著者

長谷川 結城

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.18, no.2, pp.35-45, 2012

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Nowadays, the improvement of teaching toward students oriented activity in the high school mathematics class is strongly required to enhance the mathematical thinking of students. The author has a belief that the essences of teaching matters should be found out by students themselves in the class, and the teacher should prepare appropriate situations to realize such findings. In particular, it appears often at the introduction of teaching contents. In this paper, we study a practical plan to realize the aspect stated above in high school mathematics classes, through teaching materials and actual practices. We put the theoretical base of our assertion on the constructive approach proposed by Ito (1993) and other authors. The main findings are as follows: (1) We show that it is effective to provide opportunities that students consult or discuss with neighboring students each other about the focused problem they are facing in the class, and that such effective communication by students is possible if the teacher prepares proper situation and set appropriate time allocation. (2) We analyze what are the important factors on teaching materials to be used at the introduction, and we show several concrete samples of teaching materials that satisfy such factors. The samples have been examined in actual classes and proved to inspire students' mathematics thinking at introduction. (3) We report one practice of a class by the author, which incorporates the factors in (1) and (2), and analyze the assessment of it through the questionnaire to the students. As a result, high percentage of students give positive answers to the planned class activities by expressing their satisfaction for good understanding.

著者

大谷 洋貴

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.21, no.1, pp.1-10, 2015-01-31 (Released:2019-01-17)

参考文献数

28

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The purpose of this paper is to clarify curriculum issues concerning the development from descriptive statistics to inferential statistics based on an epistemological perspective of negation, and to refer to perspectives about solving the issues.  For achieving this purpose, three tasks are worked on.  The first task is to analyze history of statistics based on Otaki’s conception model so that the static conception of descriptive and inferential statistics is interpreted.  The second task is to make clear the dynamic conception of descriptive and inferential statistics based on Iwasaki’s framework on the negation theory in concept formation.  A framework about the development to inferential statistics is made by solving these tasks.  The third task is to analyze mathematics textbooks from the perspective of the framework.  As a result, it is found out that the development to inferential statistics is not necessarily intended.   In the development to inferential statistics three steps of negation are required.  The first step is negation of statistic and construction of substance of a statistic, which is not the same of one in descriptive statistics. The next step is analytic negation in which substance of a statistic is negated by viewpoint of decision, and it is regarded as a random variable in inferential statistics.  The final step is synthetic negation in which a random variable is negated by viewpoint of a parameter, and the relation between descriptive and inferential statistics is made clear.  This sequence of negation is shown below (Fig.).Fig. the development from descriptive statistics to inferential statistic

著者

岡本 光司

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.19, no.2, pp.15-26, 2013-06-22 (Released:2019-01-17)

参考文献数

26

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This article is concerned with “class culture” in mathematics learning based on the widely accepted idea of ‘culture’. We define the purpose of the article as follows: ・Building on previous studies about the characteristics and effects of culture, we will consider and discuss the status and role of “children’s questions” in mathematics learning. As a result of our discussion, we have gained several insights (from ① to ⑫). In addition, we have given several examples of children’s questions.① “Children’s questions” are generated in a “class culture.” Moreover the former in turn contributes to the shaping of the latter.② Individual “children’s questions” are often related to something universal or paradigmatic.③ “Children’s questions” can provide strong motives for cooperative learning in school.④ “Children’s questions” can help overthrow the stale procedures of learning.⑤ “Children’s questions” can stimulate the children to surpass themselves. ⑥ “Children’s questions” can enliven activities for collective learning. ⑦ “Children’s questions” can instigate interaction between tangible resources (such as textbooks, reference books, teaching materials, etc.) and intangible ones (value judgments, beliefs, thought processes, behavior, etc.).⑧ “Questioning” can be one of the effective strategies for mutual nurturing.⑨ Learning activities in which the people involved appreciate and recommend “questioning” can lay the foundation of the understanding of others.⑩ “Questioning” can encourage learners to strive for superior values.⑪ Learning activities in which “children’s questions” are appreciated and recommended may be valuable not only in terms of efficiency and economy but also in terms of spiritual and moral cultivation. ⑫ “Questioning” can trigger the formation of true identities.

