HPM视角下数学归纳法教学的设计研究
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摘要
近年来,国际HPM领域发展迅猛,越来越多的HPM研究者走出象牙塔,进入教学第一线,教学实践成为HPM领域研究中最重要的一个研究方向,成为HPM研究者建立,检验与发展理论的重要途径。
荷兰著名数学教育家弗赖登塔尔指出数学归纳法的教学存在很多严重问题,有些甚至是违反教学法,建议参照历史的发展来教学。此外,Harel研究表明,学生对数学归纳法的理解呈现历史相似性。本研究借鉴已有研究,选择以数学归纳法为载体,从数学史融入数学教学的视角,开发教学设计,并在真实的教学情境中实践教学设计。研究梳理了数学教育领域的相关理论,在此基础上,搭建HPM教学设计的理论框架,结合HPM领域的设计研究方法,建立HPM领域教学实践的三棱锥模型,以此来指导HPM视角下数学归纳法教学实践,并在研究过程中不断修正与完善教学设计,检验建立的框架与模型。
研究选取二所高中四个班进行数学归纳法教学实践,一个班级作为控制班,另外三个班作为实验班,在三棱锥模型的指导下进行三轮教学实践。本研究问题是：
·学生理解数学归纳法是否存在历史相似性?
·HPM视角下数学归纳法教学对学生理解水平层次以及情感态度价值观有什么影响?
·HPM视角下数学归纳法教学对教师专业发展有什么影响?
本研究通过访谈、问卷测试、教学实录等多种方式收集研究数据,经过对数据进量化与质性分析,解决研究问题,得出研究结论。具体通过四次教学前的问卷测试来分析学生理解数学归纳法的历史相似性；通过四次教学前后的问卷测试来分析学生学习的认知水平变化,通过单因子方差分析四个班级间的前后测认知水平差异；通过教学前后的访谈来定性分析学生的情感态度价值观的变化；通过教学实录对HPM课堂要素进行分析,并判断HPM融入的程度；通过教师的访谈来分析教师的专业化发展三个方面；最后总结以HPM视角下数学归纳法教学实践为研究载体的研究成果。
本研究表明：(1)学生理解数学归纳法呈现出明显的历史相似性,且理解的水平层次是主要是处于归纳推理水平与联接递推水平；(2)在对比分析HPM视角下数学归纳法教学与正常教学之后,发现采用HPM教学方式的学生理解水平显著高于采用正常教学方式,且学生更喜欢HPM教学方式,认为其有助于学习并能加深理解；(3)教师在参与HPM教学实践之后,教育信念发生了微变、HPM教学知识显著增加以及教学能力也得到提升。本研究中的成果有HPM领域教学实践的三棱锥模型、学生理解数学归纳法的历史相似性研究案例、HPM视角下数学归纳法教学设计案例、HPM领域教学实践的研究三原则、学生理解数学归纳法的四个水平层次、HPM领域教学实践研究之五步骤。
In recent years, with the rapid development of international Study Group on the Relations between History and Pedagogy of Mathematics, more and more researchers on HPM walk out of the ivory tower into the teaching, practice research in the field of teaching become one of the most important research directions on HPM, becoming an important way for researchers to create, test and develop the theory of HPM.
Freudenthal H, a famous Dutch mathematics educators, said that there are a lot of serious problems in the teaching of mathematical induction, some even violated the principles of pedagogy, and proposed to teach mathematical induction in the way of basing on its history. What's more, Harel's research showed that students'understanding of mathematical induction presents historical parallelism. Learned from previous studies, this study used mathematical induction as a carrier, designed instruction of mathematical induction based on HPM, did practice research in the real teaching situations. Based on the relative theories in the field of mathematics education, researcher build a theoretical framework on combed the area of mathematics education theory, on this basis, build a theoretical framework of instructional design in the field HPM, and combined with the method of design research methods to set up pyramid model in the field teaching practice on HPM. With the guidance of pyramid model, teaching practice mathematical induction based on HPM been carried out. in the process of studying, instructional design was revised and completed, and the established framework and model were tested.
