The Postgraduate Education of Teachers and its Effects on Their Instructional Activities

This study investigates the contributions of the postgraduate education to in-service training of teachers and its reflections into classroom setting. For this purpose, volunteer teachers and courses mostly chosen by them were selected and skills these courses aim to provide were determined. The sampled teachers were also observed and interviewed. The study highlighted coherence between the qualities that teachers supposed to have and the skills aimed to be gained via this education and therefore, confirmed that postgraduate education contributes to in-service training of teachers. Finally, although teachers do their best to put these skills into practice, some skills can not be applied properly due certain difficulties explained in this article

Semiotic representations skills of prospective elementary teachers related to mathematical concepts

The aim of the study is to introduce Duval's Theory of Registers of Semiotic Representations, and with the framework of this theory to analyze how students perceive and apply different usages of the representations for the same concept and transformations between these representations. The study was carried out with 28 students from Artvin Coruh University the Faculty of Education Classroom Teacher Education program in 2007-2008 educational year. For the study, activities about identities, equations and functions were prepared and applied. Students worked as groups of two. As soon as they completed the activity all groups were interviewed in unstructured way. Students particularly can be said to have advanced transformation skills in algebraic register of representation. However, the students were not observed as good at skills of passing from verbal and graphical representations to algebraic one, and from tables to verbal representation. (C) 2009 Elsevier Ltd. All rights reserve

11. Sınıf Öğrencilerinin Matematiksel Düşünmenin Aşamalarındaki Yaşantılarından Yansımalar

This study aimed to reveal the experiences of 11th grade students related to specializing, generalizing, conjecturing and proving stages of mathematical thinking. Worksheets, each consisting of 9 questions included the stages of mathematical thinking, and this pilot study was applied to 24 students. The results of the study demonstrated that student achievement decreased as the stages of mathematical thinking progressed. From this point of view, students were found to demonstrate a good performance in specializing stage and to have a big difficulty in proving stage. Moreover, the students’ answers in generalizing and conjecturing stages were observed to accumulate under verbal and algebraic codes and in proving stages the answers were found to accumulate under the codes of arithmetic, geometric and algebraic. Suggestions are made based on the results of the study.Bu çalışmada, nitel araştırma yaklaşımı kullanılarak 11. sınıf öğrencilerinin matematiksel düşünmenin özelleştirme, genelleme, varsayımda bulunma ve ispatlama aşamalarıyla ilgili yaşantılarını ortaya çıkarmak amaçlanmıştır. Matematiksel düşünmenin aşamalarını dikkate alan ve her biri dokuzar sorudan oluşan çalışma yaprakları geliştirilmiş ve pilot çalışmadan sonra 24 lise öğrencisine uygulanmıştır. Çalışmanın sonuçları matematiksel düşünmenin aşamaları ilerledikçe öğrenci başarısının düştüğünü ortaya koymuştur. Bu bakımından, öğrencilerin özelleştirmede iyi performans sergiledikleri, ispatlamada ise büyük sıkıntı çektikleri tespit edilmiştir. Ayrıca, genelleme ve varsayımda bulunma aşamalarında öğrencilerin cevaplarının sözel ve cebirsel, ispatlama aşamasında ise aritmetik, geometrik ve cebirsel kodları altında toplandıkları belirlenmiştir. Çalışmanın sonuçlarına dayanarak çeşitli öneriler sunulmuştur

Discussion contents during a lesson study conducted with knowledgeable others

Over the past two centuries, lesson study has been applied in different countries with different participant profiles. Lesson study is defined as a dynamic research-learning cycle of planning, applying, observing, and analyzing a lesson. Although the participants' discussions in lesson study practices were examined from different perspectives, they have not been analyzed in depth in terms of their contributions to teachers' and preservice teachers' professional development. In this study, the content and frequency of participants' discussions are examined. Six weeks of research were conducted by an academician, a teacher, and three mathematics pre-service teachers. The data were collected through field notes, video recordings, interviews, and focus group interviews. A total of eleven major headings were identified in the lesson study, concerning pedagogical and mathematical issues. The study also concluded that while classroom management was the most widely discussed topic, group and individual instructional techniques received less attention

