第三节相关理论及其课程模式启示

1、n一般来说，儿童早期的数学认知发展包括了数、运算、测量、一般来说，儿童早期的数学认知发展包括了数、运算、测量、空间空间/几何几何和模式和模式五个维度，每个维度之间是相互五个维度，每个维度之间是相互独立的。独立的。n美国美国2000年年学校数学的原则和标准学校数学的原则和标准中，提出了五个方面的中，提出了五个方面的内容：数和运算；模式、函数和代数；几何与空间感；度量；内容：数和运算；模式、函数和代数；几何与空间感；度量；数据分析、统计与概率。数据分析、统计与概率。n我国的幼儿园数学教育课程内容框架中，有关数概念、数的运我国的幼儿园数学教育课程内容框架中，有关数概念、数的运算、空间与时间概念的内容

2、比较明确算、空间与时间概念的内容比较明确，而相对缺乏，而相对缺乏与儿童抽象与儿童抽象逻辑思维发展密切相关的诸如模式、统计与简单的函数关系理逻辑思维发展密切相关的诸如模式、统计与简单的函数关系理解的内容。解的内容。n“问题解决问题解决”是美国数学教育界在是美国数学教育界在20世纪世纪80年代提出的年代提出的主要口号，即认为应当以主要口号，即认为应当以“问题解决问题解决”作为学校数学教育作为学校数学教育的中心，这一思想在的中心，这一思想在80年代后期兴起的美国新的数学教育年代后期兴起的美国新的数学教育改革运动中又得到了进一步的确认，并对整个世界数学教改革运动中又得到了进一步的确认，并对整个世界数学

3、教育课程产生了极大的影响育课程产生了极大的影响。n在我国，传统的数学教育过分注重知识体系、注重纯概念在我国，传统的数学教育过分注重知识体系、注重纯概念性数学，缺乏与生活实际的联系与应用。性数学，缺乏与生活实际的联系与应用。n在在学校数学的原则和标准学校数学的原则和标准中，美国第一次加入了中，美国第一次加入了2-5岁儿童数学教育的标准，这个新标准仍然坚持了岁儿童数学教育的标准，这个新标准仍然坚持了1991年年提出的数学课程的五个能力目标，即让学生：提出的数学课程的五个能力目标，即让学生：n（1）学会认识数学的价值；）学会认识数学的价值；n（2）对自己的数学能力具有信心；）对自己的数学能力具有信心

4、；n（3）具有数学地解决问题的能力；）具有数学地解决问题的能力；n（4）学会数学的交流；）学会数学的交流；n（5）学会数学的推理。）学会数学的推理。明确的细化，其中五个是关于数学活动过程的能：明确的细化，其中五个是关于数学活动过程的能：（1）问题解决问题解决（2）推理与证明（能认识到推理与证明的一般意义，能进行数的推测、）推理与证明（能认识到推理与证明的一般意义，能进行数的推测、论证、评定和数的证明并能运用多种类型的推理和证明方法）；论证、评定和数的证明并能运用多种类型的推理和证明方法）；（3）交流（能与同伴、教师和其他人进行清楚的数学方面的交流，能）交流（能与同伴、教师和其他人进行清楚的数学

5、方面的交流，能分析和评价别人的数学思考并能用数学的语言精确地表达数的概念）；分析和评价别人的数学思考并能用数学的语言精确地表达数的概念）；（4）联系（能认识并运用数概念之间的联系，并能在数学以外的情景）联系（能认识并运用数概念之间的联系，并能在数学以外的情景中认识和应用数学）；中认识和应用数学）；（5）表述（能用多种表征的手段来表达数学的概念，能运用数的表征）表述（能用多种表征的手段来表达数学的概念，能运用数的表征方式来解决问题和演示、解释物质的、社会的和数学的现象）。方式来解决问题和演示、解释物质的、社会的和数学的现象）。n情境认知理论认为，知识是文化、情境的产物，真正的学习是在情境认知理论

6、认为，知识是文化、情境的产物，真正的学习是在有意义的情境中发生的，学习情境的性质决定了所学知识在其他有意义的情境中发生的，学习情境的性质决定了所学知识在其他情境中再应用的可能性。所谓情景学习就是在所学知识的真实的、情境中再应用的可能性。所谓情景学习就是在所学知识的真实的、应用的环境中，通过目标定向的活动而进行的学习。应用的环境中，通过目标定向的活动而进行的学习。n皮亚杰建构主义理论。皮亚杰建构主义理论。n从社会建构主义的理论出发，数学学习被认为是对生活所定从社会建构主义的理论出发，数学学习被认为是对生活所定义的知识和价值的共同建构，它是通过社会建构的机会发生，义的知识和价值的共同建构，它是通过

