义。
因而，现阶段小学数学教学必须聚焦数学核心素养的有效培养，使之渗透在日常教学中的所有教学场景和环节。
课堂作业设计是数学日常教学中的一个不可或缺的重要组成部分，所以对核心素养背景下小学数学课堂作业设计予以关注并进行深入研究和探讨便显得极为必要。
一、核心素养背景下小学数学课堂作业设计的价值小学生的核心素养实际上应该囊括三方面的内容，即态度、技能与认知这三大层面的综合，这其中特别突出地强调了价值理念、情感和态度。
数学教学的目的、内容与核心素养间存在着某种必然的联系。
小学生数学核心素养形成的背后是数学学习综合能力的集中显现，而科学地进行课堂作业设计是实现核心素养培育的前提，其价值具体表现为以下两大层面：第一，强化学生的数学素养。
从本质上来说，素养对于小学数学而言意味着测量、运算及相关的过程。
这一过程不但使学生能够掌握相关的知识，还能培养相应的思维能力，提升综合能力。
体现核心素养的课堂教学设计能针对学生综合能力的提升加以重点培养，并对学生独立进行问题的思考予以鼓励，不断提升学生学习的自主性，促进学生数学学习综合能力、核心素养的有效提升。
第二，体现教学理念与目标。
能够反映数学本质与价值的只能是数学核心素养。
当前正在推进实施的素质教育从根本目标上来讲就是促进学生在义务教育阶段掌握相应的数学知识，并在这一过程中促进学生的个性化发展。
数学教学从根本上说就是促进学生数学素养的有效提升，使学生在学习中的要求得到满足。
课堂作业可以更好地体现数学学习过程的方法与思想，是实现教学理念与目标的有效途径和重要手段。
二、核心素养背景下小学数学课堂作业设计存在的不足第一，作业内容缺乏丰富性。
在进行设计数学课堂作业时，很多教师并未能更多地考虑到作业内容的丰富性和趣味性因素，所设计的作业内容不能紧密地结合小学生的日常生活实际，一般都直接利用教材后面的习题，或者依据参考书中的现行习题材料，这样的作业设计虽然起到了巩固课堂知识的作用，但是却忽略了一个重要问题，那就是数学与日常生活应有的联系，这对小学生数学知识的运用、对核心素养的培养，收效甚微。
小升初英语数学知识点总结
小升初英语数学知识点总结1. ArithmeticArithmetic is the most important branch of mathematics, and it involves the basic operations of addition, subtraction, multiplication, and division. Students should be familiar with the properties of these operations, such as commutative, associative, and distributive properties. They should also be able to perform calculations involving fractions, decimals, percentages, and ratios.2. Number theoryNumber theory is the study of the properties of integers, and it is an important part of the mathematics curriculum for junior high school. Students should be familiar with the concepts of factors, multiples, prime numbers, composite numbers, and the fundamental theorem of arithmetic. They should also be able to solve problems involving divisibility, prime factorization, and greatest common divisor.3. AlgebraAlgebra is another important branch of mathematics, and it involves the study of mathematical symbols and the rules for manipulating those symbols. Students should be able to work with algebraic expressions, solve linear equations, and understand the concepts of variables, constants, coefficients, and exponents. They should also be familiar with the properties of real numbers, such as the distributive, associative, and commutative properties.4. GeometryGeometry is the study of shapes, sizes, and properties of objects in space. Students should be familiar with the basic concepts of geometry, such as points, lines, angles, polygons, and circles. They should also be able to calculate the area, perimeter, and volume of different geometric figures, as well as solve problems involving properties of triangles, quadrilaterals, and other shapes.5. MeasurementMeasurement is an important part of mathematics, and students should be familiar with the basic units of measurement for length, area, volume, weight, and time. They should be able to convert between different units of measurement, as well as solve problems involving measurements of objects and distances.6. Data analysis and probabilityData analysis involves the collection, organization, and interpretation of data, while probability is the study of chance and uncertainty. Students should be familiar with the concepts of mean, median, mode, range, and probability. They should be able to interpretand analyze data