• The Mathematical Education
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• v.62 no.4
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• pp.457-471
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• 2023

Measurement is an imperative content area of early elementary mathematics, but it is reported that students' understanding of units in measurement situations is insufficient despite its importance. Therefore, this study examined lower-grade elementary students' quantitative reasoning of units in length measurement by identifying the levels of reasoning of units. For this purpose, we collected and analyzed the responses of second-grade elementary school students who engaged in a set of length measurement tasks using an open number line in terms of unitizing, iterating, and partitioning. As a result of the study, we categorized students' quantitative reasoning of unit levels into four levels: Iterating unit one, Iterating a given unit, Relating units, and Transforming units. The most prevalent level was Relating units, which is the level of recognizing relationships between units to measure length. Each level was illustrated with distinct features and examples of unit reasoning. Based on the results of this study, a personalized plan to the level of unit reasoning of students is required, and the need for additional guidance or the use of customized interventions for students with incomplete unit reasoning skills is necessary.

• The Mathematical Education
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• v.54 no.1
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• pp.65-81
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• 2015

The purpose of this study is to investigate the international trends of how the core competencies are reflected in mathematics curriculum, and to find the implications for the revision of Korean mathematics curriculum. For this purpose, the curriculum of the 9 countries including the U.S., Canada(Ontario), England, Australia, Poland, Singapore, China, Taiwan, and Hong Kong were thoroughly reviewed. It was found that a variety of core competencies were reflected in mathematics curricula in the 9 countries such as problem solving, reasoning, communication, mathematical knowledge and skills, selection and use of tools, critical thinking, connection, modelling, application of strategies, mathematical thinking, representation, creativity, utilization of information, and reflection etc. Especially the four most common core competencies (problem solving, reasoning, communication, and creativity) were further analyzed to identify their sub components. Consequently, it was recommended that new mathematics curriculum should consider reflecting various core competencies beyond problem solving, reasoning, and communication, and these core competencies are supposed to combine with mathematics contents to increase their feasibility. Finally considering the fact that software education is getting greater attention in the new curriculum, it is necessary to incorporate computational thinking into mathematics curriculum.

• The Mathematical Education
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• v.54 no.3
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• pp.207-222
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• 2015

Mathematics preservice teachers' Mathematical Content Knowledge ("MCK") includes not only knowledge for mathematics, but also academic knowledge for school mathematics and mathematical process knowledge. We can consider the items in PISA 2012 as suitable tools to assess process knowledge as well as mathematical content knowledge because these items are developed by competent international educational experts. Therefore, the responses to items with the low percentage of correct answers in conjunction with the mathematical contents were analyzed with focus on FMC. The results showed the reasoning competency in responses using the conditions of the problem and of understanding the conditions after reading the complex problems within the context (i.e. the reasoning and argumentation competency, and communication competency) requires improvements. Furthermore the results indicated the errors due to a lack of ability of devising strategies for problem solving. Based on the foregoing results, the implications towards the directions of the education for the preservice mathematics teachers have been derived.

• Journal of Educational Research in Mathematics
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• v.24 no.2
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• pp.103-124
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• 2014

The process of analogical reasoning can be conventionally summarized in five steps : Representation, Access, Mapping, Adaptation, Learning. The purpose of this study is to develop more detailed model for reason of analogies considering the distinct characteristics of the mathematical education based on the process of analogical reasoning which is already established. Ultimately, This model is designed to facilitate students to use analogical reasoning more productively. The process of developing model is divided into three steps. The frist step is to draft a hypothetical model by looking into historical example of Leonhard Euler(1707-1783), who was the great mathematician of any age and discovered mathematical knowledge through analogical reasoning. The second step is to modify and complement the model to reflect the characteristics of students' thinking response that proves and links analogically between the law of cosines and the Pythagorean theorem. The third and final step is to draw pedagogical implications from the analysis of the result of an experiment.

