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Ortaokul Öğrencilerinin Çok Çözümlü Problemleri Çözme Sürecindeki Matematiksel Yaratıcılıklarının İncelenmesi
Abstract
Bu araştırmanın amacı ortaokul 7. ve 8. sınıf öğrencilerinin çok çözümlü problemlere ürettikleri farklı çözüm yollarının incelenmesi ve matematiksel yaratıcılığın çok çözümlü problemler aracılığıyla değerlendirilmesidir. Bu çalışmada, nitel araştırma deseni olarak durum çalışması kullanılmıştır. Araştırma, 2023-2024 akademik yılında, Antalya’da MEB’e bağlı bir ortaokulda gerçekleştirilmiştir. Araştırmada, bu devlet ortaokulunun 7. sınıflarından 5 öğrenci ve 8. sınıflarından 5 öğrenci olmak üzere toplam 10 öğrenci yer almıştır. Her bir öğrenciye araştırmacı tarafından 5 adet açık uçlu çok çözümlü problem verilmiştir. Öğrencilerden çok çözümlü problemlere farklı yollardan çözümler üretmeleri istenmiştir. Öğrencilerle bir ders saati süresinde klinik görüşmeler yapılarak veriler toplanmıştır. Araştırmanın bulguları betimsel analiz yoluyla incelenmiştir. Araştırmanın verileri, Wallas (1926)'ın hazırlık, kuluçka aydınlanma ve doğrulama aşamalarından oluşan yaratıcılık modelinden yararlanılarak analiz edilmiştir. Çalışmanın sonuçları incelendiğinde, 7. sınıfta öğrenim gören öğrencilerin genel olarak açık uçlu çok çözümlü problemlere fazla çözüm yolu üretemedikleri görülmüştür. Ancak öğrencilerin problemin türüne göre bazı problemlerde matematiksel yaratıcılıklarını daha etkili bir şekilde sergileyebildikleri gözlemlenmiştir. 8 sınıfa devam eden öğrenciler, 7. sınıflara kıyasla genel olarak, problemlere daha fazla çözüm yolu üretebildikleri gözlemlenmiştir. Yaratıcılık aşamaları açısından yapılan incelemelerde ise 8. sınıf öğrencilerinin aydınlanma ve doğrulama aşamalarını tam ve etkili bir şekilde gerçekleştirdikleri görülmüştür. 7. sınıf öğrencileri ise hazırlık ve kuluçka aşamalarını genel olarak tamamlamış fakat aydınlanma ve doğrulama aşamalarını yeterli ölçüde gerçekleştirememişlerdir.
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The effect of interaction patterns on JS3 learners' retention in geometric construction was investigated in Anambra State, Nigeria. The researchers used a quasi-experimental approach with a non-equivalent control group for the pre-and post-test. The population consisted of 1,813 JS3 leaners. The study's subjects were a group of 155 JS3 learners drawn from two schools. Two JS3 classes in the schools were assigned to the experimental and control groups at random. The geometric construction retention test (GCRT) was used to collect data, and it was validated by three experts. The reliability coefficient of the GCRT was 0.80. The mean and standard deviation of the data were used to report the study's questions, whereas the hypotheses were tested via analysis of covariance at a 0.05 level of significance. According to the findings, students taught geometric construction utilizing interaction patterns remembered more material than those taught using the expository approach. It also found a statistically significant difference in retention between urban and rural learners, favoring urban learners. The interaction effect of group and location on student retention was not significant. One recommendation of this study is that teachers should use interaction patterns as an instructional method when teaching geometric construction.
In this study we investigated the relationship between students' creative performance and achievement in the mathematical domain across grade levels. DISCOVER math assessment was used to measure originality, flexibility, and elaboration (OFE), fluency, and total mathematical creativity (TMC) as indexes of students' mathematical creativity. The participants were 78 second-to fourth-grade students, mostly Navajo Indians, from 4 schools in low-income areas in a southwestern state in the US. Students' scores on the Iowa Tests of Basic Skills (IT BS) or Comprehensive Tests of Basic Skills (CT BS), taken as part of their schools' regular program of achievement testing were the measures of mathematical achievement. Using Pearson correlations, authors found significant relationships among all the measures of creativity (fluency, OFE, and TMC) and between all the indices of creativity and both the measures of mathematics achievement. Using multiple regression the authors found that OFE, fluency, and TMC explained from 41% to 60% of the variance in mathematical achievement scores across all indices of creativity and both achievement tests. Using ANOVA, authors found that both mathematical creativity and mathematical achievement increased across grade levels, but the Pearson correlations by grade level showed that these increases were not linear. Recommendations include a greater emphasis on the development of mathematical creativity in the elementary grades and research on the relationships between creativity and achievement using current instruments and theories with a special focus on domain-specific creativity and achievement across domains. The creativity of mathematicians has been accepted as the insurance of the growth of the field of mathematics (Sriraman, 2004).
