《확률과 통계》의 시행과 두 가지 확률에 대한 고찰 및 교육적 시사점

이기돈,Lee, Gi Don 한국수학사학회 2018 Journal for history of mathematics Vol.31 No.5

Empirical probability and classical probability, which are two interpretations of Kolmogorov's axiom, are two ways to recognize the chances of events occurring in the real world. In this paper, I analyzed and suggested the contents of the high school textbooks ${\ll}$Probability and Statistics${\gg}$, associated with two interpretations of probability and experiments on which two interpretations are based. By presenting the cases required expressly stating what the experiment is for supporting students' understanding of some concepts, it was discussed that stating or not stating what the experiment is should be carefully determined by the educational intent. Especially, I suggested that in the textbooks we contrast the good idea of calculating the ratios of two possibilities in the imaginary world of the classical probability with the normal idea of grasping the chances of events through the frequencies in the real world of the empirical probability, with distinguishing the experiments in two interpretations of probability. I also suggested that in the textbooks we make it clear that the Weak Law of Large Numbers justifies our expectations of the frequencies' reflecting the chances of events occurring in the real world under ideal conditions. Teaching and learning about the aesthetic elements and the practicality of imaginary mathematical thinking supported by these textbooks statements could be one form of Humanities education in mathematics as STEAM education.

‘닫힌 상자’에서의 복원추출에 의한 모비율 추측 활동수업 개발 및 적용

이기돈 ( Lee Gi Don ) 한국수학교육학회 2018 수학교육 Vol.57 No.4

In this study, I developed an activity oriented lesson to support the understanding of probabilistic and quantitative estimating population ratios according to the standard statistical principles and discussed its implications in didactical respects. The developed activity lesson, as an efficient physical simulation activity by sampling with replacement, simulates unknown populations and real problem situations through completely closed 'Closed Box' in which we can not see nor take out the inside balls, and provides teaching and learning devices which highlight the representativeness of sample ratios and the sampling variability. I applied this activity lesson to the gifted students who did not learn estimating population ratios and collected the research data such as the activity sheets and recording and transcribing data of students' presenting, and analyzed them by Qualitative Content Analysis. As a result of an application, this activity lesson was effective in recognizing and reflecting on the representativeness of sample ratios and recognizing the random sampling variability. On the other hand, in order to show the sampling variability clearer, I discussed appropriately increasing the total number of the inside balls put in 'Closed Box' and the active involvement of the teachers to make students pay attention to controlling possible selection bias in sampling processes.

2015 개정 수학과 교육과정 문서의 ‘학습 요소’의 확장 또는 비확장 양상에 따른 교육과정 실행에서의 난점 및 개선 방안 탐색 : 대수 영역을 중심으로

