Ortaokul öğrencilerinin cebirsel kavramları soyutlama süreçlerinin incelenmesi
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Date
2021-02-05
Authors
Gürbüz, Mustafa Çağrı
Journal Title
Journal ISSN
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Publisher
Bursa Uludağ Üniversitesi
Abstract
Öğrencilerden günümüzde çok bilgiye sahip olmalarından ziyade mevcut bilgilerini problem çözme, akıl yürütme, iletişim gibi matematiksel düşünmenin temelini oluşturan becerileri kullanarak çeşitli amaçları gerçekleştirmeleri beklenmektedir. Çalışmada matematiksel düşünme, öğrencinin matematik kavramlarını zihninde nasıl yapılandırdığını açıklamaya yönelik olarak ele alınmıştır. Bu yapılandırma süreci ise matematiksel soyutlama çerçevesinden izlenmiştir. Matematiksel soyutlama, matematik kavramının genelleştirilmesi yoluyla kavrama daha kapsamlı bir uygulama alanı oluşturulması, diğer bir ifade ile kavramın özünü ortaya çıkarması işlemidir. Öğrencilerin zihninde bilginin oluşum sürecini doğrudan gözlemlemek oldukça zor bir durum olarak karşımıza çıkmaktadır. Bilginin öğrencinin zihninde nasıl oluştuğu, soyutlandığı ve hangi içsel süreçlerden geçtiği bilinirse öğrenme sürecine etkili müdahalelerde bulunmak kolaylaşacaktır. Bu çalışmada, ortaokul öğrencilerinin temel cebir kavramlarına yönelik soyutlama süreçlerinin incelenmesi amaçlanmıştır. Soyutlama süreçlerinin incelenmesinde RBC+C (Recognizing, Building with, Construct, Consolidation) teorisinde yer alan epistemik eylemler dikkate alınmıştır. Ayrıca araştırma sürecinde öğrencilerin soyutlama süreçlerinin daha iyi gözlenmesini sağlamak için Tahmini Öğrenme Yörüngeleri kullanılmıştır. Öğrenme yörüngelerinin öğrencilerin temel cebir kavramlarındaki başarılarına etkilerinin ve soyutlama süreçlerine olan yansımalarının tespit edilmesi araştırmada ortaya koyulması amaçlanan diğer bir husustur. Araştırma, belirlenen amaçlara ulaşmak için iki aşamalı tasarlanmıştır. İlk olarak öğrencilerin cebirin iki temel kavramı olan denge ve değişken kavramlarını soyutlama süreçlerini daha net görebilmek ve süreçte onların soyutlama yapmalarını desteklemek amacıyla tasarım tabanlı araştırma modelinden faydalanılmıştır. Tasarım tabanlı araştırma, öğrencilerin soyutlama süreçlerinin daha iyi anlaşılabilmesi için öğrenme ortamına müdahale edilmesine olanak sağlamaktadır. Araştırmanın ikinci aşaması ise bir durum çalışması olarak değerlendirilmiştir. Tasarımın uygulanabilirliği ve eksiklikleri sınıf içi gözlemler yoluyla; öğrencilerin soyutlama becerileri ise yarı yapılandırılmış öğrenci görüşmelerinden elde edilen veriler ile analiz edilmiştir. Katılımcılar Bursa İli, Nilüfer İlçesi’nde bir devlet okulunda öğrenim gören 6. sınıf öğrencileri arasından amaçlı örnekleme ile seçilmiştir. Aynı öğrencilerle 7. sınıf düzeyine geçtiklerinde veri toplama sürdürülmüştür. 