著者

岡崎 正和

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.13, pp.1-13, 2007 (Released:2019-01-17)

参考文献数

46

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This paper focuses on a design experiment methodology in mathematics education which has been developed as a methodology for establishing a close and dynamic relationship between theory and practice, and discusses the comprehensive characteristics of the methodology. The design experiment methodology intends to develop (local) theories in mathematics education through designing, practicing and systematically analyzing daily classroom lessons over a relatively long period, where a researcher is responsible for students' mathematical learning in collaboration with a teacher. However, the methodology has also been questioned as to its scientific quality by the positivist scholars, since it explicitly deals with classroom practices that can be characterized as complex phenomena. Thus, this paper tries to place the design experiment methodology especially from a scientific point of view. The points discussed in this paper are the following. 1. The design experiment is an effective methodology for realizing mathematics education as a design science, and it intends to create a fruitful relationship between theory and practice. 2. The design experiment aims to construct a class of theories about the process of learning and the means that are designed to support that learning through (a) designing and planning the learning environments, (b) experimenting the design and the ongoing analysis, and (c) the retrospective analysis. 3. The design experiment is an interventionist methodology that goes through an iterative design process featuring cycles of invention and revision. 4. The design experiment has its intention of producing a theory which premises a social and cultural nature of the classroom, active roles of teacher and students, and learning ecologies and complexities of the community. Thus, it is opposed to an orientation of theory-testing that the positivist studies adopt. 5. The design experiment has been critically discussed in terms of the traditional scientific criteria like generalizability, reliability, replicability and others. 6. We can indicate four points as our tasks for enhancing the scientific qualities of the design experiment; ・Implementing consciously both processes from scholarly knowledge to teaching, and conversely from craft knowledge to researching and scholarly knowledge, ・Analyzing practical data in a systematic way and unfolding a logic of the analysis, ・Assessing and evaluating the design experiment using the revised scientific criteria, and ・Placing some philosophy which the design experiment is based on.

著者

荻原 文弘 両角 達男

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.22, no.2, pp.11-24, 2016-08-30 (Released:2019-01-17)

参考文献数

17

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The purpose of this research is to consider students’ mathematical ideas of how the mensuration formula is deduced and how these ideas are applied in the process of first revising how to deduce the area of a circle and then trying to deduce the volume of a sphere through the teaching unit of integral calculus.  In this unit students express the process of finding the volume of a sphere mathematically and interpret it by reviewing the process of obtaining the area of a circle.  Therefore we design and practice the unit of integral calculus in order to deduct mensuration formula for circle and sphere.  We consider typical student’s activities by the qualitative method in the teaching unit of integral calculus.  Then we clarify students’ mathematical ideas and how the students apply these ideas effectively. Students’ activities through classes can be summarized into four points.  First, they expressed the description of the arithmetic textbook mathematically, which helped them interpret the area formula of a circle more deeply and give new ideas for deducing the volume formula of a sphere.  Secondly, students interpreted the process of deducing the volume formula of a sphere by connecting mathematical ideas in reproducing the area formula of a circle with transition between two dimensions and three dimensions.  Third, an encounter with a circular argument made the students recognize that they should always bear precondition in mind.  At the same time, it gave them a good opportunity to explain about their ideas to others.  Finally, students attempted to apply the ideas of the area formula of a circle and the volume formula of a sphere which were produced in the previous stage by modifying and improving their ideas.

著者

二宮 裕之

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.23, no.2, pp.73-82, 2017-07-31 (Released:2019-09-09)

