예비 수학 교사들의 수학적 모델링 및 그 교육적 활용에 대한 인식

한선영 한국수학교육학회 2019 수학교육 Vol.58 No.3

본 연구는 예비 수학 교사들의 수학적 모델, 수학적 모델링 및 수학적 모델링의 교육적 활용에 대한 인식을 조사하고 그들간의 관계에 대하여 탐색하였다. 210명의 예비 수학 교사들의 설문에 대한 응답을 구조방정식 모형을 이용하여 양적 분석하였다. 연구 결과에 따르면, 예비 수학 교사들의 수학적 모델 및 모델링에 대한 인식은 수학적 모델링의 교육적 활용에 대한인식에 영향을 미치는 것으로 나타났으며, 이에 대한 연구 및 교육적 함의점을 논의하였다. Mathematical modeling has been a crucial topic in mathematics education as students’ problem solving competency are regarded as a core skill for future society. Despite of the importance of mathematical modeling in school mathematics, there have been very limited studies relating pre-service teachers’ knowledge and perceptions on mathematical modeling. In this vein, this study aimed to investigate pe-service mathematics teachers’ perceptions on mathematical model, mathematical modeling and educational use of mathematical modeling, and their relationships. The current study utilized a survey consisted of 18 items. The responses of 210 pre-service mathematics teachers to the survey items were quantitatively analyzed using descriptive statistics, analysis of variance, exploratory and confirmatory factor analysis, the structural equation model, and multi group analysis. The results of analysis of variance revealed that pre-service teachers in difference groups (majors, grades, and experiences with mathematical modeling) showed statistically significant differences in mean values. Moreover, according to the results from the structural equation modeling analysis, pre-service mathematics teachers’ perceptions on mathematical model and modeling affected their perceptions on educational use of mathematical modeling. In addition, depending on their pre-experiences with mathematical modeling, pre-service teachers represented a different relationship between perceptions on mathematical modeling and educational use of mathematical modeling. Implications for future studies and mathematics classrooms were discussed.

수학적 모델링 관점에서 바라본 고등학교 <정보> 교과서의 모델링 과제 분석

오세준(Se Jun Oh) 학습자중심교과교육학회 2023 학습자중심교과교육연구 Vol.23 No.8

목적 본 연구에서는 고등학교 <정보> 교과서를 수학적 모델링의 관점에서 분석하여, 디지털 자원을 활용한 수학적 모델링 과제 설계에 있어서 시사점을 얻고자 하였다. 방법 이를 위하여 8종의 고등학교 <정보> 교과서에서 ‘모델링’단원과 ‘프로젝트 단원’의 과제를 분석대상으로 선정하여 문헌 분석 하였다. 분석 대상인 모델링 과제가 수학 혹은 실생활과 연관되어 있는지 분석하였으며, 디지털 소양 등 미래 역량으로 주목받는 컴퓨팅 사고력의 단계, 수학적 모델링 사이클의 관점에서 각각 분석하였다. 결과 고등학교 <정보> 교과서의 ‘모델링’ 단원 과제의 소재는 ‘수학’을 많이 활용하고 있음을 확인할 수 있었다. <정보> 교과에서 ‘모델링’은 추상화 방법의 하나로 다루고 있어 ‘수학적 모델링 사이클’ 과정으로 분석하면 구성하기-단순화하기-수학화로 구성되어 있었다. 반면, <정보> 교과서의 프로젝트 과제는 수학적 모델링 사이클과 컴퓨팅 사고력 과정의 구성요소들을 대부분 포함하고 있었으며, 별개의 독립된 단원으로 구성되어 모델링 과정을 체계적으로 경험할 수 있게 구성되어 있었다. 결론 2022 교육과정에서 제안한 미래사회의 역량 중 하나인 디지털 소양을 수학적 모델링을 학습하며 함양하기 위하여 ‘수학적 모델링’을 <정보> 교과의 모델링 과제와 연계하여 지도하는 체계적인 논의가 필요할 것이다. 또한 <정보> 교과서의 모델링 과제, 프로젝트 과제를 수학교과에 맞게 수정 보완하여 활용할 수 있으며, 수학적 모델링과 관련된 프로젝트 과제를 독립적인 단원으로 구성하여 수학 교과서를 구성하는 방안에 대한 검토가 필요할 것이다. Objectives The purpose of this study was to analyze high school information textbooks from the perspective of mathematical modeling, with the goal of gaining insights for designing mathematical modeling tasks using digital resources. Methods To achieve this objective, the study selected the ‘modeling’ unit and ‘project unit’ tasks from eight high school information textbooks for analysis, and conducted a literature review. The study examined whether the modeling tasks were related to mathematics or real-life scenarios, and analyzed them through the lens of the mathematical modeling cycle and computational thinking skills, which are increasingly recognized as crucial competencies for the future workforce, including digital literacy. Results The material in the ‘Modeling’ unit of the high school <information> textbook was heavily based on ‘mathematics’. <Information> ‘Modeling’ is treated as a method of abstraction in the textbook, and when analyzed as a process of ‘mathematical modeling cycle’, it consists of constructing, simplifying, and mathematizing. The results of the analysis showed that the treatment of ‘modeling’ in the information textbooks was distinct from ‘mathematical modeling,’ as it was seen primarily as a method of abstraction. However, the project tasks in the information textbooks included most of the components of the mathematical modeling cycle and the computational thinking process, and were organized as independent units for a systematic experience of the modeling process. Conclusions Based on these findings, the study concludes that a systematic discussion is needed to teach ‘mathematical modeling’ in connection with modeling tasks in the information textbooks in order to foster digital literacy, which is a crucial competency for future societies. Additionally, the modeling and project tasks in the information textbooks can be modified and supplemented to be incorporated into mathematics courses. Finally, the study recommends organizing project tasks related to mathematical modeling into independent units to better structure mathematics textbooks.

