수학적 신념체계에 따른 수학 문제해결 활동 : 사례연구

김윤민 이화여자대학교 교과교육연구소 2014 교과교육학연구 Vol.18 No.3

수학 문제해결은 풍부한 사고 활동 경험이 요구되고, 학생들이 직접 문제를 해결하고 분석하는 학생 자신의 문제해결 과정이다. 수학 문제해결 행동에 있어서 자원, 발견술, 자기통제, 신념이 문제해결 성공과 실패를 좌우하는 핵심적 요인이 된다. 또한, 학생들이 지니는 수학적 신념체계는 학생들이 수학을 바라보는 관점이 될 수 있고, 수학 문제해결 과정에서 학생이 수학적 상황을 어떻게 볼것인지를 결정하는 여과기의 역할을 하고, 학생들이 지니는 수학적 신념체계는 수학 문제해결 활동의 정도를 예상하게 해준다. 이에 본 연구에서는 학생들이 지니는 수학적 신념체계에 따라 수학 문제해결 활동이 어떠한지를 탐색하고자 한다. 이를 위해 각기 다른 수학적 신념체계를 지닌 고등학생 3명을 의도적 표집하여 사례연구를 실시하였다. 표집된 학생들은 보다 면밀하게 수학적 신념들을 조사 분석하였고, 각각의 학생들이 어떠한 수학 문제해결 과정 및 행동을 보여주는지 관찰 하고 분석하였다. 표집된 학생 3명은 각기 다른 수학적 신념체계를 지녔고, 수학 문제해결 활동 또한 각기 다른 양상을 나타내었다. 수학 문제해결 활동 성패에는 수학적 신념체계가 상당한 영향을 미치고 있음을 살펴볼 수 있었다. 비정형 문제를 해결하는데 인내하고 의지를 나타내는데 수학 문제해결 신념이 영향을 주고 있음을 보여주었고, 수학 문제해결 성패에 수학적 신념이 상당한 영향력을 보여주었다. 특히 수학적 자신감은 학생들이 수학 문제해결 활동에서 지속적으로 문제를 해결해 나갈 수 있도록 하기도 하지만, 문제해결을 포기하는데도 영향을 미쳤다. Mathematics problem-solving requires an experience in thought activity and is a process for students to solve their problems. In actions for problem-solving, resource, heuristic, self-control and beliefs are critical factors that determine the success or failure of problem-solving. Moreover, the mathematical belief system that students hold can serve as a perspective towards math and also as a filter that determines how the student views the problem. It also allows us to anticipate the degree of mathematical problem-solving activity. As such, this study seeks to investigate the mathematical problem-solving activities according to students’ belief systems. To that end, three high school students with different mathematical belief systems were recruited to conduct a case study. The sampled students’ mathematical belief system were all different from one another and their mathematical problem-solving method, too, took on different types. It was verified that the success or failure of mathematical problem-solving was significantly influenced by the mathematical belief system. Mathematical beliefs had an effect in particular on persistence and the willingness to solve non-routine problems and had an overall effect on the success or failure of problem-solving. In addition, mathematical confidence allowed students to continuously work on problem-solving activities, but also had an effect on when they gave up on the problem-solving.

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.

