This study deals with the following three matters: 1) The development of a method of measuring the progress in test marks and of a method of entire evaluation treating the progress in study records. 2) The development of a method of a CAI problem-selection according to the progress in marks. 3) The effect of the learning by using a CAI learning system. The progress in marks generally is formulated for tests taken twice, but not for tests taken three or more times. First, I propose a method of measuring the progress in marks for tests taken three or more times by using fuzzy regression line. Futher, I propose an entire evaluation which means that the teacher adds the average marks to the extended marks accompanied with fuzziness. I call this Fuzzy N-Evaluation. In a CAI learning system the students may often be given the same problem or problems only according to the rate of right answers. But it is necessary to give each student suitable problems in order to encourage him or her for better understanding and further study. Therefore I offer a method of problem-selection in a CAI learning system which gives individual students the suitable problem by applying Fuzzy N-Evaluation. Here I discuss the threshold value that raises or lowers the problem-level according to the student's ability. I make actual CAI courseware adopting this facility and show that this CAI learning system is effective in order to advancing the learning of each student.

View full abstract

The aim of this paper is to verify, in connection with mathmatics education, that the teacher's authorative discourse privileged in the educational institutions during the Meiji Era. J. V. Wertsch considers that the style of authorative discourse depends on the point of view named "The Decontexturalized Voice of Rationality," which relates concrete instances with a generalized law, or a teacher's category of discourse derives from so called "sign type-sign type relationships." The most well known example of this is the "planning model of action." In this paper I touched briefly on the psychological aspect of this theory, however, the main focus was on the socio-historical interpretation, -especially the structural aspect of it-, because "a socio-historical approach to mind" starts from a hypothesis which mediated action can not separate from environment. That is, the teacher's characteristic authorative discourse was established with the modern teaching plans, mass lessons and examinations of the Meiji Era educational institutions. Finally, I suggested as "Bakhtin's Dialogic Principle" an alternative to authorative discourse for the research of mathmatics lessons.

View full abstract

The purpuse of this study is to construct the theory for making of mathematical classes. As the first step of it, the author established the tentative theory of it (Shiigi, 1996c). My tentative theory has a characteristic that students do the instructional activity with small group (in pairs). The purpose of this paper is to modify the tentative theory based on four teaching experiments, and reconstruct the theory. The points of modification by doing these teaching experiments are the following: (1) The instructional activity with small group is categorized by three kinds of problems; 1) problems that the end is open and multiple solutions exist, 2) problems that the end is open and two answers (yes or no) exist, 3) problems that an end is closed. So, students do the different instructional activity with small group by the three kinds of problems. (2) Teacher's roles are not only "presenting of the introductory problem", "helping of establishment of desirable social norms and sociomathematical norms", "transmitting of the social knowledge", but also "implicit orientation on students' social interaction".

View full abstract

The purpose of this study is to establish a theory for teaching of proof by constructivistic approach. Watanabe (1996a) theoretically considered a principle for it. This paper aims to verify the efficiency of this principle by means of a teaching experiment. As a result, it is valid to teach proof by constructivistic approach under the following conditions. 1) Teacher should basically play a role as a mediator or observer, and should also didactically interpose in student's discussion if required. 2) At an introduction of the unit of proof, the main purpose of class is that students can construct and understand the universalness and deduction of proof. Then, at a development stage, it is to construct and understand such an idea that there are various ways of proving. It is suggested that under these conditions, teaching of proof by constructivistic approach is feasible and efficient. This theory for teaching of proof by constructivistic approach enables students to construct and understand not only how to solve a proof problem but also the significance of proof.

View full abstract

The former some understanding models have the defect that we can't use them in daily teaching-learnig mathematics. For the poupose of removing this defect, Author establishes an Understanding Model as following. "Analogous Understanding" is the mental form that we are aware of understand B or the relation A and B through recognition of it that there is an analogous correspondence between A and B. "Procedural Understanding" is mental form be aware of understand B or the relation A and B through certain procedure. A: an objective representation of a well-known thing. B: an objective representation of a well-known or unknown thing. Students and pupils understand mathemasical concepts through the interaction between "Analogous Understanding" and "Procedural Understanding".

