The aim of this research is examining the pedagogical content knowledge of high school mathematics teachers with regard to derivations and their implementations in context of subject matter, student and teaching strategies knowledge. In this research, case study method of the patterns of qualitative research methodology was used. Participants of this research conducted at the end of the fall semester and at the beginning of the spring semester in 2015-2016 school year were four high school mathematics teachers officiated in general high schools under the Ministry of National Education in a city in Black Sea Region in Turkey. Data collection tools of this research were interview, observation and document analysis. Descriptive analysis and content analysis techniques were used to analyze the data of this research. It was seen that the teachers mostly had difficulty in explaining the geometrical meaning of the derivative. It can be said that all of the teachers determined the errors made by the student except the derivative of the closed function completely and correctly. It can be said that high school mathematics teachers who answered the questions in the context of student knowledge mostly succeeded in determining the mistake and source and taking precautions to correct the mistake. It was seen that teachers mostly use direct instruction method, question-answer technique and expository teaching strategy in teaching of derivatives and their implementations. The subject matter knowledge of high school mathematics teachers regarding derivatives and their implementations differentiate with respect to professional experience, it did not differ with respect to gender and the type of faculty that they graduated from. According to this, it can be said that the knowledge level of high school mathematics teachers regarding the subject matter of derivatives and their implementations increases as their professional experience increases. It was seen that the mathematics teachers' knowledge about student and teaching strategies with regard to derivatives and their implementations do not differentiate in terms of gender, type of the faculty that they graduated from and professional experience. It was seen that the subject matter knowledge of high school mathematics teachers regarding derivatives and their implementations is interrelated with their knowledge of students and teaching strategies.

This study attempted to investigate a pre-service mathematics teacher’s Technological Pedagogical Content Knowledge regarding different views of 3-D figures in geometry. Data were collected from a pre-service teacher, enrolled in one of the large public universities in Ankara while she was teaching different views of 3-D figures in seventh-grade level in one of the private schools. Interviews, observations, field notes, GeoGebra files, and lesson plans were the data collection tools used in the study. Data analysis revealed that the participant’s TPACK level increased during school experience course. State differently, there was a development in the pre-service teacher’s TPACK level. While there was just a slight increase in technology integration to the curriculum and access of students to technology levels, there was a significant rise in teaching and learning with technology levels of the participant during school experience course. Findings of the study show that there is a need for complementary technology courses during school experience course in teacher education programs because teacher candidates need necessary competencies in using technology like selection and usage of appropriate technology to provide effective teaching in mathematics classes. Keywords: Pre-service Middle School Mathematics Teacher, Technological Pedagogical Content Knowledge, Different Views of 3-D Figures

Unlike high school teachers, studies for pedagogical content knowledge were often examined with prospective mathematics teachers in the literature. In spite of taking course for the formation towards teaching profession and doing internship at schools during undergraduate education, prospective mathematics teachers do not gain experience sufficiently. Therefore, the informations obtained from the prospective mathematics teachers do not allow to be determined the pedagogical content knowledge in the context of subject matter knowledge of high school mathematics teachers. Thus, the present study was conducted with the teachers in the teaching profession in person and involved in the teaching process. The purpose of this study is to investigate high school mathematics teachers’ pedagogical content knowledge in the context of subject matter knowledge on derivatives. The case study design of the qualitative research models was used in this research. The research was carried out in a small-scaled city of the Black Sea Region of Turkey during the fall semester of 2015-2016 academic terms. The participants of the research were four mathematics teachers working in public high schools connected to the ministry of national education. The maximum variation sampling of purposeful sampling method was used when determining the teachers to join the study. The data collection tool was a semi-structural interview protocol developed by the researchers. The studies examining derivatives and pedagogical content knowledge were considered in the development process of the interview protocol. The research data was analyzed by content analysis. The researchers considered the framework specified by Zazkis and Leiken (2008) and then expanded by Gokkurt (2014) in the identification process under categories and encodes of the data obtained defining derivatives, giving examples to derivatives, visualizing the definition of derivative and explaining the relationship between derivatives and continuity. When the high school teachers’ detailed definitions to derivatives were examining, all teachers had necessary but not sufficient definitions for derivatives. When the teachers’ detailed examples to derivatives from everyday life were examining, these examples have prototype features such as instantaneous change, rate of change and exchange ratio. All given examples by the teachers are included in the field of academical instruction and mathematics education books. Howewer, the examples such as instantaneous change, rate of change given by O1, O2 and O4 teachers and again the examples such as exchange ratio given by O1 are incomplete and not fully accurate examples. On the other hand, it is just an example of a non prototype given by O3 teacher. When the teachers’ detailed drawing examples for the definition of derivatives were examining, all examples drawn by the teachers are the prototype drawings included in the field of mathematics textbooks and academical instruction books. The drawings made by O1 and O2 teachers are related to the relationship between derivative and rate of change. The drawings made by O3 and O4 teachers are related to the relationship between derivative and slope. When the teachers’ detailed views for the relationship between derivatives and continuity were examining, the statements made and the examples given by O1 and O2 teachers are at the sufficient level. Accordingly, O1 and O2 teachers explained the derivatives-continuity relationship and done the right drawing examples for this relationship. On the other hand, O3 and O4 teachers expressed necessary but insufficient critical features related to the derivatives-continuity relationship.The function has a significant impact in understanding the concepts of limit, continuity and derivatives. Teachers should pay attention to the preconditionality principle in explaining the issue. It should focus on the function and limit concepts intensively to tell without possibility of misconceptions the relationship between derivatives and continuity accurately. The present study was conducted with high school mathematics teachers. Hence, the qualitative and quantitative studies may be done related to the subject matter knowledges of the mathematics teachers and preservice mathematics teachers in the issues such as the function, limit, applications of derivatives and integration. The teachers’ subject matter knowledges for derivatives may be investigated with the help of more data collection tools such as the interviews, observations and document review in qualitative studies. In order to increase the levels of the teachers’ subject matter knowledges, more permanent learnings are carried out through micro teaching at the end of the final evaluation.

