中年级学生面临的数学内容和学习问题

原文作者 National Convocation on Mathematics Educ; National Research Council

摘要：在英国，学生读中年级的年龄里也刚好是他们升学或是转学的年纪。在他们升到中学后，老师从一个变成了许多个，不同老师教授不同的学科。本文主要探讨怎样的做法更有利于解决学生在中年级面临的问题。在这个年龄段的学生往往会将课堂上真正的知识遗忘却记住了生动的例子，或者他们并没有真正掌握希望他们掌握的知识。

关键词：中年级；数学；新知识

在英国，义务教育从5岁一直到16岁，政府提供持续两年的免费教育。由于学校系统的结构各不相同，孩子们在幼儿园（5-7岁），初中（8-11）和中学（11-16 / 18）中相继学习，或者在9岁和13岁时转学、跳级。所以，在英国，孩子们在中年级的时间段也恰恰是要更换或刚更换学校的时候。在幼儿园，老师是个多面手，可能要教学生们一周的所有学科。对于学生，升到中学后意味着他们一天要面对许多不同的教师，而这些教师往往只教授一个学科的知识。在小学，老师们往往更多地是通过利用一个主题（例如，维京人）的展示，来说明英语，地理，宗教，艺术，科学或者数学。在中学，这些科目被分开成为一门门单独的学科，而老师们只教授其中的一个学科。

制度，在学校变为更加注重正规的学习，在数学上则表现为学习能力和数学技能及其应用。我们总是假设目前孩子已经掌握了某项技能，所以当她发现自己并没有完全掌握时，我们可能会对她失去信心。除此之外，还有儿童的生理变化、新的兴趣发现，不过还好他们愿意学习任何事情。

正规化操作

皮亚杰对教育理论的影响意味着许多孩子的小学教育，11岁以下的孩子的学习是在具体的层面上的。数学教育工作者多年来都认为在这个阶段，教师需要一个具体的实施方案或教具来促进学生掌握抽象的数学思想。然而，教具和抽象之间的鸿沟一直没有得到解决。“儿童数学框架”（CMF）研究项目试图探讨儿童从“具体的”经验公式到数学推广活动时所做的过渡。当一个数学教师在申请硕士学位时，在他们入学所需要完成以下模块：

a. 确定一个主题,他们通常会介绍如何使用教具，通过这个教具让学生可以学会一个公式、算法或其他数学的。

b.准备一系列用来教育学生的课程。

c. 在教学开始之前，允许CMF研究小组在正规化教学之前、之后、三个月后面试六个孩子。

d.提醒研究人员用形式化的教学方法，并且在课堂上允许观察者进行观察和用磁带录音。

e.采访班上的两个孩子，并结合磁带对录课内容进行分析，将结果整理汇报。

在具体完成实施方案的研究中，导致形式化描述的主题的是一个矩形的区域、长方体的体积、两位数和三位数减法中数字的分解、分数的规则是等价的、圆的周长或图形的放大图。

在教学手册等书籍中，给教师的建议往往描述了孩子们应该有的经验，然后暗示（甚至明确指出）说：“孩子们认识到公式。”在实践中，只有一部分的孩子会达到我们的预期设想，剩下的孩子要在老师的鼓励下才能接受。这位老师认为，这样做时间短，也应该继续这样做下去。三个月的后续面试的一部分是要求学生们能接受这些经验。只有一个受访者能够想起之前的学习经验，并为之后的学习提供基础。大多数是由女孩总结出“求和是求和，求砖是求砖。”如果具体经验已经能成功被使用的，那么遗忘是不重要的，但具体经验的形成往往是费事又费力，而学生又没有这种具体的经验。教训告诉何老师，成绩单分析观测到的是非常不同的。老师知道数学的公式或规则，并制定了一套“操控”动作来试图说服孩子。孩子不知道应该对他/她的红砖木做什么能让他们得到表扬，而老师可以假设说长宽高分别是“x，2，4”，甚至说“让我们假设面积是17。”