문장제에서 수학 문제해결력과 언어능력의 상관관계 연구 : 성별에 따른 비교

이소영 한남대학교 교육연구소 2009 교육연구 Vol.17 No.-

본 연구는 집단 간의 수학 문장제 문제해결력과 언어능력을 분석하여 보고 두 능력 사이의 상관관계를 분석하여 봄으로써 수학 문장제를 해결하는데 있어 언어능력이 중요함을 인지하고 수학적 문제해결력과 언어능력의 신장을 위해 언어교육자와 수학교육자들 간의 협조와 공동연구의 필요성을 강조하는데 목적이 있다. 이러한 연구목적을 달성하기 위하여 남·여학생의 언어능력과 수학 문장제 문제해결력 간의 차이와 상관관계를 분석하였다. 분석한 결과, 학생들의 사고가 수학 문장제 문제해결력에서 하위 영역과 문제해결력에서 남학생 집단과 여학생 집단 사이에 차이가 없음을 알 수 있었으나, 언어능력에 있어서 여학생들이 남학생들보다 뛰어남을 알 수 있다. 수학 문장제 문제해결력, 언어능력 검사 도구에 대한 하위요인간의 상관관계를 분석한 결과 수학 문장제 문제해결력과 언어능력 간에 정적인 상관관계가 있음을 알 수 있었다. In this study, Relationship between mathematical problem-solving and language skills were analyzed, through analysis, understand the importance of language skills in solving mathematical problems. And mathematical problems-solving and language ability of the kidneys between language educators and mathematics educators emphasize the need for cooperation and joint research objectives are to. The purpose of these studies in order to achieve gender context of the language skills and mathematical problem-solving in solving mathematical problems differences were analyzed and correlated. Analyzing results, students' mathematical thinking in the sub-region problem-solving in context mathematical problems and issues in problem-solving difference between male groups and female groups, but there is no unknown, and language skills in girls than boys are really good to know . mathematical problem-solving in context, language proficiency testing tool for the sub-correlation analysis of the human yo mathematical sentences in language skills and mathematical problem-solving static correlation between the could see it.

인지적 전략교수가 경도정신지체학생의 수학적 문제해결 전략 수행능력과 참여행동에 미치는 효과

김현진 국립특수교육원 2007 특수교육연구 Vol.14 No.1

본 연구는 경도정신지체 학생들에게 인지적 전략교수를 적용한 후 이들의 수학적 문제해결 전략 수행능력과 참여행동에 있어서 어떻게 변화하였는지 알아보기 위하여 대상자간 중다간헐기초선 설계(multiple probe baseline across participants)를 사용하여 수학적 문제해결 전략 수행능력과 문제해결 참여에 대한 행동적 특성을 평가하였다. 실험은 기초선, 중재, 유지의 순서로 진행되었으며 수학적 문제해결 수행능력과 참여행동은 인지적 전략을 제공한 수업시간 후에 수행평가를 실시하여 전 회기와의 차이로 효과를 평가하였다. 수학적 문제해결 전략 수행능력 검사는 문제해결전략 수행을 측정할 수 있도록 문제해결의 단계에 따라 1) 문제이해 2) 전략 선택 3) 문제해결 4) 점검단계를 거치토록 구성하였으며 문제해결 참여행동은 문제해결을 하려는 자발성과 문제해결 과제에 집중하는 행동의 두 가지 범주로 나누어 행동특성을 관찰하도록 하였다. 연구결과 중재 회기 중의 향상 정도를 살펴보면 아동 4명 모두 수학적 문제해결 전략과제 수행이 평균 31%가 향상 되었다. 문제해결 참여 행동 발생의 변화는 중재회기 중에 아동 4명 모두 문제해결 참여 행동발생비율이 평균 12%가 향상 되었다. 이는 결과적으로 인지적 전략을 아동에게 교수하였을 때 대상아동의 수학적 문제해결 전략 수행 성취도가 높아지고, 문제해결 참여행동의 발생율이 대체적으로 높아졌음을 알 수 있었다. The purpose of the study was to examine the effect of a cognitive strategy programs on the mathematical problem solving, participating act in solving the problem. The cognitive strategy programs offered the students who have mild mental retardation at the grade 4, 5, 6 in elementary school. Using the multiple probe baseline across participants, I estimate the mathematical problem solving and participating act in solving problem. The experiment includes the baseline, mediate, contingency. During pre-sessions, the students solved the mathematical problem with their own ways. In the main sessions, students learned strategies regarding how to solve the problem, how to improve the problem solving strategy. Feedback was provided in order to help the participants apply the strategies to real situations. The main goal of this study was to learn the strategy of problem solving in mathematics and to improve the participating act. The result is that, all of the students in the main session show the average 31% improvement in mathematical problem solving, the average 11% improvement in participating act in solve the problem. The program of this study revealed a significance of learning the cognitive strategy. It was revealed that the training cognitive strategy was effective to solve the mathematical problem. This study suggested that cognitive strategy increases the mathematic problem solving, participating act in solve problem.