Four classes from two high schools were selected to be taught mathematical induction, a class is a control class, the others are experimental classes. With the guidance of pyramid model, the instructional design were carried out three times. The research questions are: Whether historical parallelism on Students' understanding of mathematical induction exists?
What are the impact on the levels of students' understanding of mathematical induction after the HPM teaching, and emotion, values and attitude? What is the impact on teachers'professional development after the HPM teaching?
The study collected related data through interviews, questionnaires testing and teaching memoir, analyzed the data by the way of quantitative and qualitative to solve research questions, and drew conclusions. More specifically, historical parallelism on students' understanding of mathematics education were analyzed by comparing with four pre-tests to; and varieties of the level of understanding on mathematical induction were analyzed by comparing with four after-tests; the differences between four pre-tests and after-tests were analyzed by one-way ANOVA; varieties in the aspects of emotion, attitude and value were analyzed through interviews; the degrees of integration of HPM were analyzed through teaching memoir; teachers' professional development were analyzed through teacher interviews; lastly, the conclusions on teaching practice of mathematical induction were made.
This study showed that:(1)there are significant historical parallelism on the students' understanding of mathematical induction, and the level of understanding is the level of inductive reasoning or in connection recursive level;(2)comparative analysis of two ways of teaching, found that the levels of students taught by the way of HPM teaching were significantly higher than students taught by using the normal teaching way, and the HPM teaching way is likable by students, because this HPM teaching way could contribute to learning and deepening their understanding;(3)teachers involved in the process of HPM teaching practice, their education conviction occurred slightly changed, instructional knowledge on HPM is significantly increased, and teaching abilities has been improved.
The results of this study includes that pyramid model in the field of HPM teaching practice, case study of historical parallelism on students' understanding of mathematical induction, case study of HPM instructional design on mathematical induction perspective, three research principles of HPM teaching practice, four levels of understanding of mathematical induction, and five research Steps of HPM teaching practice.
荷兰著名数学教育家弗赖登塔尔指出数学归纳法的教学存在很多严重问题,有些甚至是违反教学法,建议参照历史的发展来教学。此外,Harel研究表明,学生对数学归纳法的理解呈现历史相似性。本研究借鉴已有研究,选择以数学归纳法为载体,从数学史融入数学教学的视角,开发教学设计,并在真实的教学情境中实践教学设计。研究梳理了数学教育领域的相关理论,在此基础上,搭建HPM教学设计的理论框架,结合HPM领域的设计研究方法,建立HPM领域教学实践的三棱锥模型,以此来指导HPM视角下数学归纳法教学实践,并在研究过程中不断修正与完善教学设计,检验建立的框架与模型。
研究选取二所高中四个班进行数学归纳法教学实践,一个班级作为控制班,另外三个班作为实验班,在三棱锥模型的指导下进行三轮教学实践。本研究问题是：
·学生理解数学归纳法是否存在历史相似性?
·HPM视角下数学归纳法教学对学生理解水平层次以及情感态度价值观有什么影响?
·HPM视角下数学归纳法教学对教师专业发展有什么影响?