초등수학에 적용된 수학적 모델링 과제 유형 탐색

오영열,박주경 서울교육대학교 초등교육연구원 2019 한국초등교육 Vol.30 No.1

Today, adapting actively to a rapidly changing society has become increasingly difficult to achieve satisfactory results in the traditional way of teaching and learning mathematics. In this context, mathematical modeling is increasingly becoming an integral part of school mathematics. However, many teachers have difficulty in applying to the class together with lack of understanding about mathematical modeling. One of the main reasons teachers have difficulty understanding and applying mathematical modeling is that it is difficult to understand the nature of the mathematical modeling task. Until now, mathematical modeling has been mainly dealt with at the middle and high school levels, and the modeling tasks for them have become common, so teachers have difficulties in understanding and applying mathematical modeling in elementary school. In this study, the case of mathematical modeling applied to elementary school was examined and the applied task type was analyzed. As a result, the types of problem-centered, table-centered, and complex-type were derived as characteristics of mathematical modeling tasks applied to elementary school. Based on this study, it is expected that mathematical modeling can be applied to elementary school, and mathematical modeling can be applied to elementary school mathematics classroom with improvement of teachers' perception about mathematical modeling. 오늘날 빠르게 변화하는 사회에 능동적으로 적응하기 위해서는 수학을 가르치고 배우는 방법에 있 어서 새로운 접근이 요구되며, 이러한 배경에서 출발한 수학적 모델링은 점차 학교수학의 필수 영역이 되고 있다. 하지만 많은 교사들은 수학적 모델링에 대한 이해 부족과 함께 수학적 모델링을 수업에 적 용하는데 어려움을 겪고 있다. 교사들이 수학적 모델링을 이해하고 적용하는데 어려움을 느끼는 가장 큰 이유 가운데 하나는 수학적 모델링 과제의 특성을 이해하기 어렵다는 데에 있다. 지금까지 수학적 모델링은 주로 중·고등학교 수준에서 다루어져 왔으며 이들을 위한 모델링 과제가 보편화되어 있어 교 사들은 초등학교에서 수학적 모델링의 교육적 활용에 대한 어려움을 갖고 있다. 이에 본 연구에서는 초등학교에 적용된 수학적 모델링의 사례를 검토하고 적용된 과제의 유형을 분석하였다. 그 결과 초등 학교에 적용된 수학적 모델링 과제의 특징으로 문제 중심, 표 중심, 복합 유형이 도출되었다. 문제 중 심 유형은 하나의 문제로부터 과제가 통합적으로 다루어지며, 표 중심 유형은 과제의 정보가 표를 중 심으로 전개되어 표에 포함된 정보를 처리하는 능력을 요구한다. 복합 유형은 서술된 문장, 표, 그림의 형태가 두 가지 이상 제시되는 형태로 대부분의 수학적 모델링 과제가 이에 포함된다. 모든 학생들에 게 수학적 모델링 과제는 항상 어려움을 야기하지만 그 어려움은 수학적 모델링 고유의 인지적 복잡성 에서 기인하며 바로 그 지점으로부터 모델링 역량을 발달시킬 수 있다. 본 연구 결과 수학적 모델링을 초등학교에 적용할 수 있는 과제 유형을 밝힘으로써 수학적 모델링에 대한 교사들의 인식 개선과 더불 어 수학적 모델링을 초등학교 수학 수업에 적용하는데 도움을 줄 수 있을 것으로 기대한다.

