수학적 모델링의 과제공간과 과제복잡성의 평가척도(rating scheme)설정 - 예비수학교사를 대상으로

국문 초록 (Abstract)

본 연구는 수학적 모델링의 과제공간을 설정하고 이를 기반으로 모델링 과제의 복잡성을 나타내는 평가척도(rating scheme)를 설계하여 그것의 타당도를 확인하였다. 예비 수학교사를 대상으로 두 단계의 과정을 거쳐 실험을 진행하였고 다음과 같은 결과를 얻을 수 있었다. 첫째는 종전의 문제구조를 표현한 문제공간을 발전시켜 모델링 과제에 맞는 과제공간을 설정하고, 모델링 과제의 복잡성을 수치로 나타내는 계량적인 평가척도를 설계하여 의미 있는 타당도를 확인하였다. 둘째로는 모델링 과제 복잡성의 평가척도, 학생 성취수준, 과제 특수 발견 전략의 개수 사이에는 일관된 패턴이 있음을 발견하였다.

다국어 초록 (Multilingual Abstract)

The purpose of this study was to decide the task space and Rating Scheme of task difficulty in complicated mathematical modelling situations. One of main objective was also to conform the validation of Rating Scheme to determine the degree of difficulty by comparing the student performance with the statement of the theoretical model. In spring 2014, the experimental setting was in Modelling Cource for 38 in-service teachers in mathematics education. In conclusions, we developed the Model of Task Space based on their solution paths in mathematical modelling tasks and Rating Scheme for task difficulty. The Validity of Rating Scheme to determine the degree of task difficulty based on comparing the student performance gave us the meaningful results. Within a modelling task the student performance verifies the degree of difficulty in terms of scoring higher using solution approaches determined as easier and vice versa. Another finding was some relations among three research topics, that is, degree of task difficulty on rating scheme, levels of students performance and numbers of specific heuristic. Those three topics showed the impressive consistence pattern.

목차 (Table of Contents)

• 요약
• Ⅰ. 서론
• Ⅱ. 이론적 배경
• 1. 문제의 난이도와 구조
• 2. 수학적 모델링 과제의 복잡성
• Ⅲ. 연구설계
• Ⅳ. 연구결과
• 1. 과제공간에서 과제복잡성의 평가척도(rating scheme)설계
• 2. 과제공간과 과제 특수 발견 전략의 관계
• Ⅴ. 결론
• 참고문헌
• Abstract

참고문헌 (Reference)

1 신현성, "한국과 미국의 교과서 체제 비교분석" 한국학교수학회 12 (12): 309-325, 2009

2 김민경, "초등교사들의 수학적 모델링에 대한 인식 조사 연구" 한국학교수학회 12 (12): 411-431, 2009

3 신현성, "실세계 상황에서 수학적 모델링 과제설정 효과" 한국학교수학회 14 (14): 423-442, 2011

4 신현성, "수학적 모델링을 통한 교육과정의 구성원리" 4 (4): 27-32, 2001

5 Stavy, R., "When analogy is perceived as such" 30 (30): 1229-1239, 1993

6 Caldwell, J. H., "Variables affecting word problem difficulty in elementary school mathematics" 323-336, 1979

7 Reit, X. R., "Thought structures as an instrument to determine the degree of difficulty of modelling tasks" 917-922, 2015

8 Ferri, R. B., "Theoretical and empirical differentiations of phases in the modelling process" 38 (38): 86-95, 2006

9 Linville, W. J., "The Effects of Syntax and Vocabulary upon the Difficulty of Verbal Arithmetic Problems with Fourth Grade Students"

10 Cauzinille-Marmèche, E., "Studies of micro-genetic learning brought about by the comparison and solving of isomorphic arithmetic problems" 8 (8): 253-269, 1998

11 Silver, E. A., "Recall of mathematical problem information: Solving related problems" 54-64, 1981

12 Kantowski, M. G., "Processes involved in mathematical problem solving" 163-180, 1977

13 Goldin, G. A., "Problem structure and problem solving behavior" 1 : 924-931, 1975

14 Jerman, M., "Problem length as a structural variable in verbal arithmetic problems" 5 (5): 109-123, 1974

15 National Council of Teachers of Mathematics [NCTM], "Principle and Standards for School Mathematics" NCTM 2000

16 vom Hofe, R., "On the Development of Mathematical Modelling Competencies The PALMA Longitudinal Study" 47 : 2009

17 Zawojewski, J. S., "Numeracy in Practice" 7 (7): 32-37, 2007

18 Galbraith, P. L., "Modelling and applications in mathematics education: The 14th ICMI study (Vol. 10)" Springer Science & Business Media 2007

19 Greer, B., "Modelling and applications in mathematics education" Springer US 89-98, 2007

20 Lesh, R., "Modeling students' mathematical modeling competencies" Springer US 5-10, 2010

21 Kaiser, G., "Modeling students' mathematical modeling competencies" Springer Netherlands 1-2, 2013

22 Cohors-Fresenborg, E., "Mathematische Kompetenzen von Schülerinnen und Schülern in Deutschland" VS Verlag für Sozialwissenschaften 109-144, 2004

23 Graumann, G., "Mathematikunterricht in der Grundschule" Julius Klinkhardt 2002

24 Lester, F. K., "Mathematical problem solving in the elementary school: Some educational and psychological considerations" ERIC Center Columbus 53-86, 1978

25 Novick, L. R., "Mathematical problem solving by analogy. Journal of experimental psychology : Learning, memory" 17 (17): 398-, 1991

26 Blum, W., "Mathematical modelling in mathematics education and instruction"

27 Silver, E. A., "Mathematical and scientific problem solving : Findings, issues, and instructional implications" 1 : 265-290, 1990

28 Varaki, B. S., "Math modeling in educational research : An approach to methodological fallacies" 31 (31): 3-, 2006

29 Niss, M., "Issues and Problems of Research on the Teaching and Learning of Applications and Modelling" loc. cit 72-88, 2001

30 Newell, A., "Human problem solving" Prentice-Hall 1972

31 Knifong, J. D., "An analysis of children's written solutions to word problems" 106-112, 1976

32 Blum, W., ""Filling Up"-the problem of independence-preserving teacher interventions in lessons with demanding modelling tasks" 1623-1633, 2005