• Journal of the Korean School Mathematics Society
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• v.14 no.4
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• pp.477-490
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• 2011

The concept of pedagogical content knowledge (PCK) has been developed and expanded to identify essential components of mathematical knowledge for teaching (MKT) by Ball and her colleagues (2008). This study proposes an alternative perspective to view MKT focusing on key developmental understandings (KDUs) that carry through an instructional sequence, that are foundational for learning other ideas. In this study we provide constructive components of KDUs in fraction multiplication by focusing on the constructs of 'three-level-of-units structure' and 'recursive partitioning operation'. Expecially, our participating preservice elementary teacher, Jane, demonstrated that recursive partitioning operations with her length model played a significant role as a KDU in fraction multiplication.

• The Mathematical Education
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• v.61 no.1
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• pp.1-28
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• 2022

The purpose of this study is to explore how students' fractional knowledge is related to their solving of proportion problems. To this end, 28 clinical interviews with four middle-grade students, each lasting about 30~50 minutes, were carried out from May 2021 to August 2021. The present study focuses on two 7th grade students who exhibited their ability to coordinate three levels of units prior to solving whole number problems. Although the students showed interiorization of three levels of units in solving whole number problems, how they coordinated three levels of units were different in solving proportion problems depending on whether the problems required reasoning with whole numbers or fractions. The students could coordinate three levels of units prior to solving the problems involving whole numbers, they coordinated three levels of units in activity for the problems involving fractions. In particular, the ways the two students employed partitioning operations and how they coordinated quantitative unit structures were different in solving proportion problems involving improper fractions. The study contributes to the field by adding empirical data corroborating the hypotheses that students' ability to transform one three levels of units structure into another one may not only be related to their interiorization of recursive partitioning operations, but it is an important foundation for their construction of splitting operations for composite units.

• Journal of Broadcast Engineering
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• v.27 no.1
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• pp.104-123
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• 2022

Style transfer based on neural network provides very high quality results by reflecting the high level structural characteristics of images, and thereby has recently attracted great attention. This paper deals with the problem of resolution limitation due to GPU memory in performing such neural style transfer. We can expect that the gradient operation for style transfer based on partial image, with the aid of the fixed size of receptive field, can produce the same result as the gradient operation using the entire image. Based on this idea, each component of the style transfer loss function is analyzed in this paper to obtain the necessary conditions for partitioning and padding, and to identify, among the information required for gradient calculation, the one that depends on the entire input. By structuring such information for using it as auxiliary constant input for partition-based gradient calculation, this paper develops a recursive algorithm for super high-resolution image style transfer. Since the proposed method performs style transfer by partitioning input image into the size that a GPU can handle, it can perform style transfer without the limit of the input image resolution accompanied by the GPU memory size. With the aid of such super high-resolution support, the proposed method can provide a unique style characteristics of detailed area which can only be appreciated in super high-resolution style transfer.