本研究通过访谈、问卷测试、教学实录等多种方式收集研究数据,经过对数据进量化与质性分析,解决研究问题,得出研究结论。具体通过四次教学前的问卷测试来分析学生理解数学归纳法的历史相似性；通过四次教学前后的问卷测试来分析学生学习的认知水平变化,通过单因子方差分析四个班级间的前后测认知水平差异；通过教学前后的访谈来定性分析学生的情感态度价值观的变化；通过教学实录对HPM课堂要素进行分析,并判断HPM融入的程度；通过教师的访谈来分析教师的专业化发展三个方面；最后总结以HPM视角下数学归纳法教学实践为研究载体的研究成果。
本研究表明：(1)学生理解数学归纳法呈现出明显的历史相似性,且理解的水平层次是主要是处于归纳推理水平与联接递推水平；(2)在对比分析HPM视角下数学归纳法教学与正常教学之后,发现采用HPM教学方式的学生理解水平显著高于采用正常教学方式,且学生更喜欢HPM教学方式,认为其有助于学习并能加深理解；(3)教师在参与HPM教学实践之后,教育信念发生了微变、HPM教学知识显著增加以及教学能力也得到提升。本研究中的成果有HPM领域教学实践的三棱锥模型、学生理解数学归纳法的历史相似性研究案例、HPM视角下数学归纳法教学设计案例、HPM领域教学实践的研究三原则、学生理解数学归纳法的四个水平层次、HPM领域教学实践研究之五步骤。
In recent years, with the rapid development of international Study Group on the Relations between History and Pedagogy of Mathematics, more and more researchers on HPM walk out of the ivory tower into the teaching, practice research in the field of teaching become one of the most important research directions on HPM, becoming an important way for researchers to create, test and develop the theory of HPM.
Freudenthal H, a famous Dutch mathematics educators, said that there are a lot of serious problems in the teaching of mathematical induction, some even violated the principles of pedagogy, and proposed to teach mathematical induction in the way of basing on its history. What's more, Harel's research showed that students'understanding of mathematical induction presents historical parallelism. Learned from previous studies, this study used mathematical induction as a carrier, designed instruction of mathematical induction based on HPM, did practice research in the real teaching situations. Based on the relative theories in the field of mathematics education, researcher build a theoretical framework on combed the area of mathematics education theory, on this basis, build a theoretical framework of instructional design in the field HPM, and combined with the method of design research methods to set up pyramid model in the field teaching practice on HPM. With the guidance of pyramid model, teaching practice mathematical induction based on HPM been carried out. in the process of studying, instructional design was revised and completed, and the established framework and model were tested.
Four classes from two high schools were selected to be taught mathematical induction, a class is a control class, the others are experimental classes. With the guidance of pyramid model, the instructional design were carried out three times. The research questions are: Whether historical parallelism on Students' understanding of mathematical induction exists?
What are the impact on the levels of students' understanding of mathematical induction after the HPM teaching, and emotion, values and attitude? What is the impact on teachers'professional development after the HPM teaching?
The study collected related data through interviews, questionnaires testing and teaching memoir, analyzed the data by the way of quantitative and qualitative to solve research questions, and drew conclusions. More specifically, historical parallelism on students' understanding of mathematics education were analyzed by comparing with four pre-tests to; and varieties of the level of understanding on mathematical induction were analyzed by comparing with four after-tests; the differences between four pre-tests and after-tests were analyzed by one-way ANOVA; varieties in the aspects of emotion, attitude and value were analyzed through interviews; the degrees of integration of HPM were analyzed through teaching memoir; teachers' professional development were analyzed through teacher interviews; lastly, the conclusions on teaching practice of mathematical induction were made.
This study showed that:(1)there are significant historical parallelism on the students' understanding of mathematical induction, and the level of understanding is the level of inductive reasoning or in connection recursive level;(2)comparative analysis of two ways of teaching, found that the levels of students taught by the way of HPM teaching were significantly higher than students taught by using the normal teaching way, and the HPM teaching way is likable by students, because this HPM teaching way could contribute to learning and deepening their understanding;(3)teachers involved in the process of HPM teaching practice, their education conviction occurred slightly changed, instructional knowledge on HPM is significantly increased, and teaching abilities has been improved.
The results of this study includes that pyramid model in the field of HPM teaching practice, case study of historical parallelism on students' understanding of mathematical induction, case study of HPM instructional design on mathematical induction perspective, three research principles of HPM teaching practice, four levels of understanding of mathematical induction, and five research Steps of HPM teaching practice.
引文
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