수학적 모델링 과정을 반영한 교과서 문제 재구성 예시 및 적용
Reconstruction and application of reforming textbook problems for mathematical modeling process
1
조회
0
다운로드
15
인용
0
참고문헌
키워드
초록
There has been a gradually increasing focus on adopting mathematical modeling techniques into school curricula and classrooms as a method to promote students' mathematical problem solving abilities. However, this approach is not commonly realized in today's classrooms due to the difficulty in developing appropriate mathematical modeling problems. This research focuses on developing reformulation strategies for those problems with regard to mathematical modeling. As the result of analyzing existing textbooks across three grade levels, the majority of problems related to the real-world focused on the Operating and Interpreting stage of the mathematical modeling process, while no real-world problem dealt with the Identifying variables stage. These results imply that the textbook problems cannot provide students with any chance to decide which variables are relevant and most important to know in the problem situation. Following from these results, reformulation strategies and reformulated problem examples were developed that would include the Identifying variables stage. These reformulated problem examples were then applied to a 7th grade classroom as a case study. From this case study, it is shown that: (1) the reformulated problems that included authentic events and questions would encourage students to better engage in understanding the situation and solving the problem, (2) the reformulated problems that included the Identifying variables stage would better foster the students’ understanding of the situation and their ability to solve the problem, and (3) the reformulated problems that included the mathematical modeling process could be applied to lessons where new mathematical concepts are introduced, and the cooperative learning environment is required. This research can contribute to school classroom’s incorporation of the mathematical modeling process with specific reformulating strategies and examples.
Abstract
There has been a gradually increasing focus on adopting mathematical modeling techniques into school curricula and classrooms as a method to promote students' mathematical problem solving abilities. However, this approach is not commonly realized in today's classrooms due to the difficulty in developing appropriate mathematical modeling problems. This research focuses on developing reformulation strategies for those problems with regard to mathematical modeling. As the result of analyzing existing textbooks across three grade levels, the majority of problems related to the real-world focused on the Operating and Interpreting stage of the mathematical modeling process, while no real-world problem dealt with the Identifying variables stage. These results imply that the textbook problems cannot provide students with any chance to decide which variables are relevant and most important to know in the problem situation. Following from these results, reformulation strategies and reformulated problem examples were developed that would include the Identifying variables stage. These reformulated problem examples were then applied to a 7th grade classroom as a case study. From this case study, it is shown that: (1) the reformulated problems that included authentic events and questions would encourage students to better engage in understanding the situation and solving the problem, (2) the reformulated problems that included the Identifying variables stage would better foster the students’ understanding of the situation and their ability to solve the problem, and (3) the reformulated problems that included the mathematical modeling process could be applied to lessons where new mathematical concepts are introduced, and the cooperative learning environment is required. This research can contribute to school classroom’s incorporation of the mathematical modeling process with specific reformulating strategies and examples.
참고문헌 (45)
- 강옥기, "중학교 수학 1, 2, 3" (2015), 동아출판
- 교육부, "수학과 교육과정. 교육부 고시 제2015-74호", ~ (2015)
- 김민경, "수학적 모델링 사례 분석을 통한 초등 수학에서의 지도 방안 연구", 수학교육, 48(4), pp.365~385 (2009), 한국수학교육학회
- 김해윤, "수학적 모델링의 적용 사례연구", 수학교육학논총, 47(2), pp.159~163 (2015)
- 서지희, "중학교 3학년 수학 영재 학생들을 위한 수학적 모델링 교수․학습 자료의 개발 및 적용: 쓰나미를 소재로", 학교수학, 15(4), pp.785~799 (2013), 대한수학교육학회
- 신은주, "탐구지향 수학적 모델링에 관한연구", 수학교육학연구, 11(1), pp.157~177 (2001)
- 신현성, "실세계 상황에서 수학적 모델링 과제설정 효과", 한국학교수학회논문집, 14(4), pp.423~442 (2011), 한국학교수학회
- 이대현, "수학 문제 해결의 역사와 모델링 관점", 한국수학사학회지, 17(4), pp.123~131 (2004), 한국수학사학회