参考文献数

19

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Hirabayashi(2006) discusses a traditional aspect of mathematics education in Japan from the concepts of GEI(art), JUTSU(technique), and DO(way).  These are traditional Japanese cultural concepts and they seem to be the fundamental philosophy of Japanese people; however, it is not easy to define these concepts with words. We can point out, at least, that Japanese mathematics education surely has its own cultural aspect, and some parts seem to not easily be understood by foreign researchers, partially because these cultural aspects are not easily described in words, and no explicit definition of these cultural ideas exists.   Japanese mathematics teachers have accumulated and cultivated a lot of teaching abilities, which are surely existing in teaching practices but no theoretical framework to explain does not exist.  Ninomiya & Corey(2016) examined the framework of “Implicit Abilities of Teaching”.  In this paper, the framework of “Implicit Abilities of Teaching” and “Teaching Abilities Model”, in Ninomiya & Corey(2016), are used to examine Shido-an, or “Lesson Plans”, in mathematics education.  Reviewing the history of Shido-an in Japan, two major aspects about making Shido-an, which are “Formal-Explicit Aspect” and “Substantial-Implicit Aspect”, are found out.  “Formal-Explicit Aspect” of Shido-an making gives the information of what and how is the lesson, like Lesson Scenarios, whereas “Substantial-Implicit Aspect” of Shido-an making describes the results of the investigation for the lesson by teachers, and gives the focusing points for the discussion during Lesson Study.   Examining Japanese 9th grade mathematics lesson and its Shido-an about quadratic function, some results of discussion from “Substantial-Implicit Aspect” of Shido-an are found out.  Finally, it is found that Japanese teachers have cultivated not only “Explicit Abilities of Teaching” but also “Implicit Abilities of Teaching”, partially from “Substantial-Implicit Aspect” of Shido-an through the results of planning and investigating the lesson by teachers themselves.

著者

松島 充

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.19, no.2, pp.117-126, 2013-06-22 (Released:2019-01-17)

参考文献数

34

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The purpose of this research throws new light upon a connection between research of the Jigsaw Method on Mathematics Education and research of mathematical communication based on these comparisons. The greatest feature of Jigsaw Method on Mathematics Education is securing the opportunity of an argumentation to all the children. This is a foundation of the structural aspect of the Jigsaw Method on Mathematics Education as a group learning method. The foundation of this structural aspect has the feature of four more points. They are “a setup of two or more expert subjects”, “a setup of the integrative viewpoint of an argumentation”, “problem solving according to people two or more”, and “serious consideration of justification.” Two points are mentioned as a practical feature for a teacher to actually practice the Jigsaw Method on Mathematics Education. They are “support in an expert group”, and “ascertaining of group support.” As for these, both are concerned with momentary judgment within a lesson. This is the contents relevant to teacher education. In my general survey of the previous study of mathematical communication, it was arranged by four views which were aim of education, way of education, development of mathematical communication, and structure of mathematical communication. The research as aim of education was able to find out the common feature with the structural aspect of the Jigsaw Method on Mathematics Education. The research as way of education was able to point out the common feature with the Jigsaw Method on Mathematics Education about growth of the knowledge by an argumentation from the epistemological feature based on social constructivism. The research as development of mathematical communication pointed out that two points, a setup of the viewpoint of the integrative argumentation from the structural aspect and the practical feature, had a common feature. However, two points of the practical feature showed the necessity of making it concerned with a theory of teacher education. The research as the structure of mathematical communication pointed out that the epistemology in mathematics education is related with the Jigsaw Method on Mathematics Education. The next question of this research was pointed out four things. First of all, it is constructs the creation principle of the teaching materials that is based on the structural aspect of the Jigsaw Method on Mathematics Education. Secondly, it is constructs of the creation principle of the lesson that is based on the practical feature of the Jigsaw Method on Mathematics Education. Thirdly , the research on the relation of the practical feature of the Jigsaw Method on Mathematics Education and teacher education are needed to investigate. Finally, the new practice of the Jigsaw Method on Mathematics Education which utilized ICT are also needed to investigate.

著者

佐々木 徹郎

出版者

Japan Academic Society of Mathematics Education

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.11, pp.25-31, 2005 (Released:2019-01-17)

参考文献数

15

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数学の架空性は,数学が発展する上で,重要な役割を担っている。そのため,子どもが数学を学習するときに,理解できなくなる難所になることが多い。数学学習の過程において,どこでどのように架空性が生まれるのかを考察した。中学校1年生が「一次方程式」を学習する中で,「架空性」の問題が生じた事例を取り上げた。「創発モデル」の理論における「意味の連鎖」を用いて,それを分析した。その結果,「記母」が「記子」に結びつく過程,つまり記号化の過程の中で,架空性が生まれることがわかった。つまり,それぞれの記号化の中では,何らかの架空性が生じているのである。したがって,学習内容が現実的か架空的かは,本質的に個々人が,そのことを認識するかどうかに依存している。つまり,相対的なものである。また,架空性そのものが,必ずしも理解困難とは限らない。記号化が理解の助けとなることと同様に,数学的理解の助けになることもある。さらに,数学の学習において必ず現実的なモデルから始める必要はない。創発モデルは,個々の単元の中だけで,構想されるべきではない。数学の長期にわたる学習を全体論的に想定すべきである。

