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Applications and modelling have gained a prominent role in mathematics education reform documents and curricula. Thus, there is a growing need for studies focusing on the effective use of mathematical modelling in classrooms. Assessment is an integral part of using modelling activities in classrooms, since it allows teachers to identify and manage problems that arise in various stages of the modelling process. However, teachers' difficulties in assessing student modelling work are a challenge to be considered when implementing modelling in the classroom. Thus, the purpose of this study was to investigate how teachers' knowledge on generating assessment criteria for assessing student competence in mathematical modelling evolved through a professional development programme, which is based on a lesson study approach and modelling perspective. The data was collected with four teachers from two public high schools over a five-month period. The professional development programme included a cyclical process, with each cycle consisting of an introductory meeting, the implementation of a model-eliciting activity with students, and a follow-up meeting. The results showed that the professional development programme contributed to teachers' knowledge for generating assessment criteria on the products, and the observable actions that affect the modelling cycle.

The purpose of this study is to assess preservice teachers’ mathematical knowledge for teaching statistics (MKT-S) and identify the relationship between the components of this knowledge. For this purpose, MKT-S instrument consisting of two dimensions, ‘content knowledge’ (CK) and ‘pedagogical content knowledge’ (PCK) was developed, and applied to 659 preservice middle school mathematics teachers (PTs). Confirmatory factor analysis showed that CK and PCK are two different dimensions of mathematical knowledge of teaching statistics. Third year and fourth year preservice teachers’ factor scores were significantly different and fourth year preservice teachers’ factor scores were slightly higher. It was found that CK factor scores were highly correlated with PCK factor scores. The reliability levels were 0.65 for CK factor scores and 0.76 for PCK factor scores. MKT-S instrument developed in this study has several implications for teacher education. MKT-S instrument can be used to evaluate efficiency of PTs’ mathematical knowledge for teaching statistics. Instrument can be adapted for in-service teachers., Bu çalışmanın amacı matematik öğretmen adaylarının istatistik öğretimine yönelikmatematiksel bilgilerinin (MKT-S) ölçülmesi ve bu bilginin yapıtaşları arasındaki ilişkininbelirlenmesidir. Bu amaçla, ‘alan bilgisi’ (CK) ve ‘pedagojik alan bilgisi’ (PCK) olmak üzereiki bölümden oluşan MKT-S ölçeği geliştirilmiş ve 659 ilköğretim matematik öğretmeni adayı(ÖA) üzerinde uygulanmıştır. Doğrulayıcı faktör analizi, CK ve PCK faktörlerinin, istatistiköğretimine yönelik matematiksel bilginin iki ayrı boyutu olduğunu göstermiştir. Üçüncü vedördüncü sınıf öğretmen adaylarının faktör puanları arasında istatistiksel olarak anlamlı farkbulunmuştur ve dördüncü sınıf öğretmen adaylarının faktör puanlarının az da olsa dahayüksek olduğu görülmüştür. CK puanlarının PCK puanlarıyla yüksek derecede ilişkisininolduğu saptanmıştır. Ayrıca CK faktör puanlarının güvenirliği 0,65 düzeyinde iken PCKfaktör puanlarının güvenirliği 0,76 düzeyindedir. Bu çalışmada geliştirilen MKT-S ölçeğininsonuçlarının öğretmen eğitimi için çeşitli çıkarımları bulunmaktadır. Geliştirilen ölçekÖA’larının istatistik öğretimine yönelik matematiksel bilgilerinin etkinliğinideğerlendirmede kullanılabilir ya da çalışan öğretmenlere yönelik adapte edilebilir.

Forming part of a wider research study, the current study investigated prospective middle school mathematics teachers' ways of covariational reasoning on tasks involving simultaneously changing quantities. As the introductory theme of a larger unit on derivative, a model development sequence on covariational reasoning was designed and experimented with 20 participants in a mathematical modeling course offered to prospective teachers. The participants' developing abilities of covariational reasoning were documented under three categories: (i) identifying the variables, (ii) ways of coordinating the variables, and (iii) ways of quantifying the rate of change. The results revealed significant improvement in the prospective teachers' ways of identifying and coordinating the variables, and in quantifying the rate of change. Moreover, the results indicated that preference for a particular way of thinking in identifying and coordinating the variables determined the prospective teachers' way of quantifying the rate of change and thereby their level of covariational reasoning.

