• The Mathematical Education
• /
• v.46 no.2 s.117
• /
• pp.173-192
• /
• 2007

We define mathematical learning as a process of overcoming framing difference of teachers and students, two main subjects in a mathematics class. We have reached this definition to the effect that we can grasp a mathematical classroom per so and understand students' mathematical learning in the context. We could clearly understand the process in which the framing differences are overcome by analyzing mutual negotiation of informants in specific cultural models, both in its form as well as in its meaning. We review both of the direct and indirect forms of negotiation while keeping track of 'evolution of subject' in terms of content of negotiation. More specifically, we discuss direct negotiation briefly and review indirect negotiation from three distinct themes of (1) argument structure, (2) revoicing, and (3) development patterns and narrative structure of proof. In addition, we describe the content of negotiation under the title of 'Evolution of Subject.' We found that major modes of mutual negotiation are inter-reference and appropriation while the product of continued negotiation is inter-resemblance.

• The Mathematical Education
• /
• v.48 no.2
• /
• pp.149-167
• /
• 2009

This study was aimed to examine problem-solving ability of fifth graders on two types of mathematical essay problems, and to analyze the process of mathematical justification in solving the essay problems. For this purpose, a total of 14 mathematical essay problems were developed, in which half of the items were single tasks and the other half were data-provided tasks. Sixteen students with higher academic achievements in mathematics and the Korean language were chosen, and were given to solve the mathematical essay problems individually. They then were asked to justify their solution methods in groups of 4 and to reach a consensus through negotiation among group members. Students were good at understanding the given single tasks but they often revealed lack of logical thinking and representation. They also tended to use everyday language rather than mathematical language in explaining their solution processes. Some students experienced difficulty in understanding the meaning of data in the essay problems. With regard to mathematical justification, students employed more internal justification by experience or mathematical logic than external justification by authority. Given this, this paper includes implications for teachers on how they need to teach mathematics in order to foster students' logical thinking and communication.

• Journal of Educational Research in Mathematics
• /
• v.17 no.4
• /
• pp.395-418
• /
• 2007

The purpose of this study was to analyze the influence of mathematical tasks on mathematical communication. Mathematical tasks were classified into four different levels according to cognitive demands, such as memorization, procedure, concept, and exploration. For this study, 24 students were selected from the 5th grade of an elementary school located in Seoul. They were randomly assigned into six groups to control the effects of extraneous variables on the main study. Mathematical tasks for this study were developed on the basis of cognitive demands and then two different tasks were randomly assigned to each group. Before the experiment began, students were trained for effective communication for two months. All the procedures of students' learning were videotaped and transcripted. Both quantitative and qualitative methods were applied to analyze the data. The findings of this study point out that the levels of mathematical tasks were positively correlated to students' participation in mathematical communication, meaning that tasks with higher cognitive demands tend to promote students' active participation in communication with inquiry-based questions. Secondly, the result of this study indicated that the level of students' mathematical justification was influenced by mathematical tasks. That is, the forms of justification changed toward mathematical logic from authorities such as textbooks or teachers according to the levels of tasks. Thirdly, it found out that tasks with higher cognitive demands promoted various negotiation processes. The results of this study implies that cognitively complex tasks should be offered in the classroom to promote students' active mathematical communication, various mathematical tasks and the diverse teaching models should be developed, and teacher education should be enhanced to improve teachers' awareness of mathematical tasks.

• Journal of Educational Research in Mathematics
• /
• v.13 no.4
• /
• pp.463-483
• /
• 2003

In the learning of mathematics, students experience the semiotic activities of representing and interpreting mathematical signs. We called these activities as the representing and interpreting of mathematical signs. On the foundation of Peirce's three elements of the sign, we analysed that students constructed the representamen to interpret the concept of correlation as for the object, "as one is taller, one's size of foot is larger" 4 middle school students who participated the gifted center in Seoul, arranged the statistical data, constructed their own representamen, and then learned the conventional signs as a result of the whole class discussion. In the process, students performed the detailed representing and interpreting of signs, depended on the templates of the known signs, and interpreted the process voluntarily. As the semiotic activities were taken place in this way, it was needed that mathematics teacher guided the representing and interpreting of mathematical signs so that the representation and the meaning of the sign were constructed each other, and that students endeavored to get the negotiation of the interpretants and the representamens, and to reach the conventional representing.