- 황혜정, "수학적 모델링의 이해 - 국내 연구 결과 분석을 중심으로 -", 학교수학, 9(1), pp.65~97 (2007), 대한수학교육학회
- Asempapa, R. S., "Mathematical modeling : essential for elementary and middle school students", Journal of Mathematics Education, 8(1), pp.16~29 (2015)
- Blum, W., "ICMI Study 14 : Applications and modelling in mathematics education–Discussion document", Educational Studies in Mathematics, 51(1), pp.149~171 (2002)
- Blum, W., "Quality Teaching of Mathematical Modelling: What Do We Know, What Can We Do?", The Proceedings of the 12th International Congress on Mathematical Education, pp.73~96 (2015)
- Blum, W., "Mathematical modelling : Can it be taught and learnt?", Journal of Mathematical Modelling and Application, 1(1), pp.45~58 (2009)
- Blum, W., "Mathematical modeling and modeling mathematics", pp.65~76 (2016), The National Council of Teachers of Mathematics
- Blum, W., "Modelling and applications in mathematics education" (2007), Springer
- Cavey, L. O., "Mathematical modeling and modeling mathematics", pp.131~142 (2016), The National Council of Teachers of Mathematics
- Cirillo, M., "Mathematical modeling and modeling mathematics", pp.3~16 (2016), NCTM
- Council of Chief State School Officers, "Common core state standards initiative" (2010)
- Doerr, H., "Mathematical modeling and modeling mathematics", pp.197~205 (2016), The National Council of Teachers of Mathematics
- Doerr, H., "A modeling perspective on students' mathematical reasoning about data", Journal for Research in Mathematics Education, pp.110~136 (2003)
- English, L. D., "Mathematical modelling in the early school years", Mathematics Education Research Journal, 16(3), pp.58~79 (2004)
- Gann, C., "Mathematical modeling and modeling mathematics", pp.97~106 (2016), The National Council of Teachers of Mathematics
- Gould, H., "Mathematical modeling and modeling mathematics", pp.179~186 (2016), The National Council of Teachers of Mathematics
- Gravemeijer, K., "Solving word problems : A case of modelling?", Learning and Instruction, 7(4), pp.389~397 (1997)
- Groshong, K., "Mathematical modeling and modeling mathematics", pp.17~25 (2016), The National Council of Teachers of Mathematics
- Hatch, J. A., "Doing qualitative research in education settings" (2002), Suny Press
- Hamilton, E., "Model-eliciting activities (MEAs) as a bridge between engineering education research and mathematics education research", Advances in Engineering Education, 1(2), pp.1~25 (2008)
- Kaiser, G., "Modelling in lower secondary mathematics classroom—problems and opportunities", Modelling and Applications in Mathematics Education, pp.99~108 (2007)
- Kang, O. K, "Teaching mathematical modelling in school mathematics", The Proceedings of the 12th International Congress on Mathematical Education, pp.8~15 (2012)
- Maaß, K., "Classification scheme for modelling tasks", Journal für Mathematik–Didaktik, 31(2), pp.285~311 (2010)
- Merriam, S. B., "Case study research in education: A qualitative approach" (1988), Jossey-Bass
- Meyer, D., "Missing the Promise of Mathematical Modeling", Mathematics Teacher, 108(8), pp.578~583 (2015)
- National Council of Teachers of Mathematics, "Principles and Standards for School Mathematics" (2000), NCTM
- Niss, M., "Modeling and application in mathematics education", pp.1~ (2007), Springer
- Palm, T., "Theory of authentic task situations, Words and worlds: Modelling verbal descriptions of situations", pp.q3~19 (2009)
- Pollak, H., "A History of School Mathematics", pp.1647~1672 (2003)
- Pollak, H., "Modelling and applications in mathematics education: the 14th ICMI Study", pp.109~120 (2007), Springer
- Puchalska, E., "Children's reactions to verbal arithmetical problems with missing, surplus or contradictory dat", For the Learning of Mathematics, 7(3), pp.9~16 (1987)
- Stake, R. E., "The case study method in social inquiry", Educational Researcher, 7(2), pp.5~8 (1978)
- Stein, M. K., "Mathematical tasks as a framework for reflection : From research to practice", Mathematics Teaching in the Middle School, 3(4), pp.268~275 (1998)
- Sutherland, R., "Teaching for learning mathematics" (2006), McGraw-Hill Education
- Tran, D., "Authenticity of mathematical modeling", Mathematics Teacher, 107(9), pp.672~678 (2014)
- Usiskin, Z., "Mathematical modeling and pure mathematics", Mathematics Teaching in the Middle School, 20(8), pp.476~482 (2015)
- Verschaffel, L., "Making sense of word problems" (2000), Lisse
- Zbiek, R. M., "Beyond motivation : Exploring mathematical modeling as a context for deepening students' understandings of curricular mathematics", Educational Studies in Mathematics, 63(1), pp.89~112 (2006)