인지와 메타인지 전략교수가 경도장애학생의 수학 문장제 문제해결 수행능력·태도·귀인에 미치는 영향

김현진(Kim, Hyun jin) 한국특수교육학회 2007 한국특수교육학회 학술대회 Vol.2007 No.-

본 연구는 문제해결의 어려움이 있는 경도장애학생들을 위해서 인지와 메타인지 전략 교수 프로그램을 만들어 실제 적용한 후, 수학 문장제 문제해결 수행능력ㆍ태도ㆍ귀인에 미치는 영향을 알아보는데 있다. 연구설계는 사전-사후 검사 통제집단 설계로 하였으며, 자료분석 방법으로는 인지와 메타인지 전략이 경도장애학생의 수학 문장제 문제해결 수행능력과 문제해결 태도에 영향을 미치는지 파악하기 위해 사전ㆍ사후 검사 차이에 대한 집단간 다변량(MANOVA)분석을 실시하였고, 인지와 메타인지 전략이 학습자들의 귀인에 영향을 미치는지 파악하기 위해 실험집단의 노력 귀인 사전ㆍ사후 차이점수와 통제집단의 노력 귀인 사전 사후 차이점수에 대해 독립표본 t 검정을 실시하였다. 연구결과 첫째, 두 집단 간에는 수학 문장제 문제해결 수행능력 4개 하위차원의 조합에서 통계적으로 유의수준 .05에서 유의미한 차이가 있는 것으로 나타났으며, 하위차원에 대한 개별적인 변량분석을 실시한 결과 문제이해와 계획수립, 점검은 중재의 유무에 따라 사전ㆍ사후 점수의 차가 유의미한 차이가 있는 것으로 분석되었다. 둘째, 문제해결 태도의 4개 하위차원 조합에서 통계적으로 유의수준 .05에서 유의미한 차이가 있는 것으로 나타났으며, 하위차원에 대한 개별적인 변량분석을 실시한 결과 과제지속력, 흥미는 중재의 유무에 따라 사전ㆍ사후 차이 점수가 유의미한 차이가 있는 것으로 분석되었다. 셋째, 문제해결 노력 귀인 점수에서는 두 집단간에 유의도 .01 수준에서 통계적으로 의미 있는 차가 있는 것으로 나타났다. 이러한 결과는 인지와 메타인지 전략이 수학 문장제 문제해결 수행능력과 문제해결에 대한 태도, 노력귀인에 긍정적 변화를 가져왔음을 시사하고 있다. The purpose of the study was to examine the effect of a cognitive and metacognitive strategy programs on the mathematical word problem solving, attitudes, and attribution of students who participated in the program. Forty eight students were divided into 24 students for each group. The experimental group participated in the cognitive and metacognitive strategy programs, while the control group did not. The cognitive and metacognitive strategy program was implemented for 7 months. To prove the effects of the program, pretest-posttest control group design was applied. The differences between pre-test and post-test of two groups on mathematics word problem solving and attitudes were analyzed using multivariate analysis of variance(MANOVA). And the differences on attribution of efforts were analyzed using independent t -test. The results of the study were as follows : First, there were statistically significant differences between the experimental group and the control group in the mathematical word problem solving. Analyzing differences in mathematical word problem solving according to cognitive and metacognitive strategy was found to be higher in three sub areas - problem solving understand, problem solving plan, problem solving monitor- among four sub areas. However, the level of development in the problem solving performance was not improved. Second, there were statistically differences in the attitude of problem solving between the experimental group and the control group in the attitude level. Analyzing differences in attitude according to cognitive and metacognitive strategy was found to be higher in two sub areas - task persistency, interest - among four sub areas. However, the level of development in the confidence and spontaneity was not improved. Third, there were statistically differences between the experimental group and the control group in the attribute of efforts. Theses results show that the cognitive and metacognitive strategy increases the mathematics word problem solving, positive attitude of problem solving, and attribution of efforts. Also, this study showed that the cognitive and metacognitive strategy could be successfully applied in problem solving process and affective aspect to the students with mild disabilities.

