• The Mathematical Education
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• v.53 no.4
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• pp.525-539
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• 2014

In this study, we suggest implications for teaching mathematical justification with analysis of 6th grade students' awareness of why they needed mathematical justification and their levels of mathematics justification in Algebra and Geometry. Also how their levels of mathematical justification were related to mathematic achievement. 96% of students thought mathematical justification was needed, the reasons were limited for checking their solutions and answers. The level of mathematical justification in Algebra was higher than in Geometry. Students who had higher mathematic achievement had higher levels of mathematical justification. In conclusion, we searched the possibility of teaching mathematical justification to students, and we found some practical methods for teaching.

• Journal of Educational Research in Mathematics
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• v.9 no.1
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• pp.245-261
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• 1999

Proof is an essential characteristic of mathematics and as such should be a key component in mathematics education. But, teaching proof in school mathematics have been unsuccessful for many students. The traditional approach to proofs stresses formal logic and rigorous proof. Thus, most students have difficulties of the concept of proof and students' experiences with proof do not seem meaningful to them. However, different views of proof were asserted in the reassessment of the foundations of mathematics and the nature of mathematical truth. These different views of justification need to be reflected in demonstrative geometry classes. The purpose of this study is to characterize the types of students' justification in demonstrative geometry classes taught using dynamic software. The types of justification can be organized into three categories : empirical justification, deductive justification, and authoritarian justification. Empirical justification are based on evidence from examples, whereas deductive justification are based logical reasoning. If we assume that a strong understanding of demonstrative geometry is shown when empirical justification and deductive justification coexist and benefit from each other, then students' justification should not only some empirical basis but also use chains of deductive reasoning. Thus, interaction between empirical and deductive justification is important. Dynamic geometry software can be used to design the approach to justification that can be successful in moving students toward meaningful justification of ideas. Interactive geometry software can connect visual and empirical justification to higher levels of geometric justification with logical arguments in formal proof.

• East Asian mathematical journal
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• v.29 no.2
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• pp.109-135
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• 2013

The purpose of the study was to investigate the reality of middle school mathematics teachers' subject matter knowledge for teaching mathematical conjecture and justification. Data in the study were collected through interviewing nine Chinese and ten Korean middle school mathematics teachers. The teachers responded to the question that was designed in the form of a scenario that presents a teaching task related to a geometrical topic. The teachers' oral responses were audiotaped and transcribed, and their written notes were collected. The results of the study were compared to the analysis of American and Chinese elementary and secondary teachers' responses to the same task in Ball (1988) and Ma (1999). The findings of the study suggested that teachers' approaches to explaining and demonstrating a mathematical topic were significantly influenced by their knowledge of learners and knowledge of the curriculum they teach. One of the practical implications of the study is that teachers should recognize the advantages of learning the conceptual structure of a mathematical topic. It allows the teachers to have the flexibility to come up with meaningful mathematical approaches to teaching the topic, which are comprehensible to the learners whatever the grade levels they teach, rather than rule-based algorithms.

• Research in Mathematical Education
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• v.8 no.1
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• pp.1-18
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• 2004

Current interest in mathematics learning that focuses on understanding, mathematical reasoning and meaning making underscores the need to develop ways of analyzing classrooms that foster these types of learning. In this paper, the author show that the constructs of social and socio-mathematical norms, which grew out of taking a symbolic interactionist perspective, and Toulmins scheme for argumentation, as elaborated for mathematics education by Krummheuer [The ethnology of argumentation. In: The emergence of mathematical meaning: Interaction in classroom cultures (1995, pp. 229-269). Hillsdale, NJ: Erlbaum], provide us with means to analyze aspects of explanation, justification and argumentation in mathematics classrooms, including means through which they can be fostered. Examples from a variety of classrooms are used to clarify how these notions can inform instruction at all levels, from the elementary grades through university-level mathematics.