向大人学习

从CMF的意见（以及一些后续的研究）中我们可以发现，这是普通的老师和孩子们踏上了发现老师熟知的知识的航程。教师解释了为什么没有观察到的学生愿意放弃砖，因为这是一种简单的思考方法，甚至发现孩子的方法更有利于形式化，但没有人能解释新知识。最近的声明原因是“你不想在你的余生随身携带砖。”该老师的态度是一个友好的指导，在这个意义上，我们已经知道教训是不断的检讨，而不是引入到已知。考虑到教师如何划分几个圆盘，将它分成相等的分数段的时候借鉴了主板准确的细分。插图展示的是徒手将整体迅速分裂成部分。难怪特伦斯制作这套图的时候，他试图说服面试官的9/27 = 3/9（图1）。

安德鲁是在一个小组学习减法，老师产生了三位数的加减导致零百。随后的谈话如下：

（注：T =教师; P =学生）：

P：如果我从100里拿走1，那就什么都没有了。

T：什么都没有，所以我把它放在这儿？

P：不。

T：那我能放在这儿吗？谁认为我应该放在这儿？谁不这么认为？好的，我的意思是，你可以这样，但是如果我要求你在你的书上写99，就写99，你不会写099吧？

P：不会。

T：你会只写了99，不是吗？所以，我们并不需要把0放在这儿。 1带走1给你留下一个空的空间，所以我们还不如留下一个空的空间，好吗？

安德鲁的希望在304-178中会有和他发生如下的争论：

A：如果你说，从4里拿走8，还剩4。你已经有了4，你不能从4取8，所以那里什么也没有。 。 。没有带7，你不能做到这一点，从3里拿走1还剩2。

T：我明白了。所以我不能从4里拿走8。所以，我下面写点东西还是不写？

A：不写。

他学会了“不写”，但不是在使用它。有儿童选择老师的说法的具体组成部分，概括记住了老师所制造的一个错误的说法。

分数和小数

中年级的时候算术的元素不再是整数，更多的精力和时间都花在了分数和小数的研究上。目前已经有很多关于儿童对这些数字理解的研究。一般八九岁的孩子能识别并命名的小数区域为1/2，1/4，1/3，1/8；较少学生能将区域分割成的十二分也可在六分之间进行标记。很少有孩子能够顺利学会分数。目前我们大多数教材引进的模式是通过区域（方形，圆形，线），这方便我们的讨论，但结果是一个有形的数量（披萨片，立方体的巧克力）不完全适合在加法操作，减法，乘法和除法中。你怎么平分两块比萨饼？A/B的其他含义往往不在学校的教科书中，而是将来自同一地区的模型推断划分成比。其结果是学习依赖混乱和死记硬背。许多孩子拒绝被询问关于数量和试图解决中学数学中他们的想法。本研究在中学数学概念的项目”与科学”（CSMS）通过访谈和测试获得的数据。11-16岁的一个代表性的样品被要求回答各种划分问题，并告诉他们说，如果问题是“不可能的”，他们就应该这么说。

我们还可以利用测试来检测掌握程度，要求学生写一个故事来说明8 4和128 8。几乎所有的反应都是朋友间在分享糖果。这种解释的划分标准在16divide;20明显不够朋友分糖果时有了冲突。教师应该重新定义来适应分数和小数吗？中年级时孩子试图移植新概念对绝望的不足来计数。在这些年里代数是容易被引入的，但把它作为广义算法似乎困难重重。在代数中，我们需要把现在所代表数的字母和数产生连接，并且算法的目的是为了进行操作，尽可能快地得到了一个结果。“X Y”不能被处理成为XY，而我们发现5 3可以被8取代。科利斯（1975）描述了一个水平的代数，理解为“允许关闭“的概念缺乏（ALC）。一个孩子能接受甚至学会（x y），是非常重要的一步。

让孩子能自己匹配数学

人们早就知道，在任何一类课中，不同孩子学会的可能是非常不同的知识。在CMF一组8名学生，老师用同样的方式教学，结果发现孩子学会的路径非常不同。孩子的成功取决于之前所知道有多少、多少当前内容物被理解、他的注意力，并对数学的兴趣。如果所有的数学练习，他被标记为不正确，那么孩子可能对自己的数学学习缺乏自信。这种情况到了中年级很可能变成是任何一组包含了一些“低注意”的孩子，不积极行动，不太可能变得更加“平均注意。”