Research Trends in the Ill-Structured Problem-Solving of Mathematics Education in Korea

김민경,이지영,조미경 대한수학교육학회 2023 수학교육학연구 Vol.33 No.3

In mathematics education, there has been substantial interest in mathematical problem-solving, which has led to the development of competencies of problem-solving based on the relevance and applicability of mathematical knowledge learned in school to problem-solving in one’s everyday life. This study revisits the development of ill-structured problem-solving and its pedagogical implications with the aim of providing students with a transferable problem-solving experience in which the knowledge they learn in school can be meaningfully employed in a real-world context. To examine the overall trend of research focusing on ill-structured problems or ill-structured problem-solving, this study statistically analyzed the papers published in the field of mathematics education in Korea according to the research period, topic, research method, and mathematics content strands to provide useful insights as descriptive research. More specifically, the ill-structuredness of textbook problems, the process of ill-structured problem-solving and the collaborative problem solving that emerges from ill-structured problem-solving were analyzed more closely. Based on the results, the researchers suggested the directions and implications for mathematical problem-solving in terms of ill-structured problems or problem-solving in mathematics education.

수학학습장애 위험군아동, 읽기,수학공존학습장애위험군아동, 일반아동의 수학문장제 문제해결력 비교 -수학 인지변인을 중심으로-

김동일 ( Dongil Kim ),고혜정 ( Hyejung Koh ),김이내 ( Ienai Kim ),백서연 ( Seoyeon Baek ),이해린 ( Haelin Lee ),이기정 ( Kijyung Lee ) 한국특수교육문제연구소 2013 특수교육저널 : 이론과 실천 Vol.14 No.1

수학문장제 문제해결력 문제는 읽기능력, 수리적 계산능력과 문제해결력을 동시에 요구한다. 수학문장제 문제해결력 검사의 경우 읽기 혹은 수학 단일 영역의 능력만을 측정하지 않고 두영역과 더불어 이에 영향을 미치는 인지변인들에 대한 측정이 종합적으로 이루어지게 된다. 본 연구에서는 수학문장제 문제해결력 문장제 문제의 식과 답, 총점을 측정하여 일반아동과 수학학습장애 위험군아동, 읽기ㆍ수학공존학습장애 위험군아동의 수학문장제 문제해결력을 비교하였으며 수학문장제 문제해결력에 영향을 미치는 인지변인들을 측정하여 각 집단 간의 수행력 차이를 알아보았다. 그 결과, 수학학습장애 위험군아동과 읽기ㆍ수학공존학습장애 위험 군아동은 수학문장제 문제해결력에서 일반아동들보다 낮은 수행력을 보였으며, 특히 공존학습 장애위험군 아동들이 가장 낮은 성취수준을 보였다. 각 인지변인에 있어서도, 주의력을 제외한 모든 인지변인에서 공존학습장애 위험군아동이 가장 낮은 집단으로 분류되었다. 이러한 결과를 미루어, 공존학습장애 위험군 아동들에게 실제적으로 적용할 수 있는 진단도구 개발과 교육중재 프로그램의 개발의 필요성을 제시하였다. The mathematical word problem needs reading skills, mathematical calculation abilities and problem solving skills at the same time. The mathematical word problem solving test measures not that only single reading or mathematical skills but that reading skills, mathematical abilities, and cognitive variables that influence these skills in overall. Therefore, this study is to compare the performance of mathematical word problem solving of children at risk in mathematics disabilities(MD), comorbid in reading and mathematical disabilities (MDRD), and children without disabilities(N) by measuring the mathematical formula, answer and total score of mathematical word problem. Additionally, this study is also to investigate the differences between the performance of cognitive variables through measuring the cognitive variables that influence the mathematical word problem solving. As a result, the children at risk in MD only and MDRD showed lower performance in mathematical word problem solving than the children without disabilities(N) did, especially the children at risk in MDRD revealed the lowest performance. Regard with the cognitive variables, the children at risk in MDRD are also classified with the lowest performance group in all the cognitive variables except for attention. According to the result of this study, implication and recommendation were suggested that diagnostic instrument and education program are needed to develop to applying practically for the children at risk in MDRD.