AN EXAMINATION OF SECONDARY SCHOOL MATHEMATICS TEACHERS’ OPINIONS ON MATHEMATICAL COMMUNICATION SKILLS

Matematik eğitiminin belki de en önemli amaçlarından biri sosyal bağlamda düşüncelerini ve akıl yürütmelerini aktarabilen özerk öğrenciler yetiştirmek olduğundan matematik ve matematik eğitiminin ayrılmaz bir parçası olan matematiksel iletişim becerisi, matematik öğretim programlarında da önemli bir yer tutar. Matematiksel iletişimin hem okul hayatında hem de günlük yaşamdaki önemi düşünüldüğünde öğrencilerin bu becerilerinin gelişiminin sağlanmasının gerekliliği ortaya çıkmaktadır. Bu hedefe ulaşmada ise öğretmenlerin rolü büyüktür. Dolayısıyla bu çalışmada öğretmenlerin matematiksel iletişim becerisi ile bu becerinin gelişimine ve göstergelerine yönelik düşüncelerinin incelenmesi amaçlanmıştır. Özel durum çalışması yönteminin kullanıldığı araştırma, 15 ortaokul matematik öğretmeni ile yarıyapılandırılmış görüşmeler aracılığıyla yürütülmüştür. Elde edilen veriler içerik analizi yöntemi ile analiz edilmiştir. Çalışmanın sonucunda öğretmenlerin matematiğe bir dil olarak bakabildikleri, matematiksel iletişim matematik dilini etkili ve doğru kullanması gerektiğini düşündükleri, büyük çoğunluğunun bu becerinin geliştirilmesini önemseyip öğretim ortamlarını bu beceriyi dikkate alarak düzenlediklerini ve öğretmenlerin öğrencilerinin değerlendirmek için en çok yazılı sınavları tercih ettiklerini belirttikleri görülmüştür. Ayrıca, öğretmenlerin matematiksel iletişim becerisinin göstergesi olan alt beceriler arasında en fazla; verilen matematiksel ifadeyi (tanımları, terimleri, işlemleri, sembolleri vb.) anlamayı vurguladıkları, bunun yanı sıra sembolleri anlama, sembolleri yazılı ve sözlü açıklama ile sembolleri doğru şekilde kullanmanın üzerinde önemle durdukları ortaya çıkmıştırAs possibly one of the most important purposes of the mathematics education is to raise autonomous students who can convey their thoughts and reasoning within the social context, mathematical communication skill which is an inseparable part of mathematics and mathematics education has a place in mathematics curricula. When considering the importance of mathematical communication both in the school life and daily life, it is apparently required to help improve these skills of students’. Teachers play a key role in achieving this objective. Hence, this study aims to examine teachers’ opinions on mathematical communication skills and development and indicators of this skill. Using the case study method, the research was conducted with 15 secondary school mathematics teachers through semi-structured interviews. The data obtained were analyzed with the content analysis method. It was observed in the study that the teachers can regard mathematics as a language, think that they should use the mathematical language efficiently and properly in the first place so students can acquire the mathematical communication skills; majority of them care about the development of this skill and arrange the learning settings in accordance with this skill; and the teachers prefer the written examination the most for assessing students’ progress in their mathematical communication skills. It was also discovered what the teachers emphasize the most is understanding the given mathematical expression (definitions, terms, operations, symbols, etc.) among the subskills which are the indicators of mathematical communication skills and that they accentuate understanding the symbols, explaining the symbols in written and orally and using the symbols properly Communication is a way for sharing thoughts and clarifying comprehension (National Council of Teachers of Mathematics [NCTM], 2000). Each discipline has its idiosyncratic language in consideration that conveyed information, thoughts and skills in every discipline may vary. Mathematics is a universal language with its own symbols and terminology and significant relations between its concepts (Ministry of National Education [MoNE], 2013). Students add meaning to their experiences via language. Mathematical thinking starts developing naturally among children in the pre-primary school period. Children make sense of their surroundings by the means of observations and communications in the social environment they are in (MoNE, 2017). It is therefore impossible to isolate language in daily life and mathematics learning. Students need to use their own language to convey what they have found while exploring mathematics and thinking mathematically to others and to clarify their findings through their observations. Proper and effective use of language encourages learning in mathematics. Playing a critical role in expressing mathematical ideas alongside mathematical symbols and drawings, language functions as a bridge in the transition between abstract and concrete notations (NSW Department of School Education [NSWDoSE], 1989). In other words, mathematical language is an instrument used in mathematical communication, mathematical thinking and the process of instructing the mathematical concepts (Jamison, 2000; Mercer & Sams, 2006). As possibly one of the most important purposes of the mathematics education is to raise autonomous students who can convey their thoughts and reasoning within the social context (Pourdavood, Svec, Cowen & Genovese, 2005), mathematical communication skill (MCS) which is an inseparable part of mathematics and mathematics education is emphasized significantly in mathematics curricula in Turkey and many other countries (i.e. USA, UK, Canada [Ontario], Singapore and New