Students' creative process in mathematics is increasingly gaining significance in mathematics education research. Researchers often use Multiple Solution Tasks (MSTs) to foster and evaluate students' mathematical creativity. Yet, research so far predominantly had a product-view and focused on solutions rather than the process leading to creative insights. The question remains unclear how students' process solving MSTs looks like-and if existing models to describe (creative) problem solving can capture this process adequately. This article presents an explorative, qualitative case study, which investigates the creative process of a school student, David. Using eye-tracking technology and a stimulated recall interview, we trace David's creative process. Our findings indicate what phases his creative process in the MST involves, how new ideas emerge, and in particular where illumination is situated in this process. Our case study illustrates that neither existing models on the creative process, nor on problem solving capture David's creative process fully, indicating the need to partially rethink students' creative process in MSTs.
This study investigated creative thinking abilities among two groups of 20 children with autism spectrum disorders (ASD) compared to 20 children with typical development ages 9–11. The study compared performance on two different creativity tests: general creativity (Pictorial Multiple Solutions-PMS) test versus mathematical creativity (Creating Equal Number-CEN) test, and investigated relationships between general and mathematical creative thinking across various cognitive measures including non-verbal IQ, verbal and non-verbal working memory and Attention. Results of the study demonstrate significant correlations among the measures of creativity indicating that the PMS and the CEN tasks represent different skills, or perhaps, different domains of creativity. Findings suggest that creativity can be found among individuals with ASD.
Creativity could be interpreted as a person's cognitive abilities in solving problems by bringing up new ideas. The problems of students’ math achievement lows are math presented as a finished product, ready to use, abstract and taught mechanistically. This case can be lead to the creativity of the less developed students because students are not given the opportunity to think and use their ideas in solving mathematical problems. Problem Based Learning Model is a learning model that emphasizes the concept and information outlined from the academic discipline. The purpose of this study is to analyzed students’ creativity in solving mathematical problems through Problem Based Learning model (PBL) in class VIII-1 MTsN Model Banda Aceh. Data gained based on the students’ worksheet in groups. The data acquisition is categorized into 5 levels (highest level 4 and lowest level 0) which is analyzed descriptively. The results are three groups were at level 4 with very creative categories, one group is at level 3 with a creative category and another group is at level 2 with deeply creative enough category. To the conclusion is PBL model could cultivate the students’ creativity in solving mathematical problems.
Many researchers assume that people are creative, but their degree ofcreativity is different. The notion of creative thinking level has beendiscussed .by experts. The perspective of mathematics creative thinkingrefers to a combination of logical and divergent thinking which is basedon intuition but has a conscious aim. The divergent thinking is focusedon flexibility, fluency, and novelty in mathematical problem solving andproblem posing. As students have various backgrounds and differentabilities, they possess different potential in thinking patterns,imagination, fantasy and performance; therefore, students have differentlevels of creative thinking. A research study was conducted in order todevelop a framework for students’ levels of creative thinking inmathematics. This research used a qualitative approach to describe thecharacteristics of the levels of creative thinking. Task-based interviewswere conducted to collect data with ten 8thgrade junior secondary schoolstudents. The results distinguished five levels of creative thinking,namely level 0 to level 4 with different characteristics in each level.These differences are based on fluency, flexibility, and novelty inmathematical problem solving and problem posing.Keywords: student’s creative thinking, problem posing, flexibility,fluency, novelty DOI: http://dx.doi.org/10.22342/jme.1.1.794.17-40
Scholars began serious study into the social psychology of creativity about 25 years after the field of creativity research had taken root. Over the past 35 years, examination of social and environmental influences on creativity has become increasingly vigorous, with broad implications for the psychology of human performance, and with applications to education, business, and beyond. In this article, we revisit the origins of the social psychology of creativity, trace its arc, and suggest directions for its future.
