• School Mathematics
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• v.7 no.2
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• pp.123-137
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• 2005

Elementary combinatorial problem may be classified into three different combinatorial models(selection, distribution, partition). The main goal of this research is to determine the effect of type of combinatorial operation and implicit combinatorial model on problem difficulty. We also classified errors in the understanding combinatorial problem into error of order, repetition, permutation with repetition, confusing the type of object and cell, partition. The analysis of variance of answers from 339 students showed the influence of the implicit combinatorial model and types of combinatorial operations. As a result of clinical interviews, we particularly noticed that some students were not able to transfer the definition of combinatorial operation when changing the problem to a different combinatorial model. Moreover, we have analysed textbooks, and we have found that the exercises in these textbooks don't have various types of problems. Therefore when organizing the teaching , it is necessary to pose various types of problems and to emphasize the transition of combinatorial problem into the different models.

• Journal of the Korean School Mathematics Society
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• v.24 no.1
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• pp.19-38
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• 2021

The purpose of this study is to examine pre-service teachers' noticing when evaluating peers' mathematical problem posing tasks. To this end, 46 secondary pre-service teachers were asked to create real-world problems related to permutation and combination and randomly assigned to evaluate peers' problems. As a result, the pre-service teachers were most likely to notice the difficulty of their peers' mathematics problems. In particular, the pre-service teachers tended to notice particular conditions in order to increase the difficulty of a problem. In addition, the pre-service teachers noticed the clarity of a question and its solution, novelty of the problem, the natural connection between real-world contexts and mathematical concepts, and the convergence between mathematical concepts.

• Communications of Mathematical Education
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• v.12
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• pp.265-280
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• 2001

고등학교 수학 I 의 확률 및 통계영역의 교육내용을 정리한 후, 고등학생들에게 확률 및 통계영역에 관한 흥미를 돋구기 위하여 2002년 월드컵을 소재로 한 문제들을 활용하여 비주얼 베이직으로 프로그램한 ‘확률상자’ 라는 확률모형을 개발하였다. 확률상자에는 확률의 역사, 경우의 수, 순열, 같은 것이 있는 순열, 원순열, 조합, 이항계수, 통계적 확률, 조건부 확률, 배반사건 등 모두 10가지 모듈을 포함한다. 확률상자의 초기화면에서 메뉴를 선택하면 선택된 내용에 관한 간단한 정의와 함께 문제가 제시되어 정답을 적도록 하였고, 오답일 때는 힌트를 누르면 정답을 이해할 수 있도록 풀이과정을 제시하였다. 특히, 메뉴가운데서 경우의 수, 순열, 같은 것이 있는 순열, 원순열, 조합, 통계적 확률의 경우에는 풀이과정 중에 애니메이션 또는 시뮬레이션이 실행되도록 하여 이해를 돕도록 하였다.

• Proceedings of the Korean Information Science Society Conference
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• 2003.04c
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• pp.67-69
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• 2003

소프트웨어 시스템을 집적하는데 있어서 컴퍼넌트 중심의 소프트웨어 개발 접근 방법이 큰 경향이다. 집적 소프트웨어 시스템의 전체 신뢰도를 확보하기 위해 소프트웨어 컴퍼넌트의 테스트 및 자원의 유한성 내에서 소프트웨어가요하는 조건을 만족해야만 한다. 기지의 비용, 신뢰도, 테스트 노력, 시스템 컴퍼넌트의 기타 다른 공헌 인자를 가지고 순열조합의 최적화 문제로서 시스템 테스트의 최적화 문제 효율을 공식화할 수 있다. 본 연구에서는 그 각각이 사전에 명세화한 신뢰도 요건을 가진 단일 또는 다중 실용화 시스템에 대한“소프트웨어 컴퍼넌트 테스트 자원 할당”을 고려한다. 이것은 내고장(fault-tolerant) 시스템에도 확장해서 실용화할 수 있다. 테스트 자원 할당문제에 체계적으로 접근하는 절차를 논하고자 한다.

• Communications of Mathematical Education
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• v.25 no.3
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• pp.607-627
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• 2011

The purpose of this study is to analyse teaching and learning effects of the structural-mapping used instruction and to find out the characteristics of problem solving process in permutation and combination. For this study, two classes of 11th grade students(67 students) were randomly selected from S high school in D city. One of them was assigned to the experimental group and the other to the control group, respectively. Four lectures of the structural-mapping used instruction were carried out in the experimental group and same amount of lectures of the text book oriented instruction were carried out in the control group. The research findings are as follows. First, the structural-mapping used instruction is shown to be more effective in achievement than the traditional textbook-oriented instruction. Second, the ball-box model is found out to be easier and simpler than the selection-distribution model. Third, students who used the ball-box model are properly able to use both model.