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

• Education of Primary School Mathematics
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• v.14 no.1
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• pp.13-26
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• 2011

The purpose of this research is to analyze geometrical level and the justification process in the proofs of construction by mathematically gifted elementary students. Justification is one of crucial aspect in geometry learning. However, justification is considered as a difficult domain in geometry due to overemphasizing deductive justification. Therefore, researchers used construction with which the students could reveal their justification processes. We also investigated geometrical thought of the mathematically gifted students based on van Hieles's Theory. We analyzed intellectual of the justification process in geometric construction by the mathematically gifted students. 18 mathematically gifted students showed their justification processes when they were explaining their mathematical reasoning in construction. Also, students used the GSP program in some lessons and at home and tested students' geometric levels using the van Hieles's theory. However, we used pencil and paper worksheets for the analyses. The findings show that the levels of van Hieles's geometric thinking of the most gifted students were on from 2 to 3. In the process of justification, they used cut and paste strategies and also used concrete numbers and recalled the previous learning experience. Most of them did not show original ideas of justification during their proofs. We need to use a more sophisticative tasks and approaches so that we can lead gifted students to produce a more creative thinking.

• Communications of Mathematical Education
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• v.18 no.1 s.18
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• pp.1-18
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• 2004

Current interest in mathematics learning that focuses on understanding, mathematical reasoning and meaning making underscores the need to develop ways of analyzing classrooms that foster these types of learning. In this paper, I show that the constructs of social and sociomathematical norms, which grew out of taking a symbolic interactionist perspective, and Toulmins scheme for argumentation as elaborated for mathematics education by Krummheuer, provide us with means to analyze aspects of explanation justification and argumentation in mathematics classrooms, including means through which they can be fostered. Examples from a variety of classrooms are used to clarify how these notions can inform instruction at all levels, from the elementary grades through university-level mathematics.

• The Mathematical Education
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• v.43 no.1
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• pp.97-107
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• 2004

Current interest in mathematics learning that focuses on understanding, mathematical reasoning and meaning making underscores the need to develop ways of analyzing classrooms that foster these types of learning. In this paper, I show that the constructs of social and sociomathematical norms, which grew out of taking a symbolic interactionist perspective, and Toulmin's scheme for argumentation, as elaborated for mathematics education by Kummheuer, provide us with means to analyze aspects of explanation, justification and argumentation in mathematics classrooms, including means through which they can be fostered. Examples from a variety of classrooms are used to clarify how these notions can inform instruction at all levels, from the elementary grades through university-level mathematics.

• Journal of Educational Research in Mathematics
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• v.16 no.1
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• pp.79-94
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• 2006

The purpose of this study is to find out the characteristics of the types(levels) of justification which are appeared by elementary mathematically gifted students in solving the tasks of plane division and spatial division. Selecting 10 fifth or sixth graders from 3 different groups in terms of mathematical capability and letting them generalize and justify some patterns. This study analyzed their responses and identified their differences in justification strategy. This study shows that mathematically gifted students apply different types of justification, such as inductive, generic or formal justification. Upper and lower groups lie in the different justification types(levels). And mathematically gifted children, especially in the upper group, have the strong desire to justify the rules which they discover, requiring a deductive thinking by themselves. They try to think both deductively and logically, and consider this kind of thought very significant.

• Journal of Educational Research in Mathematics
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• v.13 no.2
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• pp.159-178
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• 2003

This study analyzes on the reasoning in the process of justification and mathematical problem solving in our primary mathematics textbooks. In our analyses, we found that the inductive reasoning based on the paradima-tic example whose justification is founnded en a local deductive reasoning is the most important characteristics in our textbooks. We also found that some propositions on the properties of various quadrangles impose a deductive reasoning on primary students, which is very difficult to them. The inductive reasoning based on enumeration is used in a few cases, and analogies based on the similarity between the mathematical structures and the concrete materials are frequntly found. The exposition based en a paradigmatic example, which is the most important characteristics, have a problematic aspect that the level of reasoning is relatively low In Miyazaki's or Semadeni's respects. And some propositions on quadrangles is very difficult in Piagetian respects. As a result of our study, we propose that the level of reasoning in primary mathematics is leveled up by degrees, and the increasing levels are following: empirical justification on a paradigmatic example, construction of conjecture based on the example, examination on the various examples of the conjecture's validity, construction of schema on the generality, basic experiences for the relation of implication.