이기돈(Lee, Gi Don) 학습자중심교과교육학회 2022 학습자중심교과교육연구 Vol.22 No.2

목적 본 연구는 2015 개정 수학과 교육과정의 ‘학습 요소’들에 의해 나타내어지는 개념들이 교육과정 문서에서 확장 또는 비확장되는 양상에 따라 발생되는 교육과정 실행 국면에서의 난점을 논의하고 이를 완화시키기 위한 방안을 교육과정 문서 서술의 개선을 중심으로 탐색하는 것을 목적으로 한다. 방법 이를 위하여 대수 영역의 ‘학습 요소’ 중 짝수, 홀수, 약수, 배수, 나눗셈, 몫, 나머지, 나누어떨어진다, 중근 등의 개념이 2015 개정 교육과정 문서에서 확장 또는 비확장되는 양상을 분석하고, 그에 따라 난점이 발생되는 교육현장의 평가 및 평가문항 등의 실제 사례를 수집하는 한편, 개선 방안 모색의 측면에서 다른 교과의 현 교육과정 문서 및 이전 수학과 교육과정 문서를 살펴보았다. 결과 위의 대수 영역 ‘학습 요소’들이 2015 개정 수학과 교육과정 문서에서 확장 또는 비확장되는 양상을 명시적 확장, 암묵적 확장, 명시적 제한, 암묵적 제한, 암묵적 비확장 등으로 확인하였고, 이러한 양상과 관련된 교육과정 실행에서의 난점을 보여주는 교육현장의 평가 및 평가문항 등의 실제 사례를 제시하였으며, 교육과정 문서에서 ‘성취기준 해설’ 항목의 활용 및 개념의 확장을 표현했던 다른 방법 등을 관찰하였다. 결론 이러한 결과를 바탕으로 위의 대수 영역 ‘학습 요소’들을 교육과정 실행 국면에서 교수학습하거나 평가할 때 발생되는 난점을 논의하는 한편, 그러한 난점을 완화시키기 위한 교육과정 문서 서술의 개선 방안을 명시성 제고의 측면에서 논의하였다. Objectives This study discusses difficulties in the implementation phase of the curriculum that arise depending on the aspects in which the ‘learning element’ of the 2015 revised mathematics curriculum is expanded or unexpanded in the curriculum document, and measures to improve the description of the curriculum document. Methods The concepts of even, odd, divisor, multiple, division, quotient, remainder, being divided without a remainder, and multiple root among the ‘learning elements’ of the algebra domain were analysed with respect to the expansion or non-expansion aspects in the 2015 revised curriculum document, and the actual cases such as evaluation events and evaluation items were collected in the educational field where difficulties occurred. The current curriculum documents of other subjects and the previous mathematics curriculum documents were also examined to explore improvement methods. Results The patterns in which the ‘learning elements’ in the above algebraic domain are expanded or unexpanded in the 2015 revised mathematics curriculum document were identified as explicit expansion, implicit expansion, explicit restriction, implicit restriction, and implicit non-extension, and the actual examples such as evaluation events and evaluation items in the educational field showing difficulties in implementing the curriculum were presented. And the use of the ‘Explanation of Achievement Standards’ item in the curriculum document and the other method of expressing the expansion of the concepts were observed. Conclusions Based on these results, we discussed the difficulties that arise when teaching, learning or evaluating the ‘learning elements’ in the above algebraic domain during the implementation phase of the curriculum. The methods to alleviate such difficulties were also suggested focusing on the improvement of explicitness of the description in the curriculum document.

초등 수학에서 경험과 선험을 연결하는 교육적 장치로서의 상상하기

이기돈(Gi Don Lee) 수학교육철학연구회 2023 수학교육철학연구 Vol.5 No.1

수학 철학에서는 선험성에 대한 논의, 즉 수학이 일정 부분 경험과 독립적인 지식인지 아니면 절대적으로 경험에 의존하는 지식인지에 대한 논의가 있어 왔다. 그러나 이 두 가지 입장은 스스로 완전한 것이 아니어서 자기주장을 논리적으로 충분히 방어하기 어렵다. 또, 두 가지 입장이 수학의 필연성 및 현실 적용 가능성에 대해 상반된 설명력을 갖기 때문에 이 중 어느 한 가지만으로는 수학의 실제를 온전하게 설명할 수 없다. 이를 고려하면 교수학습 상황에서는 수학의 선험성과 경험성이라는 두 가지 측면을 모두 열어 놓고 논의하면서 조화를 꾀하는 것이 필요하다고 판단된다. 이 논문에서는 경험을 중심으로 구성되는 초등 수학에서도 선험적 측면이 등장하는 장면으로서 삼각형의 세 각의 합과 원주율을 살펴보고, 이때 경험과 선험을 연결하는 교육적 장치의 필요성에 대해 논의한다. 또, 그러한 교육적 장치로서 상상하기를 활용하는 방안에 대해 논의한다. In the philosophy of mathematics, there have been discussions about a priority and a posteriority of Mathematics, that is, whether mathematics is the knowledge partly independent of experience or the knowledge absolutely dependent on experience. However, these two positions are imperfect, so it is not easy to logically defend their assertion completely. In addition, since the two positions have conflicting explanatory power for the necessity of mathematics and its applicability to reality, only one of them cannot completely explain the actualities of mathematics. Considering this, it would be incumbent to seek harmony of both aspects of mathematics, i.e., a priority and a posteriority when teaching and learning mathematics. Meanwhile, also in elementary mathematics, even though most contents are constructed around experience, there are some scenes in which a priori aspect appears. In this paper, as such contents, we consider the sum of the three angles of a triangle and pi, i.e., the ratio of the circumference of a circle to its diameter. Through this considering, we discuss the need for an educational device that would connect experience and a priority in those two elementary mathematics contents. We also discuss using imagining as such an educational device.

