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《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.
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.
‘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.