著者

真野 祐輔

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.22, no.2, pp.123-132, 2016-08-30 (Released:2019-01-17)

参考文献数

19

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The purpose of the study is to consider a framework of proof reading comprehension specific to mathematical induction and to illustrate levels and difficulties of mathematical induction by means of the framework.  There are two main ideas used in this study: the idea of “Mathematical Theorem” posed by Mariotti et al. (1997) and the theoretical model of proof reading comprehension formulated by Yang & Lin (2008) and Mejia-Ramos et al. (2012).  In this study, the author will combine these two theoretical ideas in order to consider a framework of proof reading comprehension specific to mathematical induction.  Data are collected through the nineteen undergraduate students’ writing responses to a set of “scripted  statements and proofs” and “scripted  dialogue”.  As a result, different difficulties are characterized in terms of the three levels: meaning of terms and statements (first level), logical status of statements and proof framework (second level), and justification of claims (third level), in the following ways.  A difficulty at first level can be seen as a presupposition that a given statement is always true.  One of the difficulties found at second level is related to a weak understanding of connections between the statement and its proof.  Lastly, at third level, there is a misunderstanding of the proof of implication statement.  Some implications for further researches and teaching practices are also discussed.

著者

石橋 一昴

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.22, no.2, pp.133-140, 2016-08-30 (Released:2019-01-17)

参考文献数

32

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The purpose of this paper is to obtain implications to develop curriculum on probability from the point of view of formation of the student’s concept of probability. Firstly, I referred to the duality of classical probability and frequentistic probability from Otaki (2011) and the relation of the concept of probability and that of randomness from Kawasaki (1990).  Then, I pointed out that, to form the concept of probability, the followings are necessary: “understanding both classical probability and frequentistic probability, and mutual connection between them” and “understanding of the concept of randomness preceding that of probability, and connection between them”. Secondly, I considered the current curriculum critically from the point of view of formation of the concept of probability, and found that the current curriculum have some problems in this point of view. Finally, I suggest “spiral curriculum” from this point of view based on “Principles in Setting and Arranging Teaching Contents” in “A Study on Constituent Principles of Curriculum in Mathematics Education” (Nakahara, 2008).  This paper shows that spiral curriculum is effective to form the student’s concept of probability and is required in the current secondary education.

著者

山本 文隆

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.19, no.1, pp.1-8, 2013

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<p> The area of the Pythagoras triangle is the sum of area of the Pythagoras triangle that is smaller than it except some exceptions. The exception is the case of M=2N (M,N is an independent variable of the solutions of Euclid).</p><p> Furthermore, these relations are expressed as the sequence and constructed in the Fibonacci series Next, the Pythagoras number is distributed on various parabolas group on the coordinate which assume two axes into two sides sandwiching the right angle. The degree of leaning of the axis of symmetry of the parabola group is 0 in case of the basic formula (Euclid solution) of the Pythagoras number. In addition, it is 0 and ∞ in case of "the unit formula"of sum of area. Furthermore, the axial degree of leaning converges to 2 at an early stage in case of "the general formula".</p>

著者

植田 敦三

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.11, pp.205-215, 2005 (Released:2019-01-17)

参考文献数

18

---

The purpose of this research is to clarify the way of the treatment of Sakumon in "life-centered arithmetic" in the early part of the Showa era. We pay attention to the arithmetic education on which Iwashita, Fujiwara and Inatsugu insist. They play a central role in the practice of "life-centered arithmetic". Through consideration, we find the following facts. (1) The chief aim of arithmetic education by Iwashita is to develop the qualitative thinking. To realize this aim, he introduces the practice to develop the qualitative thinking in daily life. For instance, gathering the qualitative facts, measurement are examples of this activity. Children pose the problem by the use of these facts or the result of measurement. Sakumon is a part of the practice to develop childeren's qualitative attitude in daily life. (2) Fujiwara restricts the position of Sakumon in his arithmetic education with the reflections that arithmetic education based on Sakumon is not able to preparete the curriculum. He changes Sakumon's position into one of methods of teaching arithmetic, namely the representation of the qualitative life. (3) Inatsugu attemptes to accord logicism with psychologism in arithmetic education. Generalization and specialization of mathematical thinking are the scaffold to accord them. Sakumon becomes the teaching and learning method to cultivate specialization. Simizu's study of curriculum development based on Sakumon has an attraction for him.