컴퓨터 기반 학업성취도 평가 도입에 따른 수학 평가 문항의 양호도 개선 가능성 탐색

이기돈(Gi Don Lee) 학습자중심교과교육학회 2023 학습자중심교과교육연구 Vol.23 No.20

목적 본 연구는 2022년 새롭게 도입된 컴퓨터 기반 학업성취도 평가에서 기존 교육 현장의 지필평가 문항보다 더 양호한 수학 평가 문항을 제작할 수 있다는 점을 실제로 드러내고 그에 따른 교육적 시사점을 논의하는 것을 목적으로 한다. 방법 이를 위하여 수학 교육 현장의 지필평가 문항들을 수집하고, 그중 컴퓨터 기반 학업성취도 평가의 문항 유형을 활용할 수 없었기 때문에 제한된 양호도를 갖게 되었던 문항들의 사례와 이를 구현 가능한 컴퓨터 기반 학업성취도 평가 문항 유형을 사용하여 변형한 사례를 비교 분석하였다. 결과 연구 결과 확장선택형, 자료연결형, 순서배열형 등의 문항 유형별로 컴퓨터 기반 학업성취도 평가에서 구현 가능한 수학 문항이 타당도를 높이고 추측도를 낮추는 등 기존 지필평가 문항의 양호도를 개선할 수 있다는 점을 문항 수준에서 구체적으로 확인하였다. 결론 연구 결과를 바탕으로 실제적인 교육 현상 상황의 측면에서 ‘맞춤형 학업성취도 자율평가’에 참여할 필요성과 장기적으로 학교 현장의 수학 지필평가가 컴퓨터 기반 평가로 전환되는 것의 장점 등에 대해 논의하였다. Objectives The purpose of this study is to reveal that the newly introduced computer based assessment of educa-tional achievement in 2022 can produce mathematics test items to improve the quality of the mathematics pa-per-and-pencil test items in the current educational field and discusses the resulting educational implications. Methods To this end, we collected paper-and-pencil test items from the current mathematics educational field. Among them, we chose examples of items with limited quality because they could not use item types of the com-puter based assessment of educational achievement and converted them using item types of the computer based assessment of educational achievement that could be implemented. After that, we compared the chosen exam-ples with the converted items. Results As a result, it was confirmed at the item level that mathematics test items that can be implemented in the computer based assessment of educational achievement can improve the quality of existing paper-and-pen-cil test items in several specific aspects, such as increasing validity and reducing guesswork. Conclusions Based on the results, we discussed the need to participate in the ‘Customized Assessment of Educational Achievement’ and the long-term advantages of converting school mathematics paper-and-pencil tests to computer based tests in terms of actual educational situations.

수학 교수학습에서 스토리텔링의 의미에 대한 탐색

이지현 ( Ji Hyun Lee ),이기돈 ( Gi Don Lee ) 한국수학교육학회 2013 수학교육 Vol.52 No.2

We explored implications of storytelling in learning and teaching mathematics and examined examples of storytelling for deep understanding of the educational meanings of storytelling and new direction of storytelling approach to mathematics teachers. Mathematics had been commonly considered as the subject irrelevant to the narrative mode of thinking and only relevant to the paradigmatic mode of thinking that has rigorous logical forms and independent from human mind. As a result, this common sense forced a transmission pedagogy of mathematics: only the teachers as owners of the objective and logical truth of mathematics could transmit mathematical truths to students. Storytelling is highlighted as an alternative to the common teaching practices of mathematics focused only on the paradigmatic mode of thinking. Although a lot of research about the educational uses of storytelling mainly focused on the development and modification of stories, we suggested that the educational interest about storytelling should move to the elements or techniques for the positive effect of storytelling.

평판 최적화 막냉각 홀을 적용한 가스 터빈 베인의 고속조건 막냉각 성능 평가

강민석(Min Seok Kang),김우준(Woo Jun Kim),곽재수(Jae Su Kwak),김기문(Gi Mun Kim),이기돈(Ki don Lee) 한국유체기계학회 2022 한국유체기계학회 학술대회 논문집 Vol.2022 No.11