• Management Science and Financial Engineering
• /
• v.17 no.2
• /
• pp.63-84
• /
• 2011

The ordinary multiple criteria decision making (MCDM) approach requires two types of input, alternative values and criterion weights, and employs two schemes of alternative prioritization, dominance and potential optimality. This paper allows for incomplete information on both types of input and gives rise to the dominance relationships and potential optimality of alternatives. Unlike the earlier studies, we emphasize that incomplete information frequently takes the form of strict inequalities, such as strict orders and strict bounds, rather than weak inequalities. Then the issues of rising importance include: (1) The standard mathematical programming approach to prioritize alternatives cannot be used directly, because the feasible region for the permissible decision parameters becomes an open set. (2) We show that the earlier methods replacing the strict inequalities with weak ones, by employing a small positive number or zeroes, which closes the feasible set, may cause a serious problem and yield unacceptable prioritization results. Therefore, we address these important issues and develop a useful and simple method, without selecting any small value for the strict preference information. Given strict information on both types of decision parameters, we first construct a nonlinear program, transform it into a linear programming equivalent, and finally solve it via a two-stage method. An application is also demonstrated herein.

• The Mathematical Education
• /
• v.59 no.3
• /
• pp.217-235
• /
• 2020

This study examined Korean fourth-grade students' performance by gender on the Trends in International Mathematics and Science Study(TIMSS) 2011 and 2015 mathematics assessment. We first identified items which had significantly higher mean scores by gender to decide which gender did better on a certain domain(domain-level analysis). Then, we examined the content of items(item-level analysis) to understand which items lead to gender differences in mathematics achievement. Our findings showed that about 80% of the items on both assessments did not show statistically significant differences between males and females. However, there were meaningful gender differences in the other 20% items. On both assessments, females had more items with significantly higher mean scores than males on the Shapes domain, and males had more those items on the Numbers and Measurement domains and all cognitive domains(Knowing, Applying, and Reasoning). In particular, females outperformed males on items related to identifying two- and three-dimensional shapes and drawing lines and angles and identifying them. Conversely, males had higher performance than females on items related to the pre-algebraic thinking, fractions and decimals, estimation of number differences, unit of length, and measuring time, height, and volume. The effect sizes for each item ranged from .12 to .33 and the mean effect size of all items across both assessments was .20, which indicated significant gender differences but small.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.3
• /
• pp.529-545
• /
• 2022

This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring R is called right CIFD if R/I is right duo by some proper ideal I of R such that I is contained in the center of R. We first see that this property is seated between right duo and right π-duo, and not left-right symmetric. We prove, for a right CIFD ring R, that W(R) coincides with the set of all nilpotent elements of R; that R/P is a right duo domain for every minimal prime ideal P of R; that R/W(R) is strongly right bounded; and that every prime ideal of R is maximal if and only if R/W(R) is strongly regular, where W(R) is the Wedderburn radical of R. It is also proved that a ring R is commutative if and only if D₃(R) is right CIFD, where D₃(R) is the ring of 3 by 3 upper triangular matrices over R whose diagonals are equal. Furthermore, we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring R is right CIFD if and only if R/I is commutative by a proper ideal I of R contained in the center of R.

• Communications of Mathematical Education
• /
• v.37 no.1
• /
• pp.65-84
• /
• 2023

The method of Lagrange multipliers, one of the most fundamental algorithms for solving equality constrained optimization problems, has been widely used in basic mathematics for artificial intelligence (AI), linear algebra, optimization theory, and control theory. This method is an important tool that connects calculus and linear algebra. It is actively used in artificial intelligence algorithms including principal component analysis (PCA). Therefore, it is desired that instructors motivate students who first encounter this method in college calculus. In this paper, we provide an integrated perspective for instructors to teach the method of Lagrange multipliers effectively. First, we provide visualization materials and Python-based code, helping to understand the principle of this method. Second, we give a full explanation on the relation between Lagrange multiplier and eigenvalues of a matrix. Third, we give the proof of the first-order optimality condition, which is a fundamental of the method of Lagrange multipliers, and briefly introduce the generalized version of it in optimization. Finally, we give an example of PCA analysis on a real data. These materials can be utilized in class for teaching of the method of Lagrange multipliers.

