• The Mathematical Education
• /
• v.47 no.2
• /
• pp.221-224
• /
• 2008

The article shows that the concept 'weight' and the concept 'heaviness' give rise to different abstractions in the sense of Piaget and that these two concepts are differentiated by set-theoretic devices. The failure of differentiation of these two concepts 'weight' and the 'heaviness' can cause the failure of learning of the difference between reflective abstraction and empirical reflective abstraction. To explain the Piagetian abstrcation in a classroom, the author suggests to use the concept 'color' instead of the concept 'weigtht'.

• Journal of Educational Research in Mathematics
• /
• v.9 no.2
• /
• pp.383-404
• /
• 1999

This study began with an epistemological question about the nature of mathematical cognition in relation to the learner's activity. Therefore, by examining Piaget's 'reflective abstraction' theory which can be an answer to the question, we tried to get suggestions which can be given to the mathematical education in practice. 'Reflective abstraction' is formed through the coordination of the epistmmic subject's action while 'empirical abstraction' is formed by the characters of observable concrete object. The reason Piaget distinguished these two kinds of abstraction is that the foundation for the peculiar objectivity and inevitability can be taken from the coordination of the action which is shared by all the epistemic subjects. Moreover, because the mechanism of reflective abstraction, unlike empirical abstraction, does not construct a new operation by simply changing the result of the previous construction, but is forming re-construction which includes the structure previously constructed as a special case, the system which is developed by this mechanism is able to have reasonability constantly. The mechanism of the re-construction of the intellectual system through the reflective abstraction can be explained as continuous spiral alternance between the two complementary processes, 'reflechissement' and 'reflexion'; reflechissement is that the action moves to the higher level through the process of 'int riorisation' and 'thematisation'; reflexion is a process of 'equilibration'between the assimilation and the accomodation of the unbalance caused by the movement of the level. The operational learning principle of the theorists like Aebli who intended to embody Piaget's operational constructivism, attempts to explain the construction of the operation through 'internalization' of the action, but does not sufficiently emphasize the integration of the structure through the 'coordination' of the action and the ensuing discontinuous evolvement of learning level. Thus, based on the examination on the essential characteristic of the reflective abstraction and the mechanism, this study presents the principles of teaching and learning as following; $\circled1$ the principle of the operational interpretation of knowledge, $\circled2$ the principle of the structural interpretation of the operation, $\circled3$ the principle of int riorisation, $\circled4$ the principle of th matisation, $\circled5$ the principle of coordination, reflexion, and integration, $\circled6$ the principle of the discontinuous evolvement of learning level.

• The Mathematical Education
• /
• v.42 no.3
• /
• pp.303-325
• /
• 2003

This study sought to provide an explanation of university students' concept understanding on the infinity and infinite process and utilized a psychological constructivist perspective to examine the differences in transitions that students make from static concept of limit to actualized infinity stage in context of problems. Open-ended questions were used to gather data that were used to develop an explanation concerning student understanding. 47 university students answered individually and were asked to solve 16 tasks developed by Petty(1996). Microgenetic method with two cases from the expert-novice perspective were used to develop and substantiate an explanation regarding students' transitions from static concept of limit to actualized infinity stage. The protocols were analyzed to document student conceptions. Cifarelli(1988)'s levels of reflective abstraction and Robert(1982) and Sierpinska(1985)'s three-stage concept development model of infinity and infinite process provided a framework for this explanation. Students who completed a transition to actualized infinity operated higher levels of reflective abstraction than students who was unable to complete such a transition. Developing this ability was found to be critical in achieving about understanding the concept of infinity and infinite process.

• Journal of Educational Research in Mathematics
• /
• v.8 no.1
• /
• pp.299-312
• /
• 1998

In this study, we have investigated the meaning and mechanism of the 'construction' in the operational constructivism and the social constructivism. According to Piaget, a mathematical concept is the operational sch me, which is constructed through the reflective abstraction from a general coordination of activities and operations. The process of the reflective abstraction consists of 'reflechissement'and 'reflexion'. The reflechissement starting from 'intriorisation' concludes with 'thematisation', and the reflexion consists in the 'equilibration' of the result of reflechissement. The 'construction' in the social constructivism includes two process. One is the process from the individual, subjective knowledge of mathematics to the social, objective knowledge of mathematics, and the other is vice versa. The emphases is placed on the 'social interaction' and the 'representation' in this two processes. In this context, if we want to apply the social constructivism, we should clarify the meaning of 'society', and consider the difference between the society of mathematicians and the society of students.

• Journal of Educational Research in Mathematics
• /
• v.11 no.1
• /
• pp.89-102
• /
• 2001

Computer has been regarded as an alternative that could overcome the difficulties in the teaching and learning of mathematics. But the didactical problems of the computer-based environment for mathematics education could give us new obstacles. In this paper, first of all, we examined the application of the learning theories of mathematics to the computer environment. If the feedbacks of the computer are too immediate, students would have less opportunity to reflect on their thinking and focus their attention on the visual aspects, which leads to the simple abstraction rather than the reflective abstraction. We also examined some other Problems related to cognitive obstacle to learn the concepts of geometric figure and the geometric knowledge. Based on the analysis on the problems related to the computer-based environment of mathematics teaching and learning, we tried to find out the direction to use computer more adequately in teaching and learning geometry.