Zealand) (MoNE, 2013, 2017; NCTM, 1989; NCTM, 2000; NSWDoSE, 1989; Singapore Ministry of Education [SMoE], 2007; The National Curriculum for England [NCfE], 1999; The New Zealand Ministry of Education [NZMoE], 2009; The Ontario Ministry of Education [OMoE], 2005). When considering the importance of mathematical communication both in the school life and daily life, it is apparently required to help improve these skills of students’. It is highlighted by several studies and institutions that teachers have a great responsibility for enabling the development of MCS among students. Cooke and Buchholz (2005) state that teachers need to help students express their thoughts in a detailed way so that they can establish a connection between mathematics and language and this connection can be reflected in the best way possible. In addition, they lay emphasis on the idea that such kind of interactions enable students to clarify their own thoughts and develop their own understanding while trying to understand to comprehend their worlds through communication. When students asked to express what they do and think of, this does not only clarify and improve their own understanding but also help them convey their levels of understanding to teachers (NSWDoSE, 1989). Hence, teachers are liable to use several methods to enable students to communicate about mathematics and assess their progress (Thompson and Chappell, 2007). Even though it is highlighted in several studies and by several institutions that teachers have great responsibility for the development of students’ MCS, it is remarkably observed in the literature that studies are needed for the teacher attention to the skill and assessment of students’ MCS and progress. In the light of these considerations, this study aims to discover how secondary school mathematics teachers assess MCS, its importance and students’ progress in MCS and to examine teacher opinions on the indicators of MCS.This research which used the case study method was conducted with 15 volunteered mathematics teachers who are serving at secondary schools. Semi-structured interviews were performed to collect data in the research. In these interviews, semi-structured questions were asked to the teachers about what mathematical communication is and why it is important, the role of teacher in having student acquire MCS, what they do for monitoring students’ progress and what subskills can be the indicators of this skill. The questions were rearranged upon three expert opinions. The data obtained in the interviews were analyzed with the content analysis method. Four themes emerged at the end of the analysis of the data obtained in the semi-structured interviews which were performed with the teachers to enlighten the research question: ‘mathematical communication skill and its importance’, ‘teachers’ role in the development of mathematical communication skill’, ‘assessment of progress in mathematical communication skill’, and ‘subskills as the indicators of mathematical communication skill.’ At the end of the study which aimed at examining teachers’ opinions on MCS, it was discovered that the teachers can regard mathematics as a language and they emphasize mathematics is a universal language; they find MCS to be important so students can maintain their daily life and associate things and solve problems in their academic life; they think that they should use the mathematical language efficiently and properly in the first place so students can acquire the mathematical communication skills; majority of them care about the development of this skill and arrange the learning settings in accordance with this skill; and the teachers prefer the written examination the most for assessing students’ progress in their mathematical communication skills. Mathematical concepts are interconnected and hierarchic as mathematics is a domain with a priority-posteriority relation. Since a concept or topic to be learned is associated with prior pieces of information, proper and effective use of mathematical language is important both for learning and mathematical thinking (Raiker, 2002). There are findings in the literature on how students have difficulty in explaining several words which their teachers often use (Otternburn and Nicholson, 1976). Then, proper use of mathematical language by students needs to be supported through educational applications designed in instructional settings. In the study, reading was the least emphasized subskill and it was observed that the teachers indirectly approach the subskill of listening. However, according to Kane, Byrne and Hater (1974), mathematical texts at school are based on explanation, description and instruction. These texts later encourage students to use them (in Thompson and Chappell, 2007). Reading and listening therefore involve both transfer and comprehension. Furthermore, reading in mathematics, or word, number, symbol and graphic reading, lead students to do mathematical things on the top level of strong mathematical comprehension (Adams and Lowery, 2007). Teachers need to pay more attention to these important skills, and activities in which students read by adding meaning rather than reading texts passivelyuse MCS effectively and properly and provide their students with this skill in the classroom, one should also attach importance to the development of the skill among preservice teachers and how they are informed of providing this skill so that qualified teachers can be trained need to be performed. Due to the importance of the fact that teachers ca