내용 : 수학적 모델링 학습이 문장제 해결에 미치는 효과

신현용 ( Hyun Yong Shin ),정인수 ( In Su Jeong ) 한국수학교육학회 2012 初等 數學敎育 Vol.15 No.2

The purpose of this study is to investigate the effectiveness of two teaching methods of word problems, one based on mathematical modeling learning (ML) and the other on traditional learning (TL). Additionally, the influence of mathematical modeling learning in word problem solving behavior, application ability of real world experiences in word problem solving and the beliefs of word problem solving will be examined. The results of this study were as follows: First, as to word problem solving behavior, there was a significant difference between the two groups, This mean that the ML was effective for word problem solving behavior, Second, all of the students in the ML group and the TL group had a strong tendency to exclude real world knowledge and sense-making when solving word problems during the pre-test. but A significant difference appeared between the two groups during post-test, classroom culture improvement efforts. Third, mathematical modeling learning (ML) was effective for improvement of traditional beliefs about word problems. Fourth, mathematical modeling learning (ML) exerted more influence on mathematically strong and average students and a positive effect to mathematically weak students. High and average-level students tended to benefit from mathematical modeling learning (ML) more than their low-level peers. This difference was caused by less involvement from low-level students in group assignments and whole-class discussions. While using the mathematical modeling learning method, elementary students were able to build various models about problem situations, justify, and elaborate models by discussions and comparisons from each other. This proves that elementary students could participate in mathematical modeling activities via word problems, it results form the use of more authentic tasks, small group activities and whole-class discussions, exclusion of teacher`s direct intervention, and classroom culture improvement efforts. The conclusions drawn from the results obtained in this study are as follows: First, mathematical modeling learning (ML) can become an effective method, guiding word problem solving behavior from the direct translation approach(DTA) based on numbers and key words without understanding about problem situations to the meaningful based approach(MBA) building rich models for problem situations, Second, mathematical modeling learning(ML) will contribute attitudes considering real world situations in solving word problems. Mathematical modeling activities for word problems can help elementary students to understand relations between word problems and the real world. It will be also help them to develop the ability to look at the real world mathematically. Third, mathematical modeling learning (ML) will contribute to the development of positive beliefs for mathematics and word problem solving. Word problem teaching focused on just mathematical operations can`t develop proper beliefs for mathematics and word problem solving. Mathematical modeling learning (ML) for word problems provide elementary students the opportunity to understand the real world mathematically, and it increases students` modeling abilities, Futhermore, it is a very useful method of reforming the current problems of word problem teaching and learning. Therefore, word problems in school mathematics should be replaced by more authentic ones and modeling activities should be introduced early in elementary school eduction, which would help change the perceptions about word problem teaching.

Assessment of Mathematical Creativity in Mathematical Modeling

( Hong Shick Jang ) 한국수학교육학회 2011 수학교육연구 Vol.15 No.2

In mathematical modeling tasks, where students are exposed to model-eliciting for real and open problems, students are supposed to formulate and use a varieties of mathematical skills and tools at hand to achieve feasible and meaningful solutions using appropriate problem solving strategies. In contrast to problem solving activities in conventional math classes, math modeling tasks call for varieties of mathematical ability including mathematical creativity. Mathematical creativity encompasses complex and compound traits. Many researchers suggest the exhaustive list of criterions of mathematical creativity. With regard to the research considering the possibility of enhancing creativity via math modeling instruction, a quantitative scheme to scale and calibrate the creativity was investigated and the assessment of math modeling activity was suggested for practical purposes.

