• The Mathematical Education
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• v.62 no.2
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• pp.257-268
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• 2023

The purpose of the study is to analyze the characteristics of open-ended tasks in terms of cognitive demands. For this purpose, we analyzed characteristics of open-ended tasks presented in the sequence units of three high school mathematics textbooks. The results of the study have revealed that low cognitive demand levels of open-ended tasks had characteristics including procedures within previous tasks or within those tasks. On the other hand, high cognitive demand levels of open-ended tasks had characteristics of actively exploring new conditions to gain access to what is being sought, requesting a basis for judgement, linking various representations to the concepts of sequences, or requiring a variety of answers. These results are significant in that they not only specified the characteristics of open-ended tasks with high cognitive demands in terms of the intended curriculum, but also provided a direction for the development of open-ended taks with high congitive demands.

• Research in Mathematical Education
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• v.22 no.1
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• pp.35-46
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• 2019

This study examines how open-ended tasks can be implemented with the support of redefined learning goals and teaching practices from a student-centered perspective. In order to apply open-ended tasks, learning goals should be adopted by individual student's cognitive levels in the classroom context rather than by designated goals from curriculum. Equitable opportunities to share children's mathematical ideas are also attainable through flexible management of lesson-time. Eventually, students can foster their meta-cognition in the process of abstraction of what they've learned through discussions facilitated by teachers. A pedagogical implication for professional development is that teachers need to improve additional teaching practices such as how to tailor tasks relevant to their classroom context and how to set norms for students to appreciate peer's mathematical ideas in the discussions.

• Communications of Mathematical Education
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• v.20 no.1 s.25
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• pp.117-145
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• 2006

Mathematically gifted children have creative curiosity about novel tasks deriving from their natural mathematical talents, aptitudes, intellectual abilities and creativities. More effect in nurturing the creative thinking found in brilliant children, letting them approach problem solving in various ways and make strategic attempts is needed. Given this perspective, it is desirable to select open-ended and atypical problems as a task for educational program for gifted children. In this paper, various types of open-ended problems were framed and based on these, teaming activities were adapted into gifted children's class. Then in the problem solving process, the characteristic of bright children's mathematical thinking ability and examples of problem solving strategies were analyzed so that suggestions about classes for bright children utilizing open-ended tasks at elementary schools could be achieved. For this, an open-ended task made of 24 inquiries was structured, the teaching procedure was made of three steps properly transforming Renzulli's Enrichment Triad Model, and 24 periods of classes were progressed according to the teaching plan. One period of class for each subcategories of mathematical thinking ability; ability of intuitional insight, systematizing information, space formation/visualization, mathematical abstraction, mathematical reasoning, and reflective thinking were chosen and analyzed regarding teaching, teaming process and products. Problem solving examples that could be anticipated through teaching and teaming process and products analysis, and creative problem solving examples were suggested, and suggestions about teaching bright children using open-ended tasks were deduced based on the analysis of the characteristic of tasks, role of the teacher, impartiality and probability of approaching through reflecting the classes. Through the case study of a mathematics class for bright children making use of open-ended tasks proved to satisfy the curiosity of the students, and was proved to be effective for providing and forming a habit of various mathematical thinking experiences by establishing atypical mathematical problem solving strategies. This study is meaningful in that it provided mathematically gifted children's problem solving procedures about open-ended problems and it made an attempt at concrete and practical case study about classes fur gifted children while most of studies on education for gifted children in this country focus on the studies on basic theories or quantitative studies.

• Education of Primary School Mathematics
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• v.25 no.1
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• pp.81-100
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• 2022

The purpose of this study is to develop a framework for analyzing the solution spaces of open-ended task and to explore their usefulness and applicability based on the analysis of solution spaces constructed by students. Based on literature reviews and previous studies, researchers developed a framework for analyzing solution spaces (OMR-framework) organized into subspaces of outcome spaces, method spaces, representation spaces which could be used in structurally analyzing students' solutions of open-ended tasks. In our research, we developed open-ended tasks which had various outcomes and methods that could be solved by using the concepts of factors and multiples and assigned the tasks to 181 elementary school fifth and sixth graders. As a result of analyzing the student's solution spaces by applying the OMR-framework, it was possible to systematically analyze the characteristics of students' understanding of the concept of factors and multiples and their approach to reversible and constructive thinking. In addition to formal mathematical representations, various informal representations constructed by students were also analyzed. It was revealed that each space(outcome, method, and representation) had a unique set of characteristics, but were closely interconnected to each other in the process. In conclusion, it can be said that method of analyzing solution spaces of open-ended tasks of this study are useful for systemizing and analyzing the solution spaces and are applicable to the analysis of the solutions of open-ended tasks.

