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This study examined a cohort of middle school pre-service mathematics teachers\" understanding of the rate of change as they engaged in a model development sequence. By adopting a design-based research perspective, a model development sequence on the concept of rate of change has been designed and implemented as part of a mathematical modeling course for pre-service teachers. The data were collected from twenty senior year middle school pre-service mathematics teachers (PSTs) through questionnaires, modeling activities, reflection papers, and semi-structured interviews. The data analysis showed that PSTs have difficulties conceptualizing the rate of change and conceiving it as a multiplicative comparison of changes in two covarying quantities. As they frequently employed its percentage interpretation, PSTs experienced additional difficulty conceiving the conventional meaning of rate of change in a population growth context. PSTs generally used motion context as a reference while explaining the rate of change in different non-motion contexts. In general, PSTs developed their conception of the additive rate of change throughout the model development sequence. However, for some PSTs, difficulty in ratio-based reasoning on the rate of change in different non-motion contexts prevailed. We provided some arguments concerning the teaching and learning of rate of change.

This study aimed to investigate secondary mathematics teachers’ noticing of students’ mathematical thinking in the context of a professional development (PD) program utilizing modeling-based teacher investigations. The PD program was conducted with two groups of mathematics teachers from two public high schools. It included seven 1-month-long cycles, each consisting of an introductory meeting, the teachers’ classroom implementations of a model-eliciting activity (MEA), and a follow-up meeting focusing on the teachers’ collaborative investigation and discussion of students’ written work, including their strategies to solve the MEA. Findings suggested that participating in the PD program promoted the teachers’ noticing of students’ mathematical thinking, albeit differently, over time. We elaborated on the components of the PD program that might have influenced the teachers’ noticing of students’ mathematical thinking. Implications of the findings on teachers’ noticing for researchers, PD leaders, and teacher educators are discussed in terms of potential directions for future research and practice.

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.

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.

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.

To take its due place in the world of education, Turkey has been through serious reform initiatives in the curriculums of various school subjects since 2003. The new Turkish elementary school curriculum was prepared considering the research studies conducted in Turkey and in other countries, as well as the educational systems of developed countries and previous experiences with mathematics education in Turkey. This study attempts to provide a perspective on the nature of the instructional tasks in the new elementary school mathematics curriculum. In particular, our focus is to explore the level of cognitive demands (LCD) in the algebra tasks provided in the national elementary mathematics curriculum guidebook. This curriculum document is a major resource for administrators, stakeholders, textbook publishers and ultimately for teachers. For every learning objective, it provides sample tasks to be used in mathematics instructions. In this study, our purpose is to explore the LCD of each of these tasks by utilizing a framework developed by Smith and Stein (Math Teach Middle School 3:344–350, 1998). The framework classifies mathematical tasks according to the level of demands: lower-level and higher-level demands. While the lower-level demands are related to memorization and procedures without connections, the higher-level demands are related to procedures with connections and doing mathematics. The findings revealed that 60% of algebra tasks for each grade level required higher LCD and a great majority of the remaining tasks were at the level of procedures without connections. The findings of the study particularly inform curriculum developers about issues regarding the quality of the tasks given in the curriculum guide and provide possible suggestions to improve the implementation of the curriculum change process.

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.

If teachers interpret students' ways of thinking appropriately, they can make adequate instructional decisions before, during, and after teaching, and can develop their teaching with respect to stu...

Ogretmenin sahip olmasi beklenen onemli yeterliliklerden birisi ogrenci dusunme sekilleri bilgisidir. Pedagojik alan bilgisinin bir alt boyutu olan bu bilginin ogretmenlere meslege baslamadan once kazandirilmasi onemlidir. Bu calismanin amaci matematiksel modelleme etkinliklerinin sinif ici uygulamalarindan elde edilen ogrenci cozum kâgitlari ve videolari incelemenin ogrenci dusunme sekillerini anlamaya olan katkisi konusunda matematik ogretmen adaylarinin dusuncelerini incelemektir. Calismanin katilimcilari ogretmen adaylarina donuk bir matematiksel modelleme dersine kayitli yirmi bes matematik ogretmen adayidir. Sekiz hafta suren calismanin verileri bireysel dusunce raporlari, birebir gorusmeler ve bir oz-degerlendirme anketi yoluyla toplanmistir. Verilerin analizleri, ogretmen adaylarinin ogrenci cozum kâgitlari ve video kesitlerini incelemenin ogrencilerin matematiksel dusunme sureclerinin farkina varma, tahmin edebilme, anlama ve yorumlamalarina yardim ettigini dusunduklerini ortaya cikarmistir. Bu calismanin sonuclari ogretmen egitimcilerine, ogretmen adaylarinin pedagojik alan bilgilerinin gelisimine destek saglamasi acisindan gercek sinif ortamlarindan alinmis ogrenci calisma urunlerinin kullanilmasini onermektedir. Anahtar Kelimeler : Pedagojik alan bilgisi, ogrenci dusunme sekilleri, ogretmen egitimi, ogrenci calismalari