수리 모형화 환경을 위한 정형화된 수식 검증 기법

김종우 충남대학교 기초과학연구소 1997 忠南科學硏究誌 Vol.24 No.1

This paper presents a formal validation method of mathematical expressions in multi-facetted mathematical modeling framework. The multi-facetted mathematical modeling framework is a mathematical modeling framework to build multi-user modeling environments, which is based on object-oriented modeling and dependency graphs in structured modeling. The validation method is developed to validate mathematical expressions at model formulation time rather than model evaluation time. Primary key of objects as well as semantics of basic mathematical operations are used to check the correctness of mathematical expressions. The method can contribute to enhance correctness of mathematical expressions, and to support model formulation activities in mathematical modeling environments.

수학적 모델링 활동에 의한 창의적 사고 촉진 사례 연구

박진형 대한수학교육학회 2017 수학교육학연구 Vol.27 No.1

One of the most important activities in the process of mathematical modeling is to build models by conjecturing mathematical rules and principles in the real phenomena and to validate the models by considering its validity. Due to uncertainty and ambiguity inherent real-contexts, various strategies and solutions for mathematical modeling can be available. This characteristic of mathematical modeling can offer a proper environment in which creativity could intervene in the process and the product of modeling. In this study, first we analyze the process and the product of mathematical modeling, especially focusing on the students' models and validating way, to find evidences about whether modeling can facilitate students’creative thinking. The findings showed that the students’ creative thinking related to fluency, flexibility, elaboration, and originality emerged through mathematical modeling. 본 연구에서는 수학적 모델링 활동이 창의적 사고를 촉진하는 것이 가능한지 이론적으로 타진하고, 가능하다면 어떤 모델링 과제를 설계하여 촉진할 수 있으며, 실제 수학적 모델링 활동에서 창의적 사고는 어떠한 방식으로 드러나는지 확인하는 데 목적을 둔다. 연구 결과, 학생들이 다양한 수학적 모델을 생성하고, 각자 생성한 수학적 모델을 검토하고 개선하면서 수학적 모델링을 진행하는 장면이 확인되었다. 그리고 이러한 수학적 모델링 과정에서 수학적 창의성의 요인들인 유창성, 유연성, 독창성, 정교성의 발현을 확인할 수 있었다.

수학적 모델링 수업에 대한 초등 교사의 인식

최지선 대한수학교육학회 2017 수학교육학연구 Vol.27 No.2

This study aims to identify prospective elementary school teachers’ perception of mathematical modeling in elementary class. Forty elementary school teachers participated in this study. Each teacher analysed the previous case studies about mathematical modeling in elementary class, developed a hypothetical learning trajectory, applied the hypothetical learning trajectory to his/her class, reflected students’ learning and his/her teaching, and made reflective journals. These journals contained teachers’ perception of mathematical modeling and the difficulties that teachers experienced in teaching mathematics as mathematical modeling. These journals were analyzed to identify teachers’ perception of mathematical modeling in elementary class. This study shows that teachers have common features of mathematical modeling but their perspectives are little bit different, are classified into four kinds. And the difficulties that teachers experienced in teaching mathematics as mathematical modeling are classified into 5 categories; Task, Students’ cognitive demand, Teacher’ monitering, All students’ participation, and Classroom culture. At last, suggestions for mathematical modeling in elementary class are done according to the result of this study. 본 연구의 목적은 수학적 모델링 수업에 대한 초등교사의 인식을 질적으로 분석하는 것으로, 40명의 현직 교사들이 참여하였다. 참여 교사들은 수학적 모델링에 대한 이론을 습득하고, 실제 교실수업에 적용하는 것을 목적으로 교수ㆍ학습 과정 안을 개발하고 적용하며 그 결과를 분석한 보고서를 작성하였다. 이 보고서에 대한 분석을 통해서 초등 교사의 인식을 분석한 결과, 수학적 모델링의 특징으로 ‘비구조화된 상황’과 ‘다양한 문제해결’을 도출할 수 있었으나 수학적 모델링 관점은 다소 차이가 있어 네 가지 형태로 분류할 수 있었다. 수학적 모델링 수업에 대한 어려움은 크게 과제, 학생의 인지적 활동, 교사의 중재, 모든 학생의 참여, 교실 문화 범주로 구체화하였다. 이러한 결과를 바탕으로 수학적 모델링 수업에 대한 시사점을 제시하였다.