from tables, graphs, and charts, as well as solve problems involving probability and combinations.7. Problem-solving strategiesIn addition to the specific mathematical concepts mentioned above, students should also be familiar with various problem-solving strategies, such as working backwards, drawing a diagram, making a table, and using logical reasoning. They should be able to apply these strategies to solve a wide range of mathematical problems, including word problems and real-life scenarios.In conclusion, the above-mentioned mathematical concepts are important for students preparing for the junior high school entrance exam. By mastering these concepts, students will be well-prepared to tackle the challenges of the mathematics curriculum in junior high school and beyond. Good luck with your studies!In conclusion, these key mathematical concepts are essential for students preparing for the junior high school entrance exam. By mastering these concepts, students will be well-prepared to tackle the challenges of the mathematics curriculum in junior high school and beyond. Good luck with your studies!。
2020mathematics subject classification -回复
2020mathematics subject classification -回复2020数学科学分类（2020 Mathematics Subject Classification）是一个根据数学学科的不同领域进行分类的系统。
该分类系统由国际数学联合会（International Mathematical Union，简称IMU）于2020年发布。
它旨在对数学的研究和教学活动进行统一和系统的分类，并帮助学者和教育工作者更好地了解和定位自己的研究领域。
一、2020数学科学分类的目的2020数学科学分类的目的是为了促进数学研究的交流和合作。
通过将不同学科领域进行分类，可以更好地组织和管理数学知识，便于学者之间的沟通和研究合作。
此外，分类还有助于学生和教育工作者了解数学的各个分支和应用，从而更好地选择学习和教学内容。
二、2020数学科学分类的结构2020数学科学分类共分为两层，第一层是大类（Primary），第二层是子类（Secondary）。
大类主要根据数学学科的一级分类分类，而子类则进一步细分不同的领域。
在大类中，数学被分为10个大类别，分别是：1. 逻辑学和基础数学逻辑2. 代数学3. 分析学4. 几何学5. 拓扑学6. 数论7. 统计学8. 运筹学、优化和控制论9. 概率论和数理统计10. 数学物理每个大类别下又细分为几个子类别，以更加具体的领域进行分类。
例如，代数学大类别下有33个子类别，包括群论、环论、域论等；分析学大类别下有23个子类别，包括实分析、复分析、泛函分析等。
三、使用2020数学科学分类的方法对于学者和研究者来说，研究成果的发布和交流是非常重要的。
使用2020数学科学分类可以帮助他们将自己的研究内容准确地归入到相应的领域中，从而吸引到对该领域感兴趣的人士。
此外，在学术会议和期刊投稿时，也可以根据分类系统的标准选择合适的会议或期刊。
对于教育工作者和学生来说，了解数学的不同领域和应用对于选择学习和教学内容非常重要。
美国2017年度四年级数学测试题赏析
美国2017年度四年级数学测试题赏析殷莉莉(编译)【摘要】美国国家教育进展评估(NAEP)是美国衡量中小学生各学科学业成就的一个评价测量体系,规模最大,持续时间最长,1969年首次在美国全国范围内实施。
NAEP的结果被教师、校长、政策制定者和研究人员用于评估美国教育的进展,改进美国的教育。
数学学科主要评估四年级和八年级学生的知识、技能,以及在解决问题时运用知识的能力。
包括五个方面的内容,即数的特征和运算,度量,几何,数据分析、统计和可能性,代数。
根据问题的复杂程度,可将题目分为低、适度和高三个层次。
【期刊名称】《小学教学：数学版》【年(卷),期】2019(000)002【总页数】5页(P60-64)【关键词】美国教育;数学学科;四年级;试题赏析;中小学生;NAEP;政策制定者;测量体系【作者】殷莉莉(编译)【作者单位】[1]教育部北京师范大学基础教育课程中心数学工作室【正文语种】中文【中图分类】G571.2美国国家教育进展评估（NAEP）是美国衡量中小学生各学科学业成就的一个评价测量体系，规模最大，持续时间最长，1969年首次在美国全国范围内实施。
NAEP 的结果被教师、校长、政策制定者和研究人员用于评估美国教育的进展，改进美国的教育。
数学学科主要评估四年级和八年级学生的知识、技能，以及在解决问题时运用知识的能力。
包括五个方面的内容，即数的特征和运算，度量，几何，数据分析、统计和可能性，代数。
根据问题的复杂程度，可将题目分为低、适度和高三个层次。
低复杂度的问题通常规定了学生要做什么，执行常规的数学程序；适度复杂的问题往往需要多步骤解决，涉及更多的灵活性思维；高复杂度的问题则对学生的思维提出了更高的要求，需要在新的情境中进行抽象的推理、分析。
2017年度，NAEP 首次采用数字化测试，将测试题呈现在平板电脑上，部分问题还为学生提供了屏幕计算器或虚拟的便签本。
以下给出题目赏析。
第1题（领域：数的特征和运算；复杂程度：低；难度水平：简单）：如图1，每个正方形由100 个大小相同的小方格组成。
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2、造成交通事故后逃逸构成犯罪的人不号灯禁止通行。（错、绿灯行）
4、立交桥上一般都是单向行驶，车辆不必减速行驶。（错、太绝对了、应看情况）
5、将点火开关转到ACC位置起动机工作。（错）
6、车速超过规定时速达到50%的一次记3分。答案：错
7、驾驶机动车在上道路行驶前驾驶人要按规定系好安全带。 答案：对
8、伪造、变造或者使用伪造、变造驾驶证的驾驶人构成犯罪的，将依法追究刑事责任。答案：对
9、发生交通事故造成人员受伤时，要保护现场并立即报警。答案：对
10、机动车在发生碰撞时，安全带可以减轻驾乘人员伤害。答案：对