2016-2017, 2017-2018 eğitim öğretim dönemlerinde araştırmacı ve öğretmen ile birlikte bu sınıfların matematik derslerinde araştırma gerçekleştirilmiştir. Araştırmada veriler; doküman, gözlem, görüşme veri toplama araçlarıyla veri çeşitlemesi yapılarak toplanmıştır. Gözlem ve görüşme verileri içerik analizine, öğrenme yörüngeleri ise geçmişe dönük analiz sürecine tabi tutulmuştur. Bu çalışma, başarı düzeyi yüksek olan öğrencilerin değişkenler arasındaki doğrusal ilişkiyi tanımlayabildikleri ve denklemleri çözebildiklerini göstermektedir. Öğrencilerin, cebirsel ifade ve doğrusal denklem oluşturma gibi genelleme gerektiren durumlar için verilen tüm bilgileri koordine edebildiği, ayrıca doğrusal model kavramını daha soyut durumlarda oluşturabildikleri ve yeni doğrusal model için bir kural ortaya koyabildikleri görülmüştür. Buna ek olarak, bağlamsal problemlere çözümler bulmak için belirledikleri yöntemleri daha tutarlı kullanabildikleri gözlemlenmiştir. Bu durum soyut düşünebilen öğrencilerin veriyi genelleştirebildiğini ve temsili olarak cebirsel ifade kullanabildiğini göstermiştir. Öğrencilerin bağlam içerisinde karşılaştıkları problemleri matematiksel bir yolla açıklamaları soyutlama sürecini analiz etmelerine yardımcı olmuştur. Bu araştırmada öğrencilerin cebir kavramlarını soyutlama becerileri, problem çözme süreçlerinin ve görüşmelerdeki açıklamalarının epistemik olarak analiz edilmesiyle ortaya çıkarılmıştır. Öğrencilerin uygulama öncesinde daha kısır bir düşünceye sahipken süreçte farklı düşünme yollarının farkına vardıkları, farklı cebirsel düşünme yolları ortaya koydukları, başlangıçta sözel veya aritmetik olarak ifade ettikleri matematiksel durumları cebirsel açıklamalara dönüştürdükleri görülmüştür. Araştırmada soyutlama becerisi ile zihnin cebirsel alışkanlıkları arasında birbirini destekleyici argümanlar bulunmuştur. Öğrencilerin cebir ilişkilerindeki gelişimlerini sağlayan iki matematiksel alışkanlık tespit edilmiştir. Bunlar, işlemleri düzenleyerek bir soyutlamaya ulaşmak ve matematiksel bir dil kullanarak genelleme yapmaktır. Bu alışkanlıklar, öğrencilerin aritmetikten cebire geçmelerini kolaylaştırmıştır. Yapma ve fonksiyonel kural oluşturma alışkanlıklarına sahip olan öğrencilerin cebir kavramlarını soyutlama süreçlerinde daha avantajlı olduğu söylenebilir. Soyutlama sürecinde yeni bir yapı ve matematiksel dilden bahsedildiği için soyutlama sürecindeki ilişkilerin anlaşılması fonksiyonel kural oluşturma alışkanlığına sahip öğrencilerin daha kolay inşa etmelerine olanak sağlamıştır. Cebirsel alışkanlıklarda ise öğrencilerin işlemlerden soyutlama girişimleri genellikle yeni bir dil yerine kısa bir yol bulmak ve açıklayabilmek üzerine inşa edilmektedir. Cebirin iki temel kavramınu öğretmeye yönelik bir yaklaşım üzerine kurulan araştırma, etkili bir cebir eğitimini teşvik etme çabalarını koordine etme ve öğrencilerin düşüncelerindeki önemli kilometre taşlarını belirlemek amacıyla önemlidir. Öğrenme yörüngeleri, öğretmenlere ve uygulayıcılara kendi eğitsel uygulamalarına entegre edilebilmesi için sistematik bir yol sunar. Öğrencilerin cebir kavramlarını soyutlamaları öğretimde etkili bir araç olarak kullanılabileceğine yönelik öğretmenlere hizmet içi eğitimler verilebilir ve soyutlama mekanizması, daha açıklayıcı ve kullanışlı bir biçimde matematik dersi öğretim programlarına yansıtılabilir.