7、社会建构的机会发生，并通过与他人和环境的互动而进行的。因此，它包含了三个并通过与他人和环境的互动而进行的。因此，它包含了三个基本因素：社会性、情境性和互动性。基本因素：社会性、情境性和互动性。n儿童正式数学能力是一种关于数学知识的书面化、法则化和儿童正式数学能力是一种关于数学知识的书面化、法则化和系统化的知识体系。系统化的知识体系。n儿童的非正式数学能力主要包括在学校教育体系之外获得的儿童的非正式数学能力主要包括在学校教育体系之外获得的关于数量的观念与方法。具有明显的生活化和情境化特点。关于数量的观念与方法。具有明显的生活化和情境化特点。n近年来国外不少的研究表明，在学前儿童的数学学习中，近年

8、来国外不少的研究表明，在学前儿童的数学学习中，儿童并非仅仅是通过儿童并非仅仅是通过“做做”来学习的，而是通过思考以及来学习的，而是通过思考以及谈论他们所做的事情来学习的谈论他们所做的事情来学习的。n同时同时，研究还表明，教师所使用的与数学有关的语言总量，研究还表明，教师所使用的与数学有关的语言总量，与儿童今后在学校中数学知识的增长有着显著的关系与儿童今后在学校中数学知识的增长有着显著的关系。n幼儿园幼儿园的数学不应当仅仅是基于操作的的数学不应当仅仅是基于操作的“哑巴数学哑巴数学”。n所谓数学学习中的多元表征主要是指对同一个数学对象，所谓数学学习中的多元表征主要是指对同一个数学对象，至少可以用数

9、和形两类表征的多种形式进行表征。对学前至少可以用数和形两类表征的多种形式进行表征。对学前儿童数学学习而言，其表征形式一般可以概括为实物情境儿童数学学习而言，其表征形式一般可以概括为实物情境表征、教具模型表征、图形或图表表征、口语表征和书面表征、教具模型表征、图形或图表表征、口语表征和书面符号表征五类。符号表征五类。n美国美国n英国英国n日本日本England(Problem Solving, Reasoning and Numeracy)nRequirementsnPractitioners must support children in developing their understan

10、ding of problem solving, reasoning and numeracy in a broad range of contexts in which they can explore, enjoy, learn, practise and talk about their developing understanding. Practitioners must offer opportunities for these skills to be practised, in order to give children confidence and competence i

11、n their use.nWhat Problem Solving, Reasoning and Numeracy means for childrennThis Area of Learning and Development includes seeking patterns, making connections, recognising relationships, working with numbers, shapes, space and measures, counting, sorting and matching. Children use their knowledge

12、and skills in these areas to solve problems, generate new questions and make connections across other Areas of Learning and Development.nHow settings can effectively implement this Area of Learning and Development many different activities, some of which will focus on mathematical development and so

13、me of which will draw out the mathematical learning in other activities, including observing numbers and patterns in the environment and daily routines; practical activities underpinned by childrens developing communication skills; activities that are imaginative and enjoyable; real-life problems, e

14、.g. “How many spoons do we need for everyone in this group to have one?”; modelling mathematical vocabulary during the daily routines and throughout practitioner-led activities; giving children sufficient time, space and encouragement to use new words and mathematical ideas, concepts and language du

15、ring child-initiated activities in their own play; encouraging children to explore problems, to make patterns and to count and match together; the balance between learning and teaching indoors and outdoors. For example, having read a story about washing clothes, there might be laundrette play indoor

16、s and washing line play outdoors, streets of clothes shops built out of recyclables, bikes and other wheeled vehicles being used as delivery vans, numbered (and lettered) parking spaces. The staff would spend time in both environments and the level of child-initiated and practitioner-led activity wo

17、uld be monitored and divided more or less equally across both environments. Displays would include examples from both environments; help for those children who use a means of communication other than spoken English in developing and understanding specific mathematical language; opportunities to obse