View full abstract

　　Many studies on mathematical problem solving focus on some activities such as "looking back", "metacognition", "self-assessment", "reflection" and so on. These activities have a point in common; that is "when problem solver engages himself/herself in the activity, he/she refers to his/her own processes or products of the problem-solving activity". But the names and conceptions of these activities are sometimes various and ambiguous. In this series of studies, we generally grasp these activities as "self-referential activities", and aim to elaborate on the conception, structure, and role of these self-referential activities during problem solving.
　　At the beginning of these studies, this article aims at constructing a framework for conceptualizing the self-referential activities systematically. First of all, we hypothesize self-referential architecture of human intelligence. This is the same concept of Skemp's reflective intelligence which is defined as the "ability to make one's own mental processes the object of conscious observation" (Skemp, 1979, p.175). This self-referential/reflective function of intelligence (we temporarily call it just "self-referential ability" or just "self-reference" in this article) could make the point in common among these self-referential activites. According to this hypothesis, we propose the framework for conceptualizing the self-referential activities as the following; i.e., each self-referential activity emerges, when this hypothesized self-referential mental function (self-referential ability) takes some variables.
　　The significant variables are drawn from preceding studies on "metacognition", "looking back", "self-assessment", "reflection" as follows.
　(1) Ongoing (OG)/Not-Ongoing (NOG): Metacogintion occurs in ongoing process of solving problems or cognitive tasks. And the ongoing process implies a phase of problem-solving behavior that is formed out of solver's intended goal.
　(2) Monitor -> Self-Evaluation (M-SE)/Self-Evaluation -> Control (SE-C): Metacognition emerges as the process "monitor -> self-evaluation -> control" and self-assessment is characterized by two components; i.e., self-awareness corresponding to the process "monitor -> self-evaluation" and self-evaluation corresponding to the process "self-evaluation -> control". They characterize both metacognitive process and self-asssessment.
　(3) Until-the-End-of-Problem-Solving (UEPS)/After-the-End-of-Problem-Solving (AEPS): Problem-solving activity is significantly distinguished between the process until production of an answer for the given problem and the process after production of it. So the point of the production of an answer for the given problem become a variable (as a divide in problem solving) of self-referential activities.
　(4) Process (PROC)/ Product (PROD): In our definition of self-referential activities, we grasp them as such activities that "problem solver refers to his/her own processes or products of the problem-solving activity". When one engages himself/herself in the self-referential activities, he/she ought to have objects that he/she refers to. Thus we draw the variable, process/product, as a distinction among the objects of self-reference.
　By these four variables, we can construct the framework for conceptualizing the self-referential activities as the following;

[figure]

　　In this article, the reasonability of this construction is shown by eight concrete self-referential activities (which is restricted to the variable "UEPS").

View full abstract

This article aims to elaborate a theoretical framework for describing mathematical problem-solving processes, which is based on Goldin's model of internal representational systems. To describe process of problem solving in terms of representation constructed by the solver, two points of view into which partition problem-solving process are prescribed. One is a phase of process as a unified representation that is constructed from the problem (statements) and correspond to a chunk of problem solving activity like "episode" in a sense of Schoenfeld's macroscopic analysis of problem-solving protocols (which is called "a cross section of problem-solving process" or "problem representation" in this article); another is the sequence of the phases (which is called "a longitudinal section of problem-solving process" in this article). Since a chunk of problem solving activity organize (correspond to) a problem representation, we can describe the sequence of a cross section of problem-solving process as sets of problem-solving activity (what kind of activity does the solver engage in?) and problem representation (what kind of representation does the solver deal with?). The basic theoretical framework constructed here is used for describing/discussing "what kinds of cross sections are constructed, what sequence of them is constructed, and how ?" in this article. This discussion make it clear that chain of problem representation (and its transfiguration) and problem-solving activity is one of the mechanism maintaining problem-solving processes, and that one of the source of generating problem-solving activity is a function of "representational system of planning, monitoring, and executive control" (i.e., "deciding the steps to be taken, or moves to be made, within all of the internal representational systems, including itself" (Goldin, 1992, p.250)). On that occasion, the conception of problem-solving strategy is extended (reconceptualized) as "goal-directed operations of activities employed to facilitate/progress problem solving" in order to make it the function of "representational system of planning, monitoring, and executive control" and a interface between problem-solving activity and problem representation.