다전략 수학 문제해결 학습이 초등학생의 수학적 창의성과 수학적 태도에 미치는 영향

김서령 ( Kim¸ Seoryeong ),박만구 ( Park¸ Mangoo ) 한국수학교육학회 2021 初等 數學敎育 Vol.24 No.4

The purpose of this study is to investigate the effects of solving multi-strategic mathematics problems on mathematical creativity and attitudes of the 6th grade students. For this study, the researchers conducted a survey of forty nine (26 students in experimental group and 23 students in comparative group) 6th graders of S elementary school in Seoul with 19 lessons. The experimental group solved the multi-strategic mathematics problems after learning mathematics through mathematical strategies, whereas the group of comparative students were taught general mathematics problem solving. The researchers conducted pre- and post- isomorphic mathematical creativity and mathematical attitudes of students. They examined the t-test between the pre- and post- scores of sub-elements of fluency, flexibility and creativity and attitudes of the students by the i-STATistics. The researchers obtained the following conclusions. First, solving multi-strategic mathematics problems has a positive impact on mathematical creativity of the students. After learning solving the multi-strategic mathematics problems, the scores of mathematical creativity of the 6th grade elementary students were increased. Second, learning solving the multi-strategy mathematics problems impact the interest, value, will and efficacy factors in the mathematical attitudes of the students. However, no significant effect was found in the areas of desire for recognition and motivation. The researchers suggested that, by expanding the academic year and the number of people in the study, it is necessary to verify how mathematics learning through multi-strategic mathematics problem-solving affects mathematical creativity and mathematical attitudes, and to verify the effectiveness through long-term research, including qualitative research methods such as in-depth interviews and observations of students' solving problems.

알고리즘 기반의 수학활동이 유아의 수학적 문제해결력에 미치는 영향

송윤나 ( Song Yun-na ),박희숙 ( Park Hee-suk ) 한국아동교육학회 2018 아동교육 Vol.27 No.2

본 연구는 알고리즘 기반의 수학활동이 유아의 수학적 문제해결력에 미치는 영향을 알아보고자 했다. 연구대상은 D지역의 I유치원 만 5세반 유아 42명으로 실험집단 유아 21명(남아 11명, 여아10명), 비교집단 유아 21명(남아 11명, 여아 10명)이다. 연구절차는 알고리즘 기반의 유아수학활동 구성, 예비검사, 실험장소의 선정 및 실험자 교육 실시, 사전검사, 실험처치, 사후검사 순으로 진행되었다. 실험처치는 2017년 8월 21일부터 9월 29일까지 6주간 진행되었으며, 연구기간 중 실험집단 유아들에게는 알고리즘 기반의 수학활동을 매주 2-3회 적용하였다. 비교집단 유아들에게는 누리과정 중심 수학활동을 실시하였다. 수집된 자료는 SPSS 24.0을 사용하여 통계 처리하였다. 연구 결과, 유아의 수학적 문제해결력 전체는 실험진단과 비교집단 간의 유의미한 차이가 나타났으며 수학적 문제해결력의 하위영역 중 측정, 대수, 기하, 통계영역의 수학적 문제해결력을 향상 시켰다. 본 연구는 알고리즘 기반의 수학활동이 유아의 수학적 문제해결력 발달을 위한 새로운 교수학습 접근법을 제시함에 그 의의가 있다. The purpose of this study is to investigate the effects of Algorithm-based mathematic activities to young children's mathematical problem solving ability. The participants were 42 of five-year old children(experimental group 21, comparative group 21) at I kindergarten located in D city. the research proceeded in order that consisted of algorithm - based early childhood mathematical activity composition, preliminary test, selection of experiment site, implementation of experimenter education, pre - test, experimental treatment and post - test. The experiment was conducted from August 21 to September 29, 2017 for 6 weeks. The experiment group performed the mathematics activities through the algorithm-based, and the comparative group performed the mathematics activities in the Nuri course based on life topic. The results of this study is that, in the mathematical problem solving ability, there was a significant difference between the two groups. Second, mathematical problem solving ability of measurement, algebra, geometry, and statistical domain was improved among sub - areas of mathematical problem solving ability. Therefore, it has been shown that algorithmic mathematical activities are effective in improving the mathematical problem solving ability of young children. This study suggests that algorithm - based mathematical activities can be proposed as a new teaching and learning method for the development of mathematical problem solving ability of young children and it is a mathematical activity of unplugged learning which learns computer science without using computer. Is expected to be developed into a new type of mathematics education program for young children.

수학적 모델링 과정을 반영한 교과서 문제 재구성 예시 및 적용

박선영 ( Park Sunyoung ),한선영 ( Han Sunyoung ) 한국수학교육학회 2018 수학교육 Vol.57 No.3

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.