18、rve, assess and plan the next stage in childrens learning; relevant training to improve practitioners knowledge, skills and understanding.Scotland1.(building the curriculum 1 3 to18)nDeveloping successful leanersnDeveloping confident individuals nDeveloping responsible citizensnDeveloping effective

19、contributorsnInformation handling:Data and analysisIdeas of chance and uncertaintynNumber, money and measurement: Basic arithmetical and other number processes, including estimation Measurement Patterns and relationships Equations and expressions nShape， Position and movement: Properties of 2D shape

20、s and 3D objects Symmetry， Angle and transformations 2. A curriculum framework for children 3 to 5ndevelop their powers of observation using their sensesnrecognise objects by sight, sound, touch, smell and tastenask questions, experiment, design and make, and solve problemsnrecognise patterns, shape

21、s and colours in the world around themnsort and categorise things into groupsnunderstand some properties of materials, for example soft/hard, smooth/roughnunderstand the routines and jobs of familiar peoplenbecome familiar with the early years setting and places in the local areanbecome aware of eve

22、ryday uses of technology and use these appropriately (scissors, waterproof clothing, fridge, bicycle)nbe aware of daily time sequences and words to describe/measure time, for example snack-time, morning, first, next, clocknbe aware of change and its effects on them, for example their own growth, cha

23、nges in weather,trees, flowersncare for living things, for example plants, pets at homenbe aware of feeling good and of the importance of hygiene, diet, exercise and personal safetyndevelop an appreciation of natural beauty and a sense of wonder about the worldnunderstand and use mathematical proces

24、ses such as matching, sorting, grouping, counting and measuringnapply these processes in solving mathematical problemsnidentify and use numbers up to ten during play experiences and counting gamesnrecognise familiar shapes during play activitiesnuse mathematical language appropriate to the learning

25、situations.nMathematics also plays an important role in learning. It provides pupils with knowledge and procedures for working with patterns and relationships in number and shape, with a powerful, concise and unambiguous way of handling and communicating information, and with a means by which the wo

26、rld can be explained and predicted. The curriculum presents mathematics as a problem-solving activity supported by a body of knowledge, involving pupils in discovering, discussing, ordering, classifying, generalising, drawing and measuring. Through working in these ways, pupils will learn the concep

27、ts, facts and techniques required to use and apply mathematics in different contexts.1.The Big Math for Little Kids curriculum (Balfanz,Ginsburg, & Greenes, 2003; Ginsburg, Greenes, & Balfanz,2003) nuses activities and storybooks to engage children first in learning key concepts of number, t

28、hen shape, pattern, measurement, operations on number, and finally space. Activities are offered for each day of the school year. Within each of the larger topics, the activities are arranged in order of difficulty, as indicated by research on the developmental trajectories of childrens mathematics

29、learning. Thus, in the case of number concepts, children first begin to learn number words, and then encounter concepts of cardinal number, representation, and next ordinal number, in that rough order.2.Building Blocks (Clements & Sarama, 2007a)ndrawsupon an extensive body of research on develop

30、mental trajectories to create materials “ designed to help children extend and mathematize their daily activities, from building blocks to art and stories” (Clements & Sarama, 2007b, p. 138). The materials are unique in integrating three types of media: computers, manipulatives, and print. The c

31、urriculum focuses on two major topics, space/geometry and number/quantity. A “small scale summative research” study showed impressive gains for low-SES children, especially in the areas of subitizing (“seeing” a number quickly, without counting), sequencing, shape identification, and the composition

32、 of shapes (Clements & Sarama, 2007b).3.The Measurement-based approach (Sophian, 2004)was developed for teaching mathematics in the Head Start program. Drawing on the work of Russian psychologists (Davydov, 1975) and developed in collaboration with teachers, the program assumes that the concept

33、of unit is crucial to the early understanding of number, measurement, and geometric shapes. The curriculum includes a weekly project activity conducted by Head Start teachers, various supplementary activities, and weekly home activities for parents to conduct with their children.4.The Number Worlds

34、curriculum (Griffi n, 2007b)ncovers basic number concepts from preschool through the sixth grade. It pays special attention to helping children navigate among the three different worlds of “real quantities that exist in space and time, the world of counting numbers and the world of formal symbols” (Griffi n,2007a, p. 375). Building on the natural developmental progression, the program attempts to teach concepts foundationalfor learning and to promote rich connections among different areas of knowledge.5.The Pre-K Mathematics