View full abstract

Student self-assessment can provide information which is not available from other assessment techniques. For teachers this assessment is useful in collecting such information. For students it seems also useful in developing their own mathematical thinking, attitudes and metacognition. This paper focuses on development of student metacognition by self-assessment in which students evaluate their own cognitive activity after problem solving, which is called "reflective self-assessment" in this paper. The two aspects of metacognition are called metacognitive knowledge and metacognitive skill. By clarifying each aspect of metacognition I indicate both similarity and difference between reflective self-assessment and metacognition in problem solving, and assert that reflective self-assessment makes students conscious of their own metacognition.

View full abstract

　　The purpose of this study is to explore experimentally the role of metacognition in mathematical problem solving.
　　The present paper aims to examine the effectiveness of metacognitive supports. For this purpose, the auther gave metacognitive or cognitive supports to each pupil at 6th grade, when she/he stuck in the process of problem solving. In this study, metacognitive supports refer to the following speakings, for example "What are you thinking?", "What are you doing now?" And cognitive supports are to present pupils mathematical knowledge and skills.
In this study, the auther used the stimulated-recall technique to interview the pupils.
　　The main findings of this case study are the followings:
　　(1) Because of metacognitive supports appropriate to the ocation, these pupils could monitor their own processes. Then they made the subgoal clear and pupils could select the knowledge necessary to solve the problem.
　　(2) When to give metacognitive supports to each pupil, it was important to examine her/his knowledge, skill and metacognitive knowledge and to give metacognitive or cognitive supports appropriate to her/his ability.

View full abstract

According to Sfard, there are two different points of view, the "structural" conceptions and the "operational" conceptions, to the mathematical concept. When attempting to adapt this points of view to the algebraic expression, it regards as one object (result) about whether or not it regards a expression as the operation (the process). These points of view are same as "structural" and "the procedure" of Kieran. Also, it is synonymous with the concept, "procepto" of Gray & Tall, too. In this study, it considers about the difficulty of which a student is understood which switches over from the operational conceptions to the operational/structural conceptions and the limit of the operational conceptions and the necessity of the structural conceptions are shown. Next, I investigated the difficulty of the algebraic expression with two sides (the process/the result) of under-standing, for junior high school 3rd grader from the 1st grader. As a result, the following became clear. ・The difficulty of the algebraic expression with duality of understanding exists in any grade. ・The difficulty of the algebraic expression with duality of understanding decreases with the rise of the grade.

View full abstract

There are three meaning structure in addition and subtraction word problems combine, change and comparison. And addition and subtraction word problems are classified into 14 types, considering meaning structures and the positions of unknown parts. According to some precede studies, we can make 4 levels of difficulty in the 14 types. The author researched the difficulty in solving addition and subtraction word problems. Research problems consist in three classes in respect of situation : caramel, water and number problems. Each of three classes have four problems from Level 1 to Level 4. The results are following. ・In caramel situation, Level 3 problem is significantly more difficult for grade 2 and 3 students to solve than Level 4 problem is. ・In water situation, Level 3 problem is more difficult than Level 4 problem, but there is no statistic significance between them in grade 2 and 3. ・In number situation, Level 4 problem is more difficult than Level 3 problem. There is a statistic significance in grade 3, but not in grade 2.

View full abstract

It is well known that many children confuse the quantitative aspect of fractions with the operational aspect of them. In this paper, three instructions for fourth and fifth graders were investigated and analyzed. The purpose of the instructions was to separate the aspects and to (re)construct the quantitative aspect of fractions. In one of the classes of fourth grade, though some children asserted that one half meter was the half of a meter, many children insisted that the half of ten meter was called not only five meter but also one half meter, so one half plus one half was one but one half meter plus one half meter was not always one meter. Since the discussion was continued without reaching an agreement, the teacher instructed the difference between one half and one half meter. In the class of five grade, children could reach smoothly an agreement of what one half meter was. The results of investigation carried on after the first instruction, however, showed that the figurative expression of liquid with using fractions was relatively poor. The confusion of the aspects of fractions is stable and it seems not possible for all children to separate them clearly within one hour instruction. Since four fundamental rules of arithmetic on fractions is taught with using the model of quantity, we must encourage children to separate them at every opportunity.