Students are expected to achieve various purposes by using skills such as problem solving, reasoning, and communication, which are the basis of mathematical thinking of their current knowledge, rather than having much knowledge today. In the study, mathematical thinking is discussed to explain how the student constructs mathematics concepts in his mind. This construction process was followed from the framework of mathematical abstraction. Mathematical abstraction is the creation of a more comprehensive application area of comprehension through the generalization of the concept of mathematics. In other words, it is to extract the essence of the concept. It is quite a difficult situation to observe the formation process of information directly in the students' minds. Effective interventions in the learning process will be easier if it is known how the knowledge is formed, abstracted, and what internal processes go through in the student's mind. In this study, it is aimed to analyze the abstraction processes of secondary school students towards basic algebra concepts. Epistemic actions in RBC + C (Recognizing, Building with, Construct, Consolidation) theory were taken into consideration in the analysis of abstraction processes. In addition, Hypothetical Learning Trajectories were used in the research process to provide better observation of students' abstraction processes. Determining the effects of the experimental application on students' achievements in basic algebra concepts and their reflections on abstraction processes is another aim to be revealed in the research. The research is designed as two-stage and longitudinal to achieve the determined goals. Firstly, design-based research model was used in order to see more clearly the processes of abstraction of the concepts of balance and variable, which are the two basic axioms of algebra, and to support them in abstraction in the process. Design-based research allows students to intervene in the learning environment in order to better understand the abstraction processes of students. In this regard, it is believed that it will be more useful than other types of research. Design-based research focuses on learning studies as well as intervention in the learning environment. The second stage of the research was evaluated as a case study. Design applicability and deficiencies through classroom observations; The abstraction skills of the students were analyzed from the data obtained from semi-structured student interviews. Participants were selected by purposeful sampling among 6th grade students studying at a public school in Nilüfer District of Bursa Province. Data collection was continued when they moved to the 7th grade level with the same students. During the 2016-2017, 2017-2018 academic years, researchers and teachers together carried out research in the mathematics classes. In the research, data was collected by data triangulation, data collection tools compatible with the nature of qualitative research, such as documents, observations, and interviews. Observation and interview data were subjected to content analysis, and learning road maps were subjected to historical analysis. This study shows that students with a high level of success can define the linear relationship between variables and solve equations. It was observed that students were able to coordinate all the information given for situations requiring generalization, such as algebraic expression and linear equation creation, as well as create the concept of linear model in more abstract situations and set a rule for the new linear model. In addition, it has been observed that they can use the methods they have determined to find solutions to contextual problems more consistently. This shows that students who can think abstractly can generalize the data and use algebraic expression as a representation. Explaining the problems faced by students in a mathematical way helped them analyze the process of abstraction. In this study, students' skills of abstracting algebra concepts were revealed by epistemic analysis of problem-solving processes and their explanations in interviews. While the students' algebraic habits of mind had a more vicious thought before the application, it was observed that they realized different ways of thinking in the process, presented different algebraic ways of thinking, and initially converted mathematical situations that they expressed verbally or arithmetically into algebraic explanations. In the study, supportive arguments were found between abstraction skills and algebraic habits. Two mathematical habits have been identified that ensure the development of students in algebra relationships. These are to achieve an abstraction by arranging operations and to generalize using a mathematical language. These habits made it easier for students to switch from arithmetic to algebra. It can be said that students who have the habit of making and creating a functional rule are more advantageous in the process of abstracting algebra concepts. As a new structure and mathematical language are mentioned in the abstraction process, understanding the relations in the abstraction process has enabled students with the habit of forming a functional rule to build more easily. In algebraic habits, students' attempts to abstraction from transactions are generally based on finding and explaining a short path rather than a new language. Research, based on an approach to teaching the two basic axioms of algebra, is important to coordinate efforts to promote effective algebra education and to identify important milestones in students' thoughts. Learning trajectories offers teachers and practitioners a systematic way to integrate them into their educational apps. In-service trainings can be given to the teachers that students can be used as an effective tool in teaching abstraction concepts and the abstraction mechanism can be reflected in mathematics curriculum in a more descriptive and useful manner.