This study investigated pre-service secondary mathematics teachers' (PSMTs') anticipation and identification of students' thinking as they engaged in examining and analyzing student work on four modelling problems. The study comprised four two-week-long cycles. The data were collected through individual reflection papers and written group reports on students' ways of thinking from 25 PSMTs enrolled in a semester-long (15-week) elective course. Data analysis focused on the PSMTs' anticipation and identification of students' ways of thinking in the context of modelling problems over the cycles. Results indicated that the PSMTs were able to anticipate and identify some of the students' possible solution approaches, errors, and difficulties for each modelling problem used in the study. However, some of their anticipations remained vague throughout the study. On the other hand, identifying students' different responses helped most of the PSMTs to increase their awareness of students' various ways of thinking throughout the study. The implications of the findings for PSMTs' anticipations, identification, and learning about students' thinking are discussed.

Adopting a multitiered design-based research perspective, this study examines pre-service secondary mathematics teachers' developing conceptions about (a) the nature of mathematical modeling in simulations of real life problem solving, and (b) pedagogical principles and strategies needed to teach mathematics through modeling. Unlike other studies that have focused on single-topic and lesson-sized research sites, a course-sized research site was used in this study. Having been through several iterations over three teaching semesters, the 15-week long course was implemented with 25 pre-service secondary mathematics teachers. Findings revealed that pre-service teachers developed ideas about the nature of mathematical modeling involving what mathematical modeling is, the relationship between mathematical modeling and meaningful understanding, and the nature of mathematical modeling tasks. They also realized the changing roles of teachers during modeling implementations and diversity in students' ways of thinking. The researchers' conceptual development, on the other hand, involved realizing the critical aspect of the teacher role played by the instructor during modeling implementations, and the need for more experience of modeling implementations for pre-service teachers.

The purpose of this study was to examine the potential interrelationships between teachers’ mathematical knowledge for teaching (MKT) the function concept and their students’ learning outcomes of this concept. Data were collected from two teachers teaching in a vocational high school and their students through a function concept test for teachers and students, follow-up interviews with teachers, and classroom observations. Findings indicated that teachers’ MKT and students’ learning outcomes were related to a degree, but this relationship was not straightforward. The data suggested that the teachers’ knowledge influenced the quality of their instructional practices, and the instructional practices played a mediating role in student learning. Different factors contributing to student learning were also evident.

Asking effective questions has been considered to be one of the most important indicators of good teaching since questioning is an important instructional tool and the ability to ask effective questions requires sophisticated pedagogical reasoning. Teachers can facilitate students' higher-level thinking through effective questions. However, teachers' utilization of effective and higher-level questions is usually limited in traditional settings. The goal of this study is to investigate how secondary mathematics teachers' knowledge and skills of questioning change when they are engaged in professional development activities based on the modeling perspective. The program included five cycles each of which lasted a month. Each cycle consisted of meetings before and after the implementation of a model-eliciting activity and its implementation. The results showed that the professional development program had a positive effect on teachers' ability to ask qualitatively different questions in order to move students' thinking forward.

Background:This paper investigates how prospective teachers develop mathematical models while they engage in modeling tasks.Material and methods:The study was conducted on an undergraduate elective course aiming to improve prospective teachers’ mathematical modeling abilities, while providing them with pedagogical knowledge for the integrating of modeling tasks into their future classroom practices. The participants of this study were six prospective teachers selected among the nineteen who enrolled in the course. Data was collected through five modeling tasks.Results:The results showed that the prospective teachers went through five main stages during the modeling process: understanding the task, devising a solution plan, performing the plan, interpreting and verifying the model, and presenting the model.Conclusions:The nature of prospective teachers’ modeling processes was rather result-oriented, consisting of a single cycle as they did not seek an improved solution by revising and refining their models.

Use of mathematical modeling in mathematics education has been receiving significant attention as a way to develop students. mathematical knowledge and skills. As effective use of modeling in classes depends on the competencies of teachers we need to know more about the nature of teachers. knowledge to use modeling in mathematics education and how this knowledge evolves. The aim of this study is to investigate how teachers. pedagogical knowledge to use modeling in the classroom is formed and modified through professional development activities based on lesson study cycle from modeling perspective. The professional development program reported in this study included five monthly cycles consisted of meeting before the implementation, implementation of the activity and a follow up meeting. The participants were four in-service mathematics teachers. Results showed that the program provided opportunities for teachers to test, revise, and refine their knowledge. More specifically, teachers generated effective strategies for proper implementation of the modeling activities in the classroom.