著者

岡崎 正和

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.13, pp.1-13, 2007

---

This paper focuses on a design experiment methodology in mathematics education which has been developed as a methodology for establishing a close and dynamic relationship between theory and practice, and discusses the comprehensive characteristics of the methodology. The design experiment methodology intends to develop (local) theories in mathematics education through designing, practicing and systematically analyzing daily classroom lessons over a relatively long period, where a researcher is responsible for students' mathematical learning in collaboration with a teacher. However, the methodology has also been questioned as to its scientific quality by the positivist scholars, since it explicitly deals with classroom practices that can be characterized as complex phenomena. Thus, this paper tries to place the design experiment methodology especially from a scientific point of view. The points discussed in this paper are the following. 1. The design experiment is an effective methodology for realizing mathematics education as a design science, and it intends to create a fruitful relationship between theory and practice. 2. The design experiment aims to construct a class of theories about the process of learning and the means that are designed to support that learning through (a) designing and planning the learning environments, (b) experimenting the design and the ongoing analysis, and (c) the retrospective analysis. 3. The design experiment is an interventionist methodology that goes through an iterative design process featuring cycles of invention and revision. 4. The design experiment has its intention of producing a theory which premises a social and cultural nature of the classroom, active roles of teacher and students, and learning ecologies and complexities of the community. Thus, it is opposed to an orientation of theory-testing that the positivist studies adopt. 5. The design experiment has been critically discussed in terms of the traditional scientific criteria like generalizability, reliability, replicability and others. 6. We can indicate four points as our tasks for enhancing the scientific qualities of the design experiment; ・Implementing consciously both processes from scholarly knowledge to teaching, and conversely from craft knowledge to researching and scholarly knowledge, ・Analyzing practical data in a systematic way and unfolding a logic of the analysis, ・Assessing and evaluating the design experiment using the revised scientific criteria, and ・Placing some philosophy which the design experiment is based on.

著者

秋田 美代 齋藤 昇

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.17, no.2, pp.55-63, 2011

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In this paper, we clarify the inhibitory factor of flexible idea in present mathematics instruction. And we propose the problems of making a student's flexible idea. We analyzed the inhibitory factor using an achievement test and a creativity test about the area of the figure. As a result, we found that the learners have functional fixedness in the mathematics problem solving. We call it "Temporary plateau of thinking in problem solving". We made two kinds of problems of breaking out of the "Temporary plateau of thinking in problem solving", problem which put restrictions on the solution and problem which required that many methods should be considered. We compared the usual problem with the problems of breaking out of the "Temporary plateau of thinking in problem solving". The results are as follows. - When compared with the usual problem, the problems of breaking out of the "Temporary plateau of thinking in problem solving" made many flexible ideas. - When compared with the usual problem, the problem which put restrictions on the solution was still difficult for students. - If the teacher gives appropriate teaching, the students can break out the "Temporary plateau of thinking in problem solving".

著者

山田 祐樹

出版者

全国数学教育学会

雑誌

数学教育学研究 : 全国数学教育学会誌 (ISSN:13412620)

巻号頁・発行日

vol.5, pp.69-75, 1999

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It is regarded as being problem that students are unable to use mathematical knowledge to the real world. My approach to this problem focuses on students' attitude toward using mathematical knowledge to the real world. Students tend to see the activity of word problem solving as a game that have nothing to do with the real world. It is necessary to bring the activities in the mathematics classroom close to the problem solving activities in the real world. One of the points that most of the problem solving activities in the mathematics classroom are different from the problem solving activity in the real world is the fact that a decision making isn't contained there. I believe that students will notice that the learning activities in the mathematics classroom closely parallel the problem solving activities in the real world, when a decision making is part of the activities in the mathematics classroom.