Students are expected to achieve various purposes by using skills such as problem solving, reasoning, and communication, which are the basis of mathematical thinking of their current knowledge, rather than having much knowledge today. In the study, mathematical thinking is discussed to explain how the student constructs mathematics concepts in his mind. This construction process was followed from the framework of mathematical abstraction. Mathematical abstraction is the creation of a more comprehensive application area of comprehension through the generalization of the concept of mathematics. In other words, it is to extract the essence of the concept. It is quite a difficult situation to observe the formation process of information directly in the students' minds. Effective interventions in the learning process will be easier if it is known how the knowledge is formed, abstracted, and what internal processes go through in the student's mind. In this study, it is aimed to analyze the abstraction processes of secondary school students towards basic algebra concepts. Epistemic actions in RBC + C (Recognizing, Building with, Construct, Consolidation) theory were taken into consideration in the analysis of abstraction processes. In addition, Hypothetical Learning Trajectories were used in the research process to provide better observation of students' abstraction processes. Determining the effects of the experimental application on students' achievements in basic algebra concepts and their reflections on abstraction processes is another aim to be revealed in the research. The research is designed as two-stage and longitudinal to achieve the determined goals. Firstly, design-based research model was used in order to see more clearly the processes of abstraction of the concepts of balance and variable, which are the two basic axioms of algebra, and to support them in abstraction in the process. Design-based research allows students to intervene in the learning environment in order to better understand the abstraction processes of students. In this regard, it is believed that it will be more useful than other types of research. Design-based research focuses on learning studies as well as intervention in the learning environment. The second stage of the research was evaluated as a case study. Design applicability and deficiencies through classroom observations; The abstraction skills of the students were analyzed from the data obtained from semi-structured student interviews. Participants were selected by purposeful sampling among 6th grade students studying at a public school in Nilüfer District of Bursa Province. Data collection was continued when they moved to the 7th grade level with the same students. During the 2016-2017, 2017-2018 academic years, researchers and teachers together carried out research in the mathematics classes. In the research, data was collected by data triangulation, data collection tools compatible with the nature of qualitative research, such as documents, observations, and interviews. Observation and interview data were subjected to content analysis, and learning road maps were subjected to historical analysis. This study shows that students with a high level of success can define the linear relationship between variables and solve equations. It was observed that students were able to coordinate all the information given for situations requiring generalization, such as algebraic expression and linear equation creation, as well as create the concept of linear model in more abstract situations and set a rule for the new linear model. In addition, it has been observed that they can use the methods they have determined to find solutions to contextual problems more consistently. This shows that students who can think abstractly can generalize the data and use algebraic expression as a representation. Explaining the problems faced by students in a mathematical way helped them analyze the process of abstraction. In this study, students' skills of abstracting algebra concepts were revealed by epistemic analysis of problem-solving processes and their explanations in interviews. While the students' algebraic habits of mind had a more vicious thought before the application, it was observed that they realized different ways of thinking in the process, presented different algebraic ways of thinking, and initially converted mathematical situations that they expressed verbally or arithmetically into algebraic explanations. In the study, supportive arguments were found between abstraction skills and algebraic habits. Two mathematical habits have been identified that ensure the development of students in algebra relationships. These are to achieve an abstraction by arranging operations and to generalize using a mathematical language. These habits made it easier for students to switch from arithmetic to algebra. It can be said that students who have the habit of making and creating a functional rule are more advantageous in the process of abstracting algebra concepts. As a new structure and mathematical language are mentioned in the abstraction process, understanding the relations in the abstraction process has enabled students with the habit of forming a functional rule to build more easily. In algebraic habits, students' attempts to abstraction from transactions are generally based on finding and explaining a short path rather than a new language. Research, based on an approach to teaching the two basic axioms of algebra, is important to coordinate efforts to promote effective algebra education and to identify important milestones in students' thoughts. Learning trajectories offers teachers and practitioners a systematic way to integrate them into their educational apps. In-service trainings can be given to the teachers that students can be used as an effective tool in teaching abstraction concepts and the abstraction mechanism can be reflected in mathematics curriculum in a more descriptive and useful manner.
Description
Keywords
Değişken, Denklem, Matematiksel soyutlama, RBC+C soyutlama teorisi, Öğrenme yörüngesi, Equation, Mathematical abstraction, Learning trajectories, RBC+C abstraction theory, Variable
Citation
Gürbüz, M. Ç. (2021). Ortaokul öğrencilerinin cebirsel kavramları soyutlama süreçlerinin incelenmesi. Yayınlanmamış doktora tezi. Bursa Uludağ Üniversitesi Eğitim Bilimleri Enstitüsü.