• KSII Transactions on Internet and Information Systems (TIIS)
• /
• v.12 no.1
• /
• pp.476-496
• /
• 2018

With the advancement and deployment of wireless communication techniques, wireless body area network (WBAN) has emerged as a promising approach for e-healthcare that collects the data of vital body parameters and movements for sensing and communicating wearable or implantable healthful related information. In order to avoid any possible rancorous attacks and resource abuse, employing lightweight ciphers is most effective to implement encryption, decryption, message authentication and digital signature for security of WBAN. As a typical lightweight cryptosystem with an extended sponge function framework, the PHOTON family is flexible to provide security for the RFID and other highly-constrained devices. In this paper, we propose a differential fault analysis to break three flavors of the PHOTON family successfully. The mathematical analysis and simulating experimental results show that 33, 69 and 86 random faults in average are required to recover each message input for PHOTON-80/20/16, PHOTON-160/36/36 and PHOTON-224/32/32, respectively. It is the first result of breaking PHOTON with the differential fault analysis. It provides a new reference for the security analysis of the same structure of the lightweight hash functions in the WBAN.

• Communications of Mathematical Education
• /
• v.29 no.3
• /
• pp.465-489
• /
• 2015

The purpose of this study is to suggest implications for the development and revision of future teaching materials for mathematically gifted students by using network text analysis of secondary mathematics materials. Subjects of the analysis were learning goals of 110 teaching materials in a gifted education center in science attached to a university from 2002 to 2014. In analysing the frequency of the texts that appeared in the learning goals, key words were selected. A co-occurrence matrix of the key words was established, and a basic information of network, centrality, centralization, component, and k-core were deducted. For the analysis, KrKwic, KrTitle, and NetMiner4.0 programs were used, respectively. The results of this study were as follows. First, there was a pivot of the network formed with core hubs including 'diversity', 'understanding' 'concept' 'method', 'application', 'connection' 'problem solving', 'basic', 'real life', and 'thinking ability' in the whole network from 2002 to 2014. In addition, knowledge aspects were well reflected in teaching materials based on the centralization analysis. Second, network text analysis based on the three periods of the Mater Plan for the promotion of gifted education was conducted. As a result, a network was built up with 'understanding', and there were strong ties among 'question', 'answer', and 'problem solving' regardless of the periods. On the contrary, the centrality analysis showed that 'communication', 'discovery', and 'proof' only appeared in the first, second, and third period of Master Plan, respectively. Therefore, the results of this study suggest that affective aspects and activities with high cognitive process should be accompanied, and learning goals' mannerism and ahistoricism be prevented in developing and revising teaching materials.

• The Mathematical Education
• /
• v.59 no.4
• /
• pp.357-371
• /
• 2020

In the 2015 revised math curriculum, core competencies have been embodied and presented as skills and achievement standards. In this study, I analyzed aspects of the link between competencies, skills and achievement standards in the 2015 revised mathematics curriculum for elementary schools. According to the study, six mathematics curriculum competencies were presented evenly as 'skills' in each content area of elementary school, but reflected some of the sub-components of the curriculum, and some of them were presented as 'skills' but not as 'achievement standards'. In addition, the types of skills reflected in the achievement standards varied greatly by content area, but a few of specific skills such as 'understand' were found to be highly emphasized. Based on this, several implications were derived to further improve the implementation of competencies. First, 'skill' should be presented in a more systematic way and with more validity of extraction. Second, the extent to which competencies are presented in the achievement standards should be discussed. Third, Mathematics skills should be presented differently by grade(cluster) in achievement standards, 'Guidelines for Teaching and Learning' and 'Guidelines for Assesment'. Fourth, competencies related to content shall be presented separately and in detail.