View full abstract

In this research we approach to the understanding processes in generalization of mathematical concepts based on the "Optimizing Equilibration Model of Mathematical Understanding". In this paper, in particular, we will clarify the processes through which children understand inclusion relations between quadrilaterals and generalize their schema of parallelogram. Difficulties in understanding inclusion relations are not consisted in mathematical logic, but in such children's tacit property that each angle of rectangle is not 90°. We have discussed means that they overcome the difficulties and understand inclusion relations, and indicated that the following means are effective. 1. conceiving geometrical figures dynamically by manipulating on operative material and seeing continual transformations from parallelogram to rectangle or rhombus 2. understanding conservation of properties in the transformations 3. understanding negative properties in the transformations 4. analogy to the inclusion relation between parallelogram and rhombus 5. analogy to the inclusion relation between rhombus and square We also clarified that after understanding the content, children's schema of parallelogram becomes to be in the states of the followings. 1. Properties of a geometrical figure are organized, and children can define a geometrical figure by using those properties. 2. Children conceive properties of a geometrical figure as common properties between geometrical figures. 3. A geometrical figure is a bearer of their properties. These states are effective for children to learn proof in lower secondary school. Therefore, children should learn inclusion relations between quadrilaterals in primary school.

View full abstract

What we wish to show in this paper is to consider the difference between valuableness in algebraic symbols and that in geometrical figures in terms of the perspective of symbols as objects and the comparison between two teaching practices. Consideration on symbols as objects in algebraic signs was already reported referring to the class-analysis of learning through problem "Numbers on the Calendar" (1995, Iwasaki & Tagashira). This paper, therefore, is intended as a consideration on symbols as objects in geometrical figures referring to the class-analysis of learning through problem "The Sum of Five Angles in Pentagram". Development of geometrical figures as valuable corresponds to the semiosis from a drawing to form in geometry learning. The process of this symbols as objects is articulated as follows like problem, background, cognitive analysis, method and consideration, and conclusion in this order: (1) Problem It was almost impossible to identify geometrical figures as valuable "N" in symbols as objects concerning geometrical teaching clearly although it had been easy to specify algebraic symbols as valuable "n" in symbols as objects concerning algebra teaching. (2) Background of the Problem According to Thom, R., Euclidean geometry is a natural intermediate stage between common language and algebraic language (1973, p.207). Moreover according to Skemp,R.R., geometrical symbols and algebraic language contrast radically in mathematical representation. Natural language, therefore, has much more influence on geometrical figures than algebraic symbols. In Japanese, there are not distinction on singular or plural form of noun, and articles like as "the" and "a" either, which are deeply concerned with the establishment of logical quantifier. (3) Cognitive Analysis of Problem Students variously explained the reason on the proposition of the sum of five angles in pentagram in the class. Their explanations could be categorized four types. They are related to symbols as objects and proper to geometry learning. These four patterns are as follows: (1) by measurement (2) inductive explanation by using a demonstrative pronoun like as "this" and "that" (3) by expansion of special case like as regular pentagram (4) deductive explanation by using alphabetical symbols like as A,B,C,D, and E (4) Method and Consideration Semiotic trichotomy in modes of representation by Peirce corresponds to change of recognition as follows: (1) Icon as semiotic firstness is equivalent to an object of observation. A geometrical figure in measurement is a kind of icon. It, therefore, is undifferentiated from an object in thinking. This means a way of thinking depended on a concrete object. (2) Index as semiotic secondness is equivalent to the development of private schema. A geometrical figure referred with a demonstrative pronoun are a kind of index. It, therefore, is considered as representation of private schema. This means the possibility of inductive thinking. (3) Symbol as semiotic thirdness is equivalent to the development of collective schema. A geometrical figure referred with alphabetical signs are a kind of symbol. It, therefore, is considered as representation of collective schema. This means the deductive thinking rather than the extension of objects. (4) A geometrical figure in expansion of special case could be metonymy for that as a symbol. It is characterized as a transitional representation from private schema to collective schema. This implies the change of thinking way. (5) Conclusion It follows from what has been said in (1), (2), (3), and (4) that symbols as objects in geometry learning would be the fundamental base for generalization of method in thinking from induction to deduction. This is critically different from the generalization in algebra learning since symbols as objects there become the base for generalization of objects in thinking.