Background:As is well known to all bridging teacher knowledge or learning with practice is not a straightforward task. This paper aims to explore this discrepancy between a mathematics teacher’s knowing and practices and to offer ways of alignment between the two based on the social/interpersonal meanings and their realization through teacher’s discourse.Material and methods:In this study we utilized discourse analysis of interpersonal meanings within a high school mathematics class mainly by focusing on the mathematics teacher’s discourseResults:We have found that an interpersonal meaning manifested by teacher’s discourse establishes counterproductive social roles and relationships for the knowing/learning to come alive.Conclusions:We conclude that the realization of interpersonal meanings can hinder or support the generation of ideas within the classroom: both for teacher’s knowing and student’s mathematical meaning making

Researchers point out the importance of teachers’ knowledge of student thinking and the role of examining student work in various contexts to develop a knowledge base regarding students’ ways of thinking. This study investigated prospective secondary mathematics teachers’ interpretations of students’ thinking as manifested in students’ work that embodied solutions of mathematical modelling tasks. The data were collected from 25 prospective mathematics teachers enrolled in an undergraduate course through four 2-week-long cycles. Analysis of data revealed that the prospective teachers interpreted students’ thinking in four ways: describing, questioning, explaining, and comparing. Moreover, whereas some of the prospective teachers showed a tendency to increase their attention to the meaning of students’ ways of thinking more while they engaged in students’ work in depth over time and experience, some of them continued to focus on only judging the accuracy of students’ thinking. The implications of the findings for understanding and developing prospective teachers’ ways of interpreting students’ thinking are discussed.

Öğretmen adaylarının matematiksel modelleme konusunda mesleki bilgi vebecerilerini geliştirmeyi hedefleyen geniş kapsamlı bir çalışmanın parçasıolarak, bu çalışmada nüfus artışını konu alan bir modelleme etkinliğibağlamında ilköğretim matematik öğretmen adaylarının “değişim oranı” kavramıüzerine düşünme biçimleri incelenmektedir. Çalışmanın katılımcılarını birdevlet üniversitesinin ilköğretim matematik öğretmenliği bölümünde son sınıfadevam eden 9 öğretmen adayı oluşturmaktadır. Bir matematiksel modelleme dersikapsamında yapılan çalışmanın veri kaynaklarını; grup çalışmalarından derlenenyazılı raporlar ve çalışma kâğıtları, etkinlik sonrası düşünce raporları vearaştırmacı alan notları oluşturmaktadır. Bulgulara göre, çalışmanınkatılımcıları “zamana bağlı nüfus artış oranı (hızı)” ifadesini yüzdelik ve eğim olmak üzere iki farklı şekilde yorumlamıştır. Öğretmenadaylarının yüzdelik yorumları dahayoğun olmakla birlikte modelleme etkinliği bağlamında nüfus verilerindeki yılaralıklarının eşit verilmemesi bazı katılımcıları eğim yorumuna yönlendirmiştir. Ortaya çıkan bu iki farklı düşünmeşekli, öğretmen adaylarının yüzdelik ve eğim yorumu arasındaki farkı ve buikisi arasındaki matematiksel ilişkiyi yorumlamada zorlandıklarınıgöstermektedir. Ayrıca “rate of change” kavramının Türkçe ifadesi ile ilgiliproblemli bir durum da gözlemlenmiştir. Çalışmanın sonuçları, öğretmenadaylarının konu ile ilgili zorluklarının muhtemel kaynağı olan birimli oran ve birimsiz oran kavramları arasındaki fark dikkate alınaraktartışılmıştır., As a part of a larger studyaiming at developing pre-service teachers’ pedagogical knowledge aboutmathematical modeling, this study investigates pre-service elementarymathematics teachers’ ways of thinking regarding rate of change in the contextof a modeling task on population growth. The participants of the study were 9prospective middle school mathematics teachers in their senior year attending apublic university. The study was conducted as a part of an undergraduate courseon mathematical modeling for prospective teachers. Data were collected throughthe prospective teachers’ written group work and reports regarding theirsolution to the modeling activity, individual reflection papers, andresearchers’ field-notes. The results showed that participants demonstrated twodifferent ways of thinking about the expression “rate of change in populationwith respect to time”: (i) percentage of change in population, and (ii) peryear change in population (slope). Even though “percentage” interpretation wasdominant, some of the participants were directed to “per year change inpopulation” interpretation as the year intervals in the problem context werenot given with equal intervals. The results revealed about prospectiveteachers’ difficulties in conceiving the difference and the mathematicalrelationship between “percentage” and “slope” interpretations. The results alsorevealed about the problematic aspect of expressing the term/concept “rate ofchange” in Turkish. As possible sources of these difficulties, the results arediscussed in light of the distinction between rate and ratio.