The present study aims to analyze pre-service elementary school mathematics teachers’ knowledge for teaching algebra in the context of algebra pedagogical content knowledge. For this purpose, a valid and reliable test (Algebra Pedagogical Content Knowledge Test) of 20 questions on the basis of the conceptual framework of knowledge for teaching algebra, as built by McCrory et al. (2012), was developed and applied with 101 senior-year pre-service teachers enrolled at a state university. The results reveal that the pre-service teachers exhibit mediocre performance in terms of algebra pedagogical content knowledge. The pre-service teachers were observed to experience difficulties regarding sub-components of algebra pedagogical content knowledge, namely student recognition.They also exhibited difficulty with the “functions and their properties: linear and nonlinear” in algebra content and the “core concepts and procedures” in the domains of mathematical knowledge. Bu çalışmada ilköğretim matematik öğretmeni adaylarının cebir öğretme bilgilerinin cebir pedagojik alan bilgisi bağlamında incelenmesi amaçlanmıştır. Bu amaçla McCrory vd. (2012) tarafından geliştirilen cebir öğretme bilgisi kavramsal çatısı temele alınarak geçerlik ve güvenirliği sağlanmış 20 soruluk bir test (Cebir Pedagojik Alan Bilgisi Testi) geliştirilmiş ve bir devlet üniversitesinin son sınıfında öğrenim görmekte olan 101 öğretmen adayına uygulanmıştır. Elde edilen sonuçlar öğretmen adaylarının cebir pedagojik alan bilgisi açısından orta düzeyde başarı sergilediklerini ortaya koymaktadır. Öğretmen adaylarının özellikle cebir pedagojik alan bilgisinin öğrenciyi tanıma, cebir içeriğinin doğrusal-doğrusal olmayan fonksiyonlar ve özellikleri ile matematiksel bilgi içeriğinin ana kavramlar ve prosedürler alt bileşenlerinde güçlükler yaşadığı tespit edilmiştir.

The latest official national curriculum published by the Turkish Ministry of Education, now formally requires high-school mathematics teachers to actively incorporate computer software in their teaching. The primary purpose of this study is to demonstrate the development of an inquiry based learning unit especially geared for high school mathematics students and teachers for the general concept of integral calculus. The main theme chosen as a case for this proposed inquiry unit, is on volumes of solids of revolution of real life daily objects. As a result, the primary purpose will provide a report about a practical example of using pedagogically driven mathematics software, 3D digital modeling coupled with hands-on real life examples all embedded in a constructivist learning environment. A secondary goal of this study is to expose pre-service high-school mathematics teachers to this inquiry unit as a teacher and student. Finally, the shared experiences about the interconnected nature of knowledge construction through a double “lens”, that is as a teacher and students through collaboratively building, testing and reflecting on their learning process will be reported. Keywords: Mathematic Education, Inquiry, Modeling, Teacher Education

The aim of this study was to investigate primary student teachers' perspectives of the teaching of fractions, i.e. their PCK of fractions. The research design used for the study was a descriptive survey method. As data collection instrument, we conducted a questionnaire composing of 14 open and closed-ended questions. The questionnaire was administered to 126 third grade primary student teachers at the department of primary teacher education in Sinop University. There were 91 females and 35 males. In the analysis of open-ended questions, the participants' responses were qualitatively analysed to characterize patterns and categorize answers. Questionnaires were analysed by using open coding. The results of the study indicated that, when saying fractions, first thing which comes in the student teachers' mind is the meaning " part-whole " of fractions. By supporting on the relevant literature, it can be asserted that, only this meaning is not sufficient to provide better understanding of fractions among students. In parallel, activities that most of the student teachers prefer for the introduction to fractions are closely related to this meaning such as dividing a cake, a breath, etc. into equals parts or shading a region.