View full abstract

The performance of geometric proof problems were analyzed using a think aloud procedure. Ten ninth graders participated in the investigation, five students were high achievers in mathematics and other five were low achievers. Their views of proof were also investigated. The results of protocol analysis showed that high achievers generated more information and more frequently made use of it. Especially they frequently reread the problem sentences. Geometric proof problem is performed based on a figure attached to the problem. The figure includes not only premise but also conclusion of the problem. Since successful performance of the proof needs to separate the premise, conclusion and other terms generated during the performance of the proof, it is required to reread the problem sentence so as to monitor her/his own process of the performance. The high achievers were to show superior ability to monitor. They regarded the purpose of proof as the verification of inference and anticipation, and the means to develop their logical thinking ability. On the other hand the low achievers regarded it as the explanation and the indication of their understanding to others. According to these findings, some issues of teaching proof are discussed.

View full abstract

"Journal Writing" is one of the methods to use writing in mathematics classes, and it has two functions; "(1) Expressive function", and "(2) Transactional function". In this paper, "(2) Transactional function" is focused on, and the case described in Ninomiya (1996 (c)) is investigated. First of all, "the framework for the investigation of Transactional Writing", which has four stages; impression, summary(1), summary(2), consideration, is established from the framework of Nakamura (1989). Since the writing activities are taken place during the second term of 1996, the difference of the paper tests' results between the first and the second terms should be considered as to be caused by the writing activities. Every student is over the national average during the second term while more than one-fourth of the students were below national average during the first term. Furthermore, the students who are not good at math improves their score a lot, which means the advantages of writing activities work very effectively. It also means that writing activities makes "the education for every student" policy possible. Writings which have written by students are analysed in two ways; by the class teacher and from "the framework for the investigation of Transactional Writing". The correlation of these two analysis' results is 0.846. Since the analysis by class teacher is reliable, the reliability of the framework is suggested. The correlation between writing and mathematical ability is also high. This means that mathematical writing is defferent from writings in language classes. Moreover, mathematical writing ability is correlated with higher-order abilities such as an attitude or way of thinking, a lot. From such outcome, mathematical writing ability should be considered as one of the higher-order abilities, which is stressed in the current curriculum for school mathematics.

View full abstract

The purpose of this study is to investigate and compare the way of teaching and learning mathematics in Japan with that in Brazil. The aim of the investigation by questionnaire is to make clear the similarity and difference of the activity of mathematics teachers in lower secondary school as well. The questionnaire consists of four categories as follows: (1) mathematics class, (2) teaching and learning mathematics, (3) evaluation on students, (4)problems and issues to be improved in mathematics education. I analyzed the above four categories by comparing results of the questionnaire with that reported in literatures. The main results are following: (1) Brazilian teachers almost always make at least one student fail. (2) While Brazilian teachers emphasize the relationship between mathematics and daily life, Japanese teachers think it is important to develop students' ability of logical thinking. (3) Most Japanese teachers feel difficulty in improving students' mathematical ability, however most Brazilian teachers feel difficulty in the lack of equipments for teaching mathematics.

View full abstract

The purpose of this study is to make an investigation into pupils' progress of mathematical ability at secondary school level, appreciate the factor that enhance or hinder mathematics teaching, and make recommendations for the improvement of mathematics teaching in Japan. In this paper, we review the results of Year 1 Potential Test and Number Test developed by KassEx Project, and make a comparison between Japan, England, Scotland and Germany. The main results are following. (1) The score of the Potential Test differed little between Japan and other countries. It should be noted that the difference of the substance and teaching method of each countries have no effect on the result of the potential test problem. (2) Japanese pupils performed significantly better in the Number Test than England, Scotland, and Germany. This result is indicative of the effectiveness of mathematics education in Japan. (3) Through an analysis of the score of the Number Test, we find that the score of the area of decimal number, fraction, and percentage which was learned in elementary school is poor contrary to our expectation. In the future, we need to make a program which improve our students ability to use the basic concepts and skills of those areas.