• Journal of Educational Research in Mathematics
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• v.26 no.4
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• pp.635-662
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• 2016

Combinatorics, the basis of probabilistic thinking, is an important area of mathematics and closely linked with other subjects such as informatics and STEAM areas. But combinatorics is one of the most difficult units in school mathematics for leaning and teaching. This study, using the designed combinatorial models and executable expression, aims to analyzes the eye movement of graduate students when they translate the written combinatorial problems to the corresponding executable expression, and examines not only the understanding process of the written combinatorial sentences but also the degree of difficulties depending on the combinatorial semantic structures. The result of the study shows that there are two types of solving process the participants take when they solve the problems : one is to choose the right executable expression by comparing the sentence and the executable expression frequently. The other approach is to find the corresponding executable expression after they derive the suitable mental model by translating the combinatorial sentence. We found the cognitive processing patterns of the participants how they pay attention to words and numbers related to the essential informations hidden in the sentence. Also we found that the student's eyes rest upon the essential combinatorial sentences and executable expressions longer and they perform the complicated cognitive handling process such as comparing the written sentence with executable expressions when they try the problems whose meaning structure is rarely used in the school mathematics. The data of eye movement provide meaningful information for analyzing the cognitive process related to the solving process of the participants.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.14 no.3
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• pp.1451-1458
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• 2013

This paper deals with the programs and algorithms that generate the full data set that satisfy the basic combinatorial requirement of combination, permutation, partial permutation or shortly r-permutation, which are used in the application of the total data testing or the simulation input. We search the programs able to meet the rules which is permutations and combinations, r-permutations, select the fastest program by field. With further study, we developed a new program reducing the time required to processing. Our research performs the following pre-study. Firstly, hundreds of algorithms and programs in the internet are collected and corrected to be executable. Secondly, we measure running time for all completed programs and select a few fast ones. Thirdly, the fast programs are analyzed in depth and its pseudo-code programs are provided. We succeeded in developing two programs that run faster. Firstly, the combination program can save the running time by removing recursive function and the r-permutation program become faster by combining the best combination program and the best permutation program. According to our performance test, the former and later program enhance the running speed by 22% to 34% and 62% to 226% respectively compared with the fastest collected program. The programs suggested in this study could apply to a particular cases easily based on Pseudo-code., Predicts the execution time spent on data processing, determine the validity of the processing, and also generates total data with minimum access programming.

• Journal of the Korea Society of Computer and Information
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• v.26 no.10
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• pp.27-35
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• 2021

Local search has been used to solve various combinatorial optimization problems. One of the most important factors in local search is the method of generating a neighbor solution. In this paper, we propose neighbor generation strategies of local search for permutation-based combinatorial optimization, and compare the performance of each strategies targeting the traveling salesman problem. In this paper, we propose a total of 10 neighbor generation strategies. Basically, we propose 4 new strategies such as Rotation in addition to the 4 strategies such as Swap which have been widely used in the past. In addition, there are Combined1 and Combined2, which are made by combining basic neighbor generation strategies. The experiment was performed by applying the basic local search, but changing only the neighbor generation strategy. As a result of the experiment, it was confirmed that the performance difference is large according to the neighbor generation strategy, and also confirmed that the performance of Combined2 is the best. In addition, it was confirmed that Combined2 shows better performance than the existing local search methods.

• Journal of the Korean School Mathematics Society
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• v.12 no.2
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• pp.261-279
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• 2009

Permutation and combination are the subjects that most teachers feel difficult to teach in Mathematics. This paper investigated evaluation items and factors of anxiety of students for permutation and combination, and further examined the way to lessen the factors of anxiety. Two high school students participated for over a year from December 2007 to February 2008. Also, two teachers joined for the analysis of evaluation items. We found that the ill-structured problems and word problems are the main factors to bring about the anxiety, whereas cooperative learning with high intelligent peers, practice to read word problems and write the process of problems solving are helpful in lessening the mathematical anxieties. Further we propose that the study of appropriate teaching and learning method for permutation and combination should be performed in the future.

• The Mathematical Education
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• v.63 no.2
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• pp.187-207
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• 2024

In this study, we explored the feasibility of automatically scoring descriptive assessment items using GPT-4 based ChatGPT by comparing and analyzing the scoring results between teachers and GPT-4 based ChatGPT. For this purpose, three descriptive items from the permutation and combination unit for first-year high school students were selected from the KICE (Korea Institute for Curriculum and Evaluation) website. Items 1 and 2 had only one problem-solving strategy, while Item 3 had more than two strategies. Two teachers, each with over eight years of educational experience, graded answers from 204 students and compared these with the results from GPT-4 based ChatGPT. Various techniques such as Few-Shot-CoT, SC, structured, and Iteratively prompts were utilized to construct prompts for scoring, which were then inputted into GPT-4 based ChatGPT for scoring. The scoring results for Items 1 and 2 showed a strong correlation between the teachers' and GPT-4's scoring. For Item 3, which involved multiple problem-solving strategies, the student answers were first classified according to their strategies using prompts inputted into GPT-4 based ChatGPT. Following this classification, scoring prompts tailored to each type were applied and inputted into GPT-4 based ChatGPT for scoring, and these results also showed a strong correlation with the teachers' scoring. Through this, the potential for GPT-4 models utilizing prompt engineering to assist in teachers' scoring was confirmed, and the limitations of this study and directions for future research were presented.