A Study on Mathematical Modeling Trends in Korea

황선영,한선영 대한수학교육학회 2023 수학교육학연구 Vol.33 No.3

Mathematical modeling refers to a core competency and teaching-learning method that is being treated as important worldwide. This study aimed at examining trends of previous studies on mathematical modeling, which were published in Korean journals. This study was conducted with the aim of introducing domestic studies on mathematical modeling to both domestic and foreign researchers. Fifty-four studies from 2013 to 2022 were selected for the current trend study and classified in terms of years, research subjects, and research methods. By year, at least one paper and up to 10 papers were published from 2013 to 2022. As a result of examining the trends of the studies by subject, we revealed that studies targeting teachers were very insufficient. Moreover, the findings show that biased research methods and quantitatively simple analysis methods were mainly used. Last, the relational trend between research topics and implications were diverse depending on the theme such as task, lesson, and teacher education. Specifically, although the studies have provided implications on teacher education steadily, research targeting the topic of teacher education for mathematical modeling have been very limited. For the future study on mathematical modeling, mathematics educators and researchers need to recognize that teacher education is significant in implementing mathematical modeling in school classrooms successively, and to try diverse studies on teacher education for mathematical modeling. This paper will contribute to helping foreign scholars to know the research on mathematical modeling being conducted in Korea, which will ultimately contribute to the literature on mathematical modeling.

중학교 3학년 수학 영재 학생들을 위한 수학적 모델링 교수․학습 자료의 개발 및 적용: 쓰나미를 소재로

서지희,윤종국,이광호 대한수학교육학회 2013 학교수학 Vol.15 No.4

The researchers developed the teaching-learning materials for 9th grade mathematically gifted students in terms of the hypothesis that the students would have opportunity for problem solving and develop various mathematical thinking through the mathematical modeling lessons. The researchers analyzed what mathematical thinking abilities were shown on each stage of modeling process through the application of the materials. Organization of information ability appears in the real-world exploratory stage. Intuition insight ability, spatialization/visualization ability, mathematical reasoning ability and reflective thinking ability appears in the pre-mathematical model development stage. Mathematical abstraction ability, spatialization/visualization ability, mathematical reasoning ability and reflective thinking ability appears in the mathematical model development stage. Generalization and application ability and reflective thinking ability appears in the model application stage. The developed modeling assignments have provided the opportunities for mathematically- gifted students' mathematical thinking ability to develop and expand. 본 연구는 수학적 모델링 수업이 수학 영재 학생들에게 문제해결의 기회를 제공하고 수학적 모델링 활동을 통해 다양한 수학적 사고력을 발전시킬 수 있다는 가정 하에 중학교 3학년 수학 영재 학생들을 위한 수학적 모델링 교수․학습 자료를 개발하였다. 개발된 교수 ․학습 자료를 적용하여 사례연구를 통해 수학적 모델링의 단계별 활동과정을 살펴보고 각 단계에서 어떠한 수학적 사고능력이 나타나는지 분석하였다. 수학적 모델링 과정에서 다양한 수학적 사고능력이 나타났는데 문제를 이해하는 실세계 탐구과정에서는 정보의 조직화 능력이, 상황모델을 개발하는 과정에서는 직관적 통찰능력, 공간화/시각화 능력, 수학적 추론 능력, 반성적 사고 능력이 나타났다. 수학모델 개발과정에서는 수학적 추상화 능력, 공간화/시각화 능력, 수학적 추론 능력, 반성적 사고가 나타났으며 모델적용 과정에서는 일반화 및 적용 능력과 반성적 사고가 나타났다. 모델링 수업이 진행됨에 따라 반성적 사고능력이 더 많이 나타나는 것을 확인할 수 있었다.