• The Journal of Korean Institute of Communications and Information Sciences
• /
• v.25 no.10B
• /
• pp.1675-1689
• /
• 2000

We present CReMeS a CORBA-compliant design and implementation of a new real-time communication service. It provides for efficient predictable and scalable communication between information producers and consumers. The CReMeS architecture is based on MidART's Real-Time Channel-based Reflective Memory (RT-CRM) abstraction. This architecture supports the separation of QoS specification between producer and consumer of data and employs a user-level scheduling scheme for communicating real-time tasks. These help us achieve end-to-end predictability and allows our service to scale. The CReMeS architecture provides a CORBA interface to applications and demands no changes to the ORB layer and the language mapping layer. Thus it can run on non real-time Off-The-Shelf ORBs enables applications on these ORBs to have scalable and end-to-end predictable asynchronous communication facility. In addition an application designer can select whether to use an out-of-band channel or the ORB GIOP/IIOP for data communication. This permits a trade-off between performance predictability and reliability. Experimental results demonstrate that our architecture can achieve better performance and predictability than a real-time implementation of the CORBA Even Service when the out-of-band channel is employed for data communication it delivers better predictability with comparable performance when the ORB GIOP/IIOP is used.

• School Mathematics
• /
• v.15 no.4
• /
• pp.785-799
• /
• 2013

The researchers developed the teaching-learning materials for 9th grade mathematically gifted students in terms of the hypothesis that the students would have opportunity for problem solving and develop various mathematical thinking through the mathematical modeling lessons. The researchers analyzed what mathematical thinking abilities were shown on each stage of modeling process through the application of the materials. Organization of information ability appears in the real-world exploratory stage. Intuition insight ability, spatialization/visualization ability, mathematical reasoning ability and reflective thinking ability appears in the pre-mathematical model development stage. Mathematical abstraction ability, spatialization/visualization ability, mathematical reasoning ability and reflective thinking ability appears in the mathematical model development stage. Generalization and application ability and reflective thinking ability appears in the model application stage. The developed modeling assignments have provided the opportunities for mathematically-gifted students' mathematical thinking ability to develop and expand.

• Journal of Educational Research in Mathematics
• /
• v.8 no.2
• /
• pp.485-494
• /
• 1998

This study is on the theory of Piaget's reflective abstraction and the mechanism of the development of knowledge and the history of algebra and its application to understand the difficulties that many students have in learning algebra. Piaget considers the development of knowledge as a linear process. The stages in the construction of different forms of knowledge are sequential and each stage begins with reorganization. The reorganization consists of the projection onto a higher level from the lower level and the reflection which reconstructs and reorganizes within a lager system that is transferred by profection. Piaget shows that the mechanisms mediating transitions from one historical period to the next are analogous to those mediating the transition from one psychogenetic stage to the next and characterizes the mechanism as the intra, inter, trans sequence. The historical development of algebra is characterized by three periods, which are intra inter, transoperational. The analysis of the history of algebra by the mechanism explains why the difficulties that students have in learning algebra occur and shows that the roles of teachers are important to help students to overcome the difficulties.

• School Mathematics
• /
• v.4 no.4
• /
• pp.643-655
• /
• 2002

In this paper we emphasize the introduction of ‘incommensurability’ on the teaching and learning the irrational number because we think of the origin of number as ‘ratio’. According to Greek classification of continuity as a ‘never ending’ divisibility, discrete number and continuous magnitude belong to another classes. That is, those components were dealt with respectively in category of arithmetic and that of geometry. But the comparison between magnitudes in terms of their ratios took the opportunity to relate ratios of magnitudes with numerical ratios. And at last Stevin coped with discrete and continuous quantity at the same time, using his instrumental decimal notation. We pay attention to the fact that Stevin constructed his number conception in reflecting the practice of measurement : He substituted ‘subdivision of units’ for ‘divisibility of quantities’. Number was the result of such a reflective abstraction. In other words, number was invented by regulation of measurement. Therefore, we suggest decimal representation from the point of measurement, considering the foregoing historical development of number. From the perspective that the conception of real number originated from measurement of ‘continuum’ and infinite decimals played a significant role in the ‘representation’ of measurement, decimal expression of real number should be introduced through contexts of measurement instead of being introduced as a result of algorithm.

• Journal of the Korean School Mathematics Society
• /
• v.12 no.1
• /
• pp.47-70
• /
• 2009

The purpose of this research was to confirm one of constructivists' assumptions that even children 조o are with low reasoning ability can make reflective abstracting ability and cognitive structures by this ability can make generation ability of new knowledge by themselves. To investigate the assumption, learner-centered instruction were implemented to 2nd grade classroom located in Suseong Gu, DaeGu City and with lesson plans which initially were developed by Burns and corrected by the researchers. Recordings videoed using 2 video cameras, observations, instructions, children's activity worksheets, instruction journals were analyzed using multiple tests for qualitative analysis. Some conclusions are drawn from the results. First, even children with low reasoning ability can construct mathematical knowledge on multiplication in their own. ways, Thus, teachers should not compel them to learn a learning lesson's goals which is demanded in traditional instruction, with having belief they have reasoning ability. Second, teachers need to have the perspectives of respects out of each child in their classroom and provide some materials which can provoke children's cognitive conflict and promote thinking with the recognition of effectiveness of learner-centered instruction. Third, students try to develop their ability of reflective and therefore establish cognitive structures such as webs, not isolated and fragmental ones.