• The Mathematical Education
• /
• v.55 no.3
• /
• pp.357-382
• /
• 2016

The purpose of this study was to report the effects of self-directed and self-regulated learning programs on elementary, middle, and high school students through meta-analysis of previous studies. For this research, 22 of previous studies were selected which were all conducted in the country, and calculated the effect size of 'standardized change of the mean difference' for many factors included in each research. The findings were as follows: first, the overall effect sizes of self-directed and self-regulated learning programs on elementary, middle, and high school students were .665 and .702 in the affective and cognitive domain, respectively, meaning that the self-directed and self-regulated learning programs had average or greater effects on elementary, middle, and high school students and exerted somewhat greater effects in the cognitive domain. Second, when the areas of moderating effects were divided into self-directed and self-regulated learning, the former and latter had more influences on the cognitive and affective domains, respectively. Third, the elementary school level recorded a larger effect size both in the affective and cognitive domains than the secondary school level. Fourth, the findings show that the characteristics of affective domain, "reflective thinking" and "self-confidence," recorded a very large effect size both at the elementary and secondary school levels. Finally, the programs were more effective when the application period was one to four weeks in the affective domain and more than four weeks in the cognitive domain. And, Significance and implications of this research were discussed.

• The Mathematical Education
• /
• v.45 no.2 s.113
• /
• pp.191-204
• /
• 2006

Computer technology has a potential to change the contents of school mathematics and the way of teaching mathematics. But in our country, the problem whether computer technology should be introduced into mathematics classroom or not was not resolved yet. As a tool, computer technology can be used by teachers who are confident of the effectiveness and who can use it skillfully and can help students to understand mathematics. The purpose of this study was to investigate the effective way to introduce and utilize computer technology based on the status quo of mathematics classroom setting. One possible way to utilize computer technology in mathematics classroom in spite of the lack of computer and the inaccessibility of useful software is using domain specific simulation software like 'Polyhedron'. 'Polyhedron', as we can guess from the name, can be used to explore regular and semi regular polyhedra and the relationship between them. Its functions are limited but it can visualize regular polyhedra, transform regular polyhedra into other polyhedra. So it is easier to operate than other dynamic geometry software like GSP. To investigate the effect of using this software in mathematics class, three classes(one in 6th grade from science education institute for the gifted, two in 7th grade) were chosen. Activities focused on the relationship between regular and semi regular polyhedra. After the class, several conclusions were drawn from the observation. First, 'Polyhedron' can be used effectively to explore the relationship between regular and semi regular polyhedra. Second, 'Polyhedron' can motivate students. Third, Students can understand the duality of polyhedra. Fourth, Students can visualize various polyhedra by reasoning. To help teachers in using technology, web sites like NCTM's illuminations and NLVM of Utah university need to be developed.

• Research in Mathematical Education
• /
• v.11 no.4
• /
• pp.247-263
• /
• 2007

Problem solving takes place not only in mathematics classes but also in real-world. For this reason, a problem and the structure of problem solving, and the enhancing of success in problem solving is a subject which has been studied by any educators. In this direction, the aim of this study is that the strategy used by students in Turkey when solving oral problems and their achievements of choosing operations when solving oral problems has been researched. In the research, the students have been asked three types of questions made up groups of 5. In the first category, S-problems (standard problems not requiring to determine any strategy but can be easily solved with only the applications of arithmetical operations), in the second category, AS-SA problems (problems that can be solved with the key word of additive operation despite to its being a subtractive operation, and containing the key word of subtractive operation despite to its being an additive operation), and in the third category P-problems (problematic problem) take place. It is seen that students did not have so much difficulty in S-problems, mistakes were made in determining operations for problem solving because of memorizing certain essential concepts, and the succession rate of students is very low in P-problems. The reasons of these mistakes as a summary are given below: $\cdot$ Because of memorizing some certain key concepts about operations mistakes have been done in choosing operations. $\cdot$ Not giving place to problems which has no solution and with incomplete information in mathematics. $\cdot$ Thinking of students that every problem has a solution since they don't encounter every type of problems in mathematics classes and course books.