• Research in Mathematical Education
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• v.14 no.1
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• pp.51-65
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• 2010

Open-ended problems can foster deeper understanding of mathematical ideas, generating creative thinking and communication in students. High-order thinking tasks such as open-ended problems involve more ambiguity and higher level of personal risks for students than they are normally exposed to in routine problems. To explore the classroom-based factors that could support or inhibit such higher-order processes, this paper also describes two cases of Singapore primary school teachers who have successfully or unsuccessfully implemented an open-ended problem in their mathematics lessons.

• The Mathematical Education
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• v.62 no.1
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• pp.57-78
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• 2023

Students' engagement in lessons not only determines the direction and result of the lessons, but also affects academic achievement and continuity of follow-up learning. In order to provide implications related to teaching strategies for encouraging students' engagement in elementary mathematics lessons, this study implemented lessons for middle-low achieving fifth graders using open-ended tasks and analyzed characteristics of students' engagement in the light of the framework descripors developed based on previous research. As a result of the analysis, the students showed behavioral engagement in voluntarily answering teacher's questions or enduring difficulties and performing tasks until the end, emotional engagement in actively expressing their pleasure by clapping, standing up and the feelings with regard to the topics of lessons and the tasks, cognitive engagement in using real-life examples or their prior knowledge to solve the tasks, and social engagement in helping friends, telling their ideas to others and asking for friends' opinions to create collaborative ideas. This result suggested that lessons using open-ended tasks could encourage elementary students' engagement. In addition, this research presented the potential significance of teacher's support and positive feedback to students' responses, teaching methods of group activities and discussions, strategies of presenting tasks such as the board game while implementing the lessons using open-ended tasks.

• The Mathematical Education
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• v.46 no.2 s.117
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• pp.207-226
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• 2007

The purpose of this study was to design and develop the processes of tasks and assessment rubrics of open-ended tasks, and those for the 5th graders of elementary school mathematics. 7 tasks were finally developed, and 'problem understanding', 'problem solving process', 'communication' were selected as the criteria for assessment rubrics. The result was that the ability of mathematical power covering problem understanding ability, problem solving ability and mathematical communication ability was low. Specifically, problem understanding ability was the highest, problem solving ability was middle, and mathematical communication ability was the lowest.

• Research in Mathematical Education
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• v.22 no.2
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• pp.123-134
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• 2019

In this paper, we articulate what is a lesson for all learners with different cognitive levels and what kind of teaching practices are required to implement this type of lesson. For all learners' own sense-making, open-ended tasks are the primary sources to bring their various mathematical ideas. These tasks can be meaningfully implemented by appropriate teaching practices: providing enough time (for thinking deeply and for preparing a reply), acting intentionally (alternative wrapping up activities and appointment of a struggling student), and cultivating collaborative classroom norms (respecting peer's thinking and learning from peers). This exploratory study has the potential to help practitioners and researchers understand the complexity of the work of teaching and clarify how to deal with such complexity.

• Research in Mathematical Education
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• v.5 no.2
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• pp.87-98
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• 2001

The Center for Science Gifted Education (CSGE) of Chongju National University of Education was established in 1998 with the financial support of the Korea. Science & Engineering Foundation (KOSEF). In fact, we had prepared mathematics and science gifted education program beginning in 1997. It was possible due to the commitment of faculty members with an interest in gifted education. Now we have 5 classes in Mathematics, two of which are fundamental, one of which is a strengthened second-grade class gifted elementary school students, and one a fundamental class, and one a strengthened class for gifted middle school students in Chungbuk province. Each class consists of 16 students selected by a rigorous examination and filtering process. Also we have a mentoring system for particularly gifted students in mathematics. We have a number of programs for Super-Saturday, Summer School, Winter School, and Mathematics and Science Gifted Camp. Each program is suitable for 90 or 180 minutes of class time. The types of tasks developed can be divided into experimental, group discussion, open-ended problem solving, and exposition and problem solving tasks. Levels of the tasks developed for talented elementary students in mathematics can be further divided into grade 5 and under, grade 6, and grade 7 and over. Types of the tasks developed can be divided into experimental, group discussion, open-ended problem solving, and exposition and problem solving task. Also levels of the tasks developed for talented elementary students in mathematics can be divided into the level of lower than grade 5, level of grade 6, and level of more than grade 7. Three tasks developed and practiced are reported in this article.

• The Mathematical Education
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• v.43 no.2
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• pp.163-175
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• 2004

We believe new assessment strategies and practices need to be developed that will enable teachers and others to assess students' performance in a manner that reflect the 7th Korean curriculum reform vision for school mathematics. This research was conducted to develop the assessment tasks based on the current literatures such as National Council of Teachers of Mathematics (1999) and Korea Institute of Curriculum & Evaluation(KICE, 2002, 2003) in quadratic functions of the secondary school and to find the effect of these tasks by classifying students' responses. The research instrument were composed of three criteria, the previous knowledge, the application of quadratic functions, and the general properties in functions. The research data were collected from 32 high school students in a suburb of Seoul and sorted by their similarities and differences in mathematical understanding. Through the research, we could know more than ever before about how the students learned mathematics and about how to improve teachers' mathematical instruction.