View full abstract

The purpose of this study is to investigate pupil' progress of mathematical ability at lower secondary school level and consider the suggestion for improving our mathematics education, by means of using the problems developed by Kassel-Exeter Project. We investigated the pupils' progress (or retrogradation) concerning "Number" test which is composed of fifty problems by examining the same pupils in Fukuoka prefecture an year later. Our pupils' progress is almost same as the progress of pupils in England, Scotland and Germany. But, since our pupils' "Number" test point is comparatively high, our pupils' progress at such a higher level can be regarded as a result of our effective teaching of mathematics. According to the longitudinal investigation about the points of each problem, we couldn't find a remarkable progress of points concerning the area of "estimation", "proportion and percentage" and "problem solving". Moreover, we divided our pupils into three groups (PH, PM, PL) by means of their points of "Potential" test. As a general finding, we can point out that the PH pupils' progress is due to the success concerning comparatively difficult problems and that the PL pupils' progress is due to the success concerning comparatively easy problems. At the same time, we could find that their retrogradation is also due to the failure concerning the same kind of problems. Such a valuable suggestion gained is this study can be considered as an important point of improving our mathematics education.

View full abstract

We need a lot of talented students in mathematics in the future society. They don't want to learn mathematics. So there are few women who study physical science. This percentage is extremely low in comparison with foreign countries. From this thing I can imagin that there are a lot of women who have the mathematical ability in Japan. However, it isn't possible to develop their ability for the internal and external factors which surround the talented woman students. Then, how will we be able to develop their talent? There are very important that home environment of the equal man and woman intention, thinking concretely their future occupation, the existence of mentor who encourage them in the physical science and the models who are active in the physical science, and so on. I want to make three proposals to the social organization. The first is to increase the women's teachers of mathematics. It is because they it become girls' students model near by. The second is to organize the science and mathematics class only by the woman party. It is because the women avoid the internal factor (Hounar effect and so on). The last is to make the social environment in which women can enjoy the mathematics learning. It is possible to expect that more women learn mathematics and then that a lot of talented students grow up.

View full abstract

The aim of this study is to clarify the professionaliam of teachers in mathematics teaching. In this paper, as the first step for it, I analyzed the studies of Leinhardt (1988, 1989) that compared the lessons of experts and novices, connecting them with STANDARDS of NCTM (1991). And I confirmed the relevancy between experts-novices and professionalism that experts were more professional than novices. And after that, with the study of Casle (1994), I suggested that the experience was considered as one of elements of professional development but it must be based on the autonomous endevor of teachers who seted their own heart on professional development.

View full abstract

In learning of trigonometric ratios in high school mathematics "Mathematics I", many text books define at first a tangent, next a sine and cosine. A reason to define in this order would come from an educational care since students are familiar with a notion of tangent as a slope, for example, in their daily life. However, in the process of the study, the sine and cosine play a central role in trigonometric ratios and functions. For example, the Sine Theorem and the Cosine Theorem are the important topics in "Mathematics I", and in fact the tangent is simply expressed as the ratio of sine to cosine. Then, we can propose a problem: To consider trigonometric ratios and functions from a viewpoint of tangent. We study this problem by using an intersection of a line through the origin and a line x=k in the orthogonal coordinate plane. We show Tangent Addition Theorem without using Sine and Cosine Addition Theorems. A derivative of tangent function tanx is obtained as 1+ tan^2x by using the Tangent Addition Theorem. A geometric interpretation is given for the property: an integration of a rational function of the sine and cosine functions of x is reduced to an integration of rational function of t by a substitution tan(x/2)=t. As a result, we see that some properties of trigonometric ratios and functions are naturally obtained from a viewpoint of tangent or line x=k.

View full abstract