课程开发项目“纳菲尔德中学数学”的目的是给中学生（年龄11-16岁）提供合适的材料，让学生都能达到应有水平。这些书是：1）主题的书籍：四本内容较短书籍；2）核心图书：一整年的书籍，让学生运用他们的数学组合能力共同努力；3）教师指导：为教师提供非常充分的信息。为了找到从哪里开始的号码链，对包括一些被确定为显示“特殊需求”的约100名小学生进行了测试。后续的采访透露，有学生即将进入中学，但掌握的技能可能非常有限。通过之前必要的技能列表，我们可以说，在开始的时候，孩子们可以做到以下几点：

bull;安排显示卡的配置点1到6。

bull;写之前的数字和后两位数。

bull;不从1数，数2条邮票。

bull;正确书写两位数时，会读法。

bull;会写小于25的数字。

bull;解释说“多”与“少”的时候，给出两组数

bull;用不同面值的计数一堆硬币（不足一磅）的准确数值。

bull;选择购买合适的单枚便士。

中学教师在职业培训课程实践中的一级数量的内容，问他们是否有学生只能“做这个。”通常他们向我们保证，学生不知道的很少，但后来被我们要求出示在这一层次的材料，我们发现在试点学校通常有大约有十个11-12岁的孩子还需要进一步学习。我们只发现一个学生在一个正常的学校，学习的难度对他们的来说太大了。该材料通过几个监管良好的试验表明，如果它证明了对于使用它的学生来说难度太大了，那我们就需要重新编写​​。我们的目的是，让孩子体验成功，让他们对自己的数学学习有信心。早期的结果是，第一本书也许需要三个月才能完成，第二个花更少的时间。随着信心的建立，编写的速度没有与孩子学习的速度相当。使用这种材料的早期类通常是学生在老师的帮助下两人或三人一组。没有孩子进展到下一本书，直到他/她通过考试证明了前一本书的内容至少已经被吸收了80％，一直考到老师们确定学生们都会通过因为他们都理解了书本的内容，失败的测试有助于老师和学生。这个概念是对一些教师非常困难，因为他们认为总会有失败。我们密切参与了所有的试验包括测试。在一所学校中，老师一致认为，之所以六个孩子还没有达到中考的“掌握”级别，是因为他们没有覆盖整本书，只是掌握了其中的一部分，这导致他们不能完整的学习所有内容。老师解释说，她没办法等到这些学生也掌握了才进行下一环节的教学。

外文文献出处：Mathematics Education in the Middle Grades: Teaching to Meet the Needs of Middle Grades Learners and to Maintain High Expectations

附外文文献原文

Mathematics Content and Learning Issues in the Middle Grades

In the United Kingdom compulsory schooling starts at the age of five and continues until 16 years of age. The provision of free education continues for another two years. The structure of the school system varies and children can proceed through infant school (age 5-7),junior school (8-11) and secondary school (11-16/18) or through a system in which they change schools at 9 and 13 years of age. So in England a child in the middle grades is probably changing (or has just changed) schools. In the first schools, the teacher is a generalist and probably teaches all the subjects the child meets during a week. For the pupil, being promoted to a secondary school (or even a middle school) means that there are many teachers to face in any one day. These teachers tend to be interested in only one curriculum subject. In the primary school, it is likely that the teacher has tried to present the curriculum through project work, which might mean that the intention was to exploit a topic (e.g., The Vikings) for its possibilities to illustrate English, geography, religion, art, science, and mathematics. In the secondary school, these subje

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Mathematics Content and Learning Issues in the Middle Grades

In the United Kingdom compulsory schooling starts at the age of five and continues until 16 years of age. The provision of free education continues for another two years. The structure of the school system varies and children can proceed through infant school (age 5-7),junior school (8-11) and secondary school (11-16/18) or through a system in which they change schools at 9 and 13 years of age. So in England a child in the middle grades is probably changing (or has just changed) schools. In the first schools, the teacher is a generalist and probably teaches all the subjects the child meets during a week. For the pupil, being promoted to a secondary school (or even a middle school) means that there are many teachers to face in any one day. These teachers tend to be interested in only one curriculum subject. In the primary school, it is likely that the teacher has tried to present the curriculum through project work, which might mean that the intention was to exploit a topic (e.g., The Vikings) for its possibilities to illustrate English, geography, religion, art, science, and mathematics. In the secondary school, these subjects are allotted separate time slots, and the teacher only teaches that subject.

