摘 要:数学史是研究数学概念、数学方法和数学思想起源与发展、及其与社会、经济和一般文化联系的一门学科,它反映了数学发展的脉络与本质。数学史的价值表现为三个方面:数学史的历史价值,数学史的数学价值,数学史的教育价值。随着数学史与数学教育关系国际研究小组成果的不断涌现,数学史对数学教育的价值被人们所认识和应用。在新一轮中学数学课程改革中,数学史被看成理解数学的一种途径。
HPM研究成果的日益丰富以及数学课程改革的理念和要求,促使我们关注数学史在数学教育的实际应用,数学史应用研究应重视从概念教学着手,因为数学概念是数学的基础,思维的细胞,是数学教学的出发点。
本研究目的是探讨数学史如何应用于概念教学,主要采用文献研究、实验研究和调查研究的研究方法。通过理论研究探索数学史应用于概念教学的切入点、原则和策略。根据理论研究结果,教师有目的的将数学史融入概念教学,通过教学实验验证理论的有效性。
本文通过理论研究得出数学史在概念教学中的切入点、遵循的原则和采用的主要教学策略:
数学史在概念教学中的切入点主要有概念产生的背景及价值、产生过程与内涵、概念表示方法的发展变化、从历史角度分析概念蕴含的数学思想等。
中学阶段,数学史在概念教学中的应用遵循的原则主要有:应用形式为“融入式”为主;根据教学内容和学生的认知选择合适的数学史内容;在应用时应注重认知纬度的教学;遵循再创造的原则获得概念;遵循直观性原则呈现数学概念的演变过程;采用灵活多样的教学方式,提高学生的参与程度。
数学史应用于概念教学的主要策略是 “问题策略”和“有指导的再创造策略”,即教师要引导学生由一系列“问题”的解决 “再创造”的获得概念。
教学实验结束,学生的问卷及访谈结果显示:学生不仅能够适应数学史融入概念教学,而且提高学习兴趣。实验班与对比班测试成绩显示:实验班与对比班存在显著性差异,由此可知数学史融入概念教学的策略有助于学生的概念理解与运用。
但是在实验中,也出现了很多的困难,如因为小部分学生由于没有做好课前准备导致上课时不能积极参与;数学教材与数学史的发展不吻合,这些都对试验造成了一定的难度,另外,由于课时较紧,作者水平的局限,所采用的课时较少,对试验的结果大面积推广有所保留。
关键词:数学史;概念教学;切入点;原则;策略
ABSTRACT :Mathematical history is a subject to research into the contact between mathematical concept, mathematical method, the origin of mathematical thought and society, economy and common culture. It reflects the skeleton and essence of mathematical development. The value of mathematical history represents the three aspects as following: the history value, the mathematical value and educational value. With all the results of mathematical history and International Study Group on the Relations between History and Pedagogy of Mathematics, the educational value of mathematical history is being understood and accepted by people. In present mathematical course reform, mathematical history is considered as a new way to understand mathematical.
The research results of HPM are becoming more and more colorful and they urge us to pay more attention to the practical application of mathematical history in mathematical education. Application research on mathematical history should attach importance to the concept because mathematical concept is the basis of mathematical, cell of thinking and jumping-off place of mathematical teaching.
The aim of this research is to discuss how the mathematical history can be used in concept teaching; seek after the cut-in points, principles and strategies about the application to the concept teaching of mathematical history through academic research. Teachers combine mathematical history with the design of concept teaching on purpose and validate the academic validity through teaching experiments.
This essay concludes the cut-in points, principles and strategies about the application to the concept teaching of mathematical history through academic research:
I think, via research, during the period of middle school, the application form of mathematical history in concept teaching should be “the form of being in harmony”; and we should choose the suitable mathematical history content according to the teaching content and the students’ perceive; we should obtain concept through paying more attention to the latitudinous teaching and recreating principles; presenting the outcoming process of mathematical history through sight principles; improving the degree of the students’ attaching themselves to the process .
The main strategies of the mathematical history’s application in concept teaching are: “problem strategy” and “the instructed recreated strategy”, that’s to say, teachers instruct students to recreate concept happily with problems.
After the experiment, the students’ questionnaire and talk results indicate: they can not only adapt to the concept teaching of mathematical history, but also improve their interest in study. The achievement test between the academic class and contrast class indicate: there is obvious difference between them. Therefore, concept teaching of mathematical history is helpful in understanding and seizing the concept.
Difficulties also appear in the experiments: some students can’t take part in the activities well because of being not flexible in using the mathematical ways; or not seizing the mathematical history well and not being prepared well enough before class. The development between mathematical teaching material and mathematical history doesn’t cooperate. All the above affect the experiments. And because of the lack of lesson hours; the limit of the writers’ level; they all lead to remain of the wide spread of the experiments.
Key words: Mathematical History;Concept Teaching;Cut-in Points; Principle; Strategy
第一章 问题提出
一、问题提出的背景
1.数学史具有重要的教育价值
数学史是研究数学概念、数学方法和数学思想起源与发展、及其与社会、经济和一般文化联系的一门学科,它反映了数学发展的脉络与本质。数学史的价值表现为三个方面:数学史的历史价值,数学史的数学价值,数学史的教育价值。随着数学史与数学教育关系国际研究小组(International Study Group on the Relations between History and Pedagogy of Mathematics)成果的不断涌现,数学史的教育价值被人们所认识和应用。
对于在数学教育中使用数学史的问题至少可以追溯到1890年代以前。如美国著名数学史家卡约黎(Florian.Cajori,1859─1930), 美国著名数学史家和数学教育家史密斯 (David Eugene Smith,1860─1944), 意大利著名数学史家洛利亚, 丹麦著名数学史家邹腾(Hieronymus Georg Zeuthen,1839─1920), 英国著名数学家德摩根(A.De Morgan,1806─1871), 荷兰著名数学史家迪克斯特休斯(E.Jan Dijksterhuis)等欧美数学家、数学史家都大力提倡数学史的教育价值。他们关于在数学教学中如何使用数学史都有不同的论述。