• Research in Mathematical Education
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• v.19 no.1
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• pp.61-79
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• 2015

This study discussed about how pseudo-thinking process actually occurs in the mind of the students, used Piaget's frame work of the assimilation and accommodation process. The data collection is conducted using Think-Out-Loud (TOL) method. The study reveals that pseudo thinking process of covariational reasoning occurs originally from incomplete assimilation, incomplete accommodation process or both. Based on this, three models of incomplete thinking structure constructions are established: (1) Deviated thinking structure, (2) Incomplete thinking structure on assimilation process, and (3) Incomplete thinking structure on accommodation process.

• Journal for History of Mathematics
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• v.22 no.1
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• pp.75-88
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• 2009

In this article it is investigated what role analysis play in the reasoning. The author suggests that the mathematical statements should be reformulated so that analysis can be activated in the reasoning.

• Research in Mathematical Education
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• v.18 no.3
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• pp.171-185
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• 2014

Mathematically deductive reasoning skill is one of the major learning objectives stated in senior secondary curriculum (CDC & HKEAA, 2007, page 15). Ironically, student performance during routine assessments on geometric reasoning, such as proving geometric propositions and justifying geometric properties, is far below teacher expectations. One might argue that this is caused by teachers' lack of relevant subject content knowledge. However, recent research findings have revealed that teachers' knowledge of teaching (e.g., Ball et al., 2009) and their deductive reasoning skills also play a crucial role in student learning. Prior to a comprehensive investigation on teacher competency, we use a case study to investigate teachers' knowledge competency on how to teach their students to mathematically argue that, for example, two triangles are congruent. Deductive reasoning skill is essential to geometry. The initial findings indicate that both subject and pedagogical content knowledge are essential for effectively teaching this challenging topic. We conclude our study by suggesting a method that teachers can use to further improve their teaching effectiveness.

• The Mathematical Education
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• v.53 no.2
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• pp.275-290
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• 2014

This study investigated the types of the 3D geometric thinking and spatial reasoning through the observation of the 2D representing activities for representing the 3D geometrical objects with preservice secondary mathematics teachers. For this purpose, the 43 sophomoric students in college of education were divided into 10 groups and observed their group task performance on the basis of the representation they used. Observed processes were all recorded and the participants were interviewed based on the task. As a result, the role of physical object that becoming the object of geometric thinking and spatial reasoning, and diverse strategies and phenomena of the process that representing the 3D geometric figures in 2D were discovered. Furthermore, these processes of representing were assumed to be influenced by experience and study practice of students, and various forms of representing process were also discovered in the process of small group activities.

• Journal of Educational Research in Mathematics
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• v.20 no.1
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• pp.57-71
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• 2010

Similarity between concept and concept, principle and principle, theory and theory is known as a strong motivation to mathematical knowledge construction. Metaphor and analogy are reasoning skills based on similarity. These two reasoning skills have been introduced as useful not only for mathematicians but also for students to make meaningful conjectures, by which mathematical knowledge is constructed. However, there has been lack of researches connecting the two reasoning skills. In particular, no research focused on the interplay between the two in didactic transposition. This study investigated the process of knowledge construction by metaphor and analogy and their roles in didactic transposition. In conclusion, three kinds of models using metaphor and analogy in didactic transposition were elaborated.

• The Mathematical Education
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• v.51 no.2
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• pp.161-171
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• 2012

Generally mathematics is regarded as a subtle subject to grasp their true meaning. And teacher's personal conceptions of mathematics influence greatly on the teaching and learning of mathematics. More over often teachers confess their difficulties in explaining the true nature of mathematics. In this paper, applying the theory of epistemology, we tried to search factors that must be counted important when trying to understand the true nature of mathematics. As results, we identified five characteristics of mathematical knowledge such as logical reasoning, abstractive concept, mathematical representation, systematical structure, and axiomatic validation. Next, we tried to investigate math education major students' conception of mathematics using these items. To proceed this research we asked 51 students from three Universities to answer their opinion on 'What do you think is mathematics?'. Analysing their answers in the light of the above five items, we got the following facts. 1. Only 38% of the students regarded mathematics as one of the five items, which can be considered to reveal students' low concern about the basic nature of mathematics. 2. The status of students' responses to the question were greatly different among the three Universities. This shows that mathematics professors need to lead students to have concern about the true nature of mathematics.