This research aims to adapt the Pedagogical Knowledge and Skills in Teaching (PKST) survey developed by Wong, Chong, Choy and Lim (2012) to Turkish. The participants of the study are 830 4th year students of education faculty studying at two different public universities. Explanatory Factor Analysis (EFA) was conducted with total 205 pre-service teachers in which 110 (53. 6%) of female and 95 (46,4 %) of male. Confirmatory Factor Analysis (CFA) was also carried out with total 625 preservice teachers in which 330 (52. 8 %) of them are female, 295 (47. 2 %) of them are male. Back translation was used to ensure language validity. EFA and CFA were conducted for the construct validity and to ensure psychometric characteristics of measurement tool. EFA show that survey has six factors and 37 items. Coefficient was 0.94 for the survey whereas it ranged between 0.70 and 0.88 for its factors. The analyses and findings show that the survey is a valid and reliable data collection tool. Keywords: Pedagogical knowledge and skills, Pre-service teachers, Reliability, Validity, Survey adaptation Pedagojik bilgi ve beceri ölçeğinin Türkçe’ye uyarlanması: Geçerlik ve güvenirlik çalışması ÖZ Araştırmada Wong, Chong, Choy ve Lim (2012) tarafından geliştirilerek, geçerlik ve güvenirlik çalışması yapılan öğretmen adaylarının Öğretimde Pedagojik Bilgi ve Beceri (ÖPBB) ölçeğinin Türkçeye uyarlanması amaçlanmıştır. Çalışma grubunu iki farklı devlet üniversitesinin eğitim fakültesinin son sınıfında öğrenim gören 830 öğretmen adayı oluşturmuştur. Açımlayıcı Faktör Analizi (AFA), 110 (53,6%)’u kadın ve 95 (46,4 %)’’i erkek olmak üzere toplam 205 öğretmen adayından elde edilen verilerle gerçekleştirilmiştir. Doğrulayıcı Faktör Analizi (DFA), 330 (52,8 %) ‘u kadın ve 295 (47,2 %)’i erkek toplam 625 öğretmen adayı üzerinde gerçekleştirilmiştir. Dil eşdeğerliğini test etmek için İngilizce- Türkçe; Türkçe-İngilizce geri çeviriler uygulanmıştır. Yapı geçerliği kapsamında AFA ve DFA’dan yararlanılmıştır. AFA sonucunda ölçeğin altı boyut ve 37 maddeden oluştuğu tespit edilmiştir. DFA sonucunda AFA’da elde edilen yapı doğrulanmıştır. Ölçme aracının güvenirliği için Cronbach Alpha güvenirlik katsayı ölçeğin tümünde 0.94 iken alt boyutlarında 0.70 ile 0.88 arasında değerler almıştır. Ölçme aracının geçerli ve güvenilir ölçüm yapabilen bir veri toplama aracı olduğu söylenebilir. Anahtar Kelimeler: Pedagojik bilgi ve beceri, Öğretmen adayları, Geçerlik, Güvenirlik, Ölçek uyarlama

Pedagogical Content Knowledge (PCK) has been considered to be the most important factor for identifying the role of the teacher in carrying out effective teaching. This paper examines, with respect to the components of PCK, the types of difficulties that a group of pre-service teachers encountered during the process of designing teaching activities. For this purpose, 8 pre-service teachers (4 elementary mathematics and 4 classroom teachers) are observed whilst teaching mathematics during the Teaching Practicum course and are interviewed. The study reveals that both groups of pre-service teachers had similar problems in teaching practices. Pre-service teachers experienced difficulties in determining the activities with regard to prior knowledge of the students, deciding the difficulty level of the activities and sorting them, informing students about the purpose of the activities, conducting activities according to their perceived purposes and making transitions between activities. ÖZ: Alanı Öğretme Bilgisi (AÖB) etkili öğretim yapan bir öğretmenin sahip olduğu en temel bilgilerdendir. Bu çalışma, AÖB'nin bileşenleri bağlamında öğretmen adaylarının karşılaştığı zorlukları incelemeyi amaçlamaktadır. Bu amaç doğrultusunda 4'ü sınıf, 4'ü ilköğretim matematik olmak üzere 8 öğretmen adayı Öğretmenlik Uygulaması dersi kapsamında gözlenmiş ve bu adaylarla mülakatlar yapılmıştır. Çalışmada her iki gruptaki öğretmen adaylarının benzer problemler yaşadıkları ortaya konulmuştur. Bu bağlamda, öğretmen adaylarının öğrenciye ulaşma, öğrenci önbilgisine uygun etkinlik belirleme ve uygulama, etkinlikleri zorluk derecesine göre sıralama, öğrencileri etkinliklerin amacından haberdar etme, etkinlikleri toparlama ve etkinler arası geçiş yapma gibi hususlarda zorluklar yaşadıkları sonucuna varılmıştır. Anahtar sözcükler: Alanı öğretme bilgisi, öğretmen adayı, öğrenci etkinliği.