The institutional life in school changes to a greater focus on formal learning, and in mathematics the concentration is on competencies and skills and their application. Often assumptions are made concerning the repertoire of skills the child already possesses, and she may lose confidence when it is shown her repertoire is limited. Add to this the changes in childrenrsquo;s physical makeup and the new interests which occupy them, and it is a wonder that they learn anything.

FORMALISATION—MANIPULATIVE LINK

The influence of Piaget on educational theory has meant that much of the childrsquo;s primary education has been imbedded in the idea that pre-11 years of age the child is operating at the

concrete level. Mathematics educators for many years have interpreted this stage as one which requires concrete embodiments or manipulatives to promote what are essentially abstract mathematical ideas. The chasm between the manipulatives and the abstraction has not been addressed very thoroughly. The research project “Childrenrsquo;s Mathematical Frameworks” (CMF) sought to investigate the transition made by children when moving from “concrete” experiences to a formula or mathematical generalisation. Teachers who were pursuing a mastersrsquo; degree in mathematics education enrolled for a module which required them to:

a. Identify a topic which they would normally introduce with the use of manipulatives, which experience was to lead to a formula, algorithm or other mathematical generalisation.

b. Prepare a series of lessons and teach them to a target class.

c. Allow the CMF research team to interview six children, before the teaching started, just before the formalisation took place, just after it, and three months later.

d. Alert the researchers to when the “formalisation” lesson or acceptance of the rule would take place and allow the lesson(s) to be observed and tape-recorded.

e. Interview two other children in the class and report on the responses. Additionally, an analysis of the tape recorded lesson would be written (the transcript of the recording being sup-plied by the researchers).

Topics which were included in the study, fulfilling the description of concrete embodiments leading to formalisation, were area of a rectangle, volume of a cuboid, subtraction of two and three digit numbers with decomposition, the rule for fractions to be equivalent, the circumference of a circle, and enlargement of a figure.

The advice given to teachers in teaching manuals etc., often describes the experiences the children should have and then implies (or even states) that “the children will come to realise” the formula. In practice, it seems that a few children in a class might come to the realisation, and the rest be encouraged to accept the findings of their fellows. The teacher feels that time is short, and the class must move on. Part of the three month follow-up interviews was to ask the pupils for the connection between the two experiences, concrete and formal. Only one of the interviewees (out of 150) remembered that one experience led to the other and provided a base for it. Most of the replies are summed up by the girl who said “Sums is sums and bricks is bricks.” The forgetting would be unimportant if the concrete experience (which is often arduous and time consuming) had resulted in a successful use of the formalisation, but it had not. The observations of lessons and analysis of the transcripts of what the teacher said brought to light how very disparate were the views of teacher and pupil. The teacher knew the mathematics, knew the formula or rule, and had devised a set of “manipulative” moves to convince the child of the truth of the rule. The child did not know where the manipulations were leading and to him/her a red brick made from two centimetres of wood was exactly that. The teacher might refer to it as “x, 2, 4” and could even say “let us pretend it is 17.”

JOINING THE GROWN-UPS

From the observations in CMF (and some subsequent research), it was plain that teachers and children embarked on a voyage of discovery to a place well known to the teacher. None of the teachers observed (some 20 experienced practitioners) explained why the pupils would want to abandon bricks, naive methods, and even invented childmethods in favour of the formalisation. Nobody explained the power of the new knowledge. The nearest statement to a reason for its adoption was “you do not want to carry around bricks for the rest of your life.” The teacherrsquo;s attitude was one of friendly guidance, more in the sense that the lessons were a review of some

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