The Impact of Book Reading on Students’ Problem Solving Skills and Their Mathematics Success

In this study, it is aimed to determine the effect of students’ reading levels on their success rates in mathematics class and on their mathematical problem solving skills. 74 students from two different secondary schools were separated into levelled reading groups after determining the number of books they have read so far and then their academic success rates in mathematics and their mathematical problem solving skills were determined. A problem solving success test consisting of 20 questions prepared with the help of a specialist, a form to determine the number of books read so far, and students’ mathematics class written exam scores were used as data gathering tools. A quantitative analysis was used in analysing the correlations of gathered data. Data analysing was made by mean of one-way independent sample variant analysis (one way ANOVA). At the end of the study, it was found out that extensive reading had no meaningful impact on students’ mathematic success rates and on their problem solving skills and steps

Examining the Effect of Lesson Study on Prospective Primary Teachers’ Knowledge of Lesson Planning

This article reflects a special part of a research conducted to examine the effect of lesson study on prospective classroom teachers’ mathematical pedagogical content knowledge (MPCK). In this article, the special part consists of prospective teachers’ knowledge of lesson planning including a mastery of planning an affective lesson taking into account student’s current knowledge, understanding and difficulties within mathematics. Therefore, the research question is how lesson study practices affect prospective classroom teachers’ knowledge of lesson planning as a sub component of MPCK. The research is conducted with 12 prospective classroom teachers, six of them have already assisted to lesson study and the others have not. Data collection tools consist of video records, class observations, field notes, interviews and lesson plans prepared and used by prospective teachers participated in lesson study. Findings indicated that the prospective classroom teachers who participated in lesson study improved their knowledge in terms of planning an affective lesson taking student’s current knowledge and understanding into consideration. They appeared to be aware of selecting and ordering appropriate activities related to the actual objectives of the mathematical topics. They also appeared to be better in lesson organization and lesson presentation comparing to the other group of prospective teachers who did not participated in lesson study

Reflections from the Experiences of 11th Graders during the Stages of Mathematical Thinking

Bu çalışmada, nitel araştırma yaklaşımı kullanılarak 11. sınıf öğrencilerinin matematiksel düşünmenin özelleştirme, genelleme, varsayımda bulunma ve ispatlama aşamalarıyla ilgili yaşantılarını ortaya çıkarmak amaçlanmıştır. Matematiksel düşünmenin aşamalarını dikkate alan ve her biri dokuzar sorudan oluşan çalışma yaprakları geliştirilmiş ve pilot çalışmadan sonra 24 lise öğrencisine uygulanmıştır. Çalışmanın sonuçları matematiksel düşünmenin aşamaları ilerledikçe öğrenci başarısının düştüğünü ortaya koymuştur. Bu bakımından, öğrencilerin özelleştirmede iyi performans sergiledikleri, ispatlamada ise büyük sıkıntı çektikleri tespit edilmiştir. Ayrıca, genelleme ve varsayımda bulunma aşamalarında öğrencilerin cevaplarının sözel ve cebirsel, ispatlama aşamasında ise aritmetik, geometrik ve cebirsel kodları altında toplandıkları belirlenmiştir. Çalışmanın sonuçlarına dayanarak çeşitli öneriler sunulmuştur.This study aimed to reveal the experiences of 11th grade students related to specializing, generalizing, conjecturing and proving stages of mathematical thinking. Worksheets, each consisting of 9 questions included the stages of mathematical thinking, and this pilot study was applied to 24 students. The results of the study demonstrated that student achievement decreased as the stages of mathematical thinking progressed. From this point of view, students were found to demonstrate a good performance in specializing stage and to have a big difficulty in proving stage. Moreover, the students’ answers in generalizing and conjecturing stages were observed to accumulate under verbal and algebraic codes and in proving stages the answers were found to accumulate under the codes of arithmetic, geometric and algebraic. Suggestions are made based on the results of the study