In this article, we focus on the instructional explanation of guessing as a heuristic for solving the Five Planes Problem (FPP), given in a lesson by George Pólya. We use the analysis of the lesson to test the applicability of what is known about instructional explanations in elementary mathematics to higher mathematics. The most salient characteristic of Pólya's lesson is the profusion of models and representations used to develop a sense of the problem and to support the instructional explanation. Pólya used analogical models to transform the complex FFT to a simpler one, and he used representations to extend the perspective on the problem. Introducing such models and representations requires keeping track of the links between them and the original problem. Pólya excelled in this endeavor, and his passage to each new representation or model was generally justified in terms of the goals of the explanation. Another very important feature of an instructional explanation is problem identification, a fragile goal state that needs to be constantly maintained when the problem being explained is complex. For this lesson, we examine how Pólya established and maintained that goal. Finally, we offer some insight on instructional explanations in a situation in which the teacher needs to fulfill two goals at the same time. Here, the first goal was to teach students both how to solve FPP and how to use guessing as a problem-solving heuristic or strategy; the second goal was to show us how to teach guessing. Keeping these goals in mind, we offer some suggestions on how to teach metaskills through mathematical problems in accordance with the current understanding of constructivist approaches to learning.
Although creativity is often viewed as being associated with the notions of "genius" or exceptional ability, it can be productive for mathematics educators to view creativity instead as an orientation or disposition toward mathematical activity that can be fostered broadly in the general school population. In this article, it is argued that inquiry-oriented mathematics instruction which includes problem-solving and problem-posing tasks and activities can assist students to develop more creative approaches to mathematics. Through the use of such tasks and activities, teachers can increase their students' capacity with respect to the core dimensions of creativity; namely, fluency, flexibility, and novelty. Because the instructional techniques discussed in this article have been used successfully with students all over the world, there is little reason to believe that creativity-enriched mathematics instruction cannot be used with a broad range of students in order to increase their representational and strategic fluency and flexibility, and their appreciation for novel problems, solution methods, or solutions.
It is widely recognized that purely cognitive behavior is extremely rare in performing mathematical activity: other factors,
such as the affective ones, play a crucial role. In light of this observation, we present a reflection on the presence of
affective and cognitive factors in the process of proving. Proof is considered as a special case of problem solving and the
proving process is studied adopting a perspective according to which both affective and cognitive factors influence it. To
carry out our study, we set up a framework where theoretical tools coming from research on problem solving, proof and affect
are present. The study is performed within a university course in mathematics education, where students were given a statement
in elementary number theory to be proved and were asked to write down their proving process and the thoughts that accompanied
this process. We scrutinize the written protocols of two unsuccessful students, with the aim of disentangling the intertwining
between affect and cognition. In particular, we seize the moments in which beliefs about self and beliefs about mathematical
activity shape the performance of our students.
A Solomon Four-Group research design was used to investigate the effectiveness of the Selective Problem Solving Model (SPS) on the development of students’ creativity skills in mathematics. The SPS is a highly structured creative problem solving model composed of six steps. The participants of the study included 201 seventh-grade students who were attending a public school. As the principles of the Solomon Four-Group research design, the study included eight classrooms randomly assigned to two experimental and two control groups. A total of 10 SPS lesson plans were developed and used in four experimental classrooms. No intervention was carried out in the control classrooms. The Analogical Problem Construction Test and the Problem Analysis Test were used as the pre-tests and post-tests to measure analogical problem construction and problem analysis skills in mathematics. The findings showed that the experimental groups had significantly higher gain scores than the control groups. Additionally, the experimental groups had higher post-test scores on both measures than did the control groups. The findings can be interpreted that the SPS Model improves students’ creativity skills in mathematics.
Handbook of Organizational Creativity is designed to explain creativity and innovation in organizations. This handbook contains 28 chapters dedicated to particularly complex phenomena, all written by leading experts in the field of organizational creativity. The format of the book follows the multi-level structure of creativity in organizations where creativity takes place at the individual level, the group level, and the organizational level. Beyond just theoretical frameworks, applications and interventions are also emphasized. This topic will be of particular interest to managers of creative personnel, and managers that see the potential benefit of creativity to their organizations. *Information is presented in a manner such that students, researchers, and managers alike should have much to gain from the present handbook *Variables such as idea generation, affect, personality, expertise, teams, leadership, and planning, among many others, are discussed *Specific practical interventions are discussed that involve training, development, rewards, and organizational development *Provides a summary of the field's history, the current state of the field, as well as viable directions for future research.
This paper discusses multiple way problem solving as a habit of mind and claims that it is an effective tool for the development and diagnostics of advanced mathematical thinking. I introduce a notion of solution spaces of a mathematical problem and exemplify the ideas using several problems. The concepts of symmetry and continuity are presented here as important ideas that should be a part of habitual mathematical behavior in the context of advanced mathematical thinking.
We consider developing mathematical creativity in each student as one of the purposes of school mathematics education. Teachers' creativity is not less important. We believe that creative teachers are essential in achieving this goal. A creative teacher advances students' motivation, encourages knowledge construction and promotes students' own creativity. This article presents a part of a larger exploratory study that analyzes teachers' conceptions of creativity in teaching mathematics. Based on the literature review and the analysis of the data we suggest a system of categories for both describing and analyzing teachers' conception of creativity in teaching mathematics. To explain the model we provide several illustrative examples from teachers' discourse about creativity in teaching and from the mathematics lesson that one of the teachers consider to be creative.
This paper describes changes in students’ geometrical knowledge and their creativity associated with implementation of Multiple Solution Tasks (MSTs) in school geometry courses. Three hundred and three students from 14 geometry classes participated in the study, of whom 229 students from 11 classes learned in an experimental environment that employed MSTs while the rest learned without any special intervention in the course of one school year. This longitudinal study compares the development of knowledge and creativity between the experimental and control groups as reflected in students’ written tests. Geometry knowledge was measured by the correctness and connectedness of the solutions presented. The criteria for creativity were: fluency, flexibility, and originality. The findings show that students’ connectedness as well as their fluency and flexibility benefited from implementation of MSTs. The study supports the idea that originality is a more internal characteristic than fluency and flexibility, and therefore more related with creativity and less dynamic. Nevertheless, the MSTs approach provides greater opportunity for potentially creative students to present their creative products than conventional learning environment. Cluster analysis of the experimental group identified three clusters that correspond to three levels of student performance, according to the five measured criteria in pre- and post–tests, and showed that, with the exception of originality, performance in all three clusters generally improved on the various criteria.
How much confidence, based on recent research evidence, can educators have in using creativity training programs in their classrooms? In general, little research has been done since the significant reviews conducted 5 years ago, and much of the research is inaccessible to teachers. Research continues to focus primarily on using strategies in classrooms. Some research is beginning to look at aspects of the creative person, product, and environment, but more could be done to look at the interactions among these. The continued emphasis on classroom strategies raises concerns about the failure by researchers to address issues of strategy transfer to other environments.
At the K-12 level one assumes that mathematically gifted students identified by out-of-level testing are also creative in their work. In professional mathematics, “creative” mathematicians constitute a very small subset within the field. At this level, mathematical giftedness does not necessarily imply mathematical creativity but the converse is certainly true. In the domain of mathematics, are the words creativity and giftedness synonyms? In this article, the constructs of mathematical creativity and mathematical giftedness are developed via a synthesis and analysis of the general literature on creativity and giftedness. The notions of creativity and giftedness at the K-12 and professional levels are compared and contrasted to develop principles and models that theoretically “maximize” the compatibility of these constructs. The relevance of these models is discussed with practical considerations for the classroom. The paper also significantly extends ideas presented by Usiskin (2000).
This study aims to deepen our understanding of the development of teacher knowledge in systematic (through learning) and craft (through teaching) modes. The main research tools of this study were multiple-solution connecting tasks. We used the notion of solution spaces to analyze the data and demonstrated that modifications of the teachers’ solution spaces were situated in their practices of varying types. We found that the implementation of multiple-solution connecting tasks in systematic mode meaningfully developed teachers’ problem-solving performance due not only to the reproduction of solutions offered during the course but to the production of new solutions. Furthermore, in creating opportunities for students to solve tasks with multiple solutions, teachers expanded their personal solution spaces. We conclude that the combining of systematic and craft modes is an optimal condition for the development of teachers’ knowledge.
Summarizes the results of 142 studies testing approaches to teaching children to think creatively. The most successful approaches seem to be those that involve both cognitive and emotional functioning; provide adequate structure and motivation; and give opportunities for involvement, practice, and interaction with teachers and other children. It is noted that motivating and facilitating conditions make a difference in creative functioning, but differences seem to be greatest and most predictable when deliberate teaching is involved. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
In spite of the neglect of the study of creativity specifically within the subject of school mathematics, the notion of creativity is apparently considered by many mathematical educators to be relevant and important in terms of children doing mathematics. Some of the research and literature associated with creativity in school mathematics, mainly from English speaking countries, is reviewed. Particular attention is given to attempts to assess creative ability in school mathematics. Two key aspects emerge: the ability to overcome fixations in mathematical problem-solving, and the ability for divergent production within mathematical situations. It is proposed that these might form the basis for a framework for fostering and rewarding mathematical creativity in schoolchildren.
This article explores aspects of a unified psychological model for mathematical learning and problem solving, based on several different types of representational systems and their stages of development. The goal is to arrive at a scientifically adequate theoretical framework, complex enough to account for diverse empirical results but sufficiently simple to be accessible and useful in mathematics education practice. Some perspectives on representational systems are discussed, and components of the model are described in relation to these ideas—including constructs related to imagistic thinking, heuristics and strategies, affect, and the fundamental role of ambiguity.
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