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The purpose of this study was twofold. First, to explore middle and high school students’ intuitively-based (mis)conceptions in probability, particularly availability and representativeness heuristics. Second, to investigate teachers’ awareness of these intuitively-based (mis)conceptions and the effectiveness of their instructional practices to support students’ understanding of probability beyond heuristics. The participants were two middle school mathematics teachers, three high school mathematics teachers and their students. Data were collected through a diagnostic test administered to students as a pretest and posttest, interviews with the teachers and two students from each class and classroom observations. The findings indicated the existence of intuitively-based (mis)conceptions regarding availability and representativeness heuristics among middle and high school students. In general, the teachers did not consider the students’ intuitions and difficulties in probability in their instructions. Not only did they rarely guide students to analyze and solve the tasks coherently and deliberately, but they also did not discuss students’ intuitively-based (mis)conceptions. The results highlighted that if teachers do not attend to student thinking and change their instructional practices accordingly, their knowledge about students’ difficulties will not necessarily help students overcome their intuitions and attain a probabilistic way of thinking.

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.

The purpose of this study was to investigate secondary mathematics teachers' conceptions of the barriers to the development of mathematical literacy. The data were collected through semi-structured interviews with sixteen in-service mathematics teachers from nine secondary schools of five different school types. The data analysis consisted of an inductive analysis process for identifying and labeling the patterns or themes through examining interview data with the help of NVivo qualitative data analysis software. The results indicated that teachers' conceptions regarding the barriers to mathematical literacy development could be categorized under: (i) conceptual challenges (What is mathematical literacy?); (ii) educational challenges (How can we develop mathematical literacy?); and (iii) systemic challenges (Where is mathematical literacy (to be) located?). The teachers' identification of different obstacles to the development of mathematical literacy can account for the fact that each teacher in this study has diverse but convergent conceptions of mathematical literacy. This fact also promises hope for assisting teachers in resolving these obstacles that have been raised from various angles.

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.

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.

Bu calismanin amaci, modelleme etkinlikleri baglaminda ogrenci calismalarini inceleyen ve ogrencilerin dusunme bicimlerindeki hatalarini tespit eden lise matematik ogretmen adaylarinin, ogrencilerin bu hatalarina yonelik ne tur pedagojik yaklasimlar sergilediklerini arastirmaktir. Calismanin verileri arastirmaya katilan 7 matematik ogretmen adayiyla gerceklestirilen bire-bir gorusmelerle toplanmistir. Calismanin bulgulari ogretmen adaylarinin ogrencilerin hatalarina yonelik mudahalelerinin bes yaklasim altinda toplanabilecegini gostermistir: soru sorma (sorgulatma), dogruyu aciklama, dogru yolu hissettirme, hatayi soyleme/gosterme ve mudahale etmeme . Calismanin bulgulari ogretmen adaylarinin ogrenci hatalarina yaklasimlarina yonelik pedagojik yeterliliklerinin zayifligina da isaret etmektedir. Bu cercevede, matematik ogretmen egitimcilerine matematik egitimi derslerinde ogretmen adaylarinin ogrenci dusunme sekillerini incelemesi ve cevap vermesine yonelik calismalara yer vermesi onerilmektedir.

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.

The purpose of this study was to investigate the extent to which personality traits, motivation, academic risk-taking, and metacognition explain the mathematical creative ability of high school students. The participants were 217 9th-grade students that were exceptionally high achievers. The participants responded to a set of measures about personality traits, motivation, academic risk-taking, metacognition, and creative ability in mathematics. The results revealed that although openness to experience and consciousness were significantly correlated with creative ability in mathematics, extraversion, agreeableness, and neuroticism were not correlated with the creativity scores. Moreover, intrinsic goal orientation followed by openness to experience was the most significant predictor of mathematical creative ability. Academic risk-taking was not significantly correlated with creative ability. Although knowledge of cognition and regulation of cognition showed moderate correlations with creative ability in ma...

Background:This study presents a methodology for investigating teacher learning in and from practice based on discourses that are in constant flux and transformation. Conceptualizing teacher learning as a frame of meaning based on knowing and doing discourses, the ideas are illustrated through data collected from a secondary mathematics teacher conducting an inquiry of self-practice.Material and methods:Narrative analysis of the data from the teacher interviews was conducted along with classroom observations of the teacher’s mathematical practiceResults:The data supported that tracing shifts in teacher discourses enables to understand the connection between the teacher’s past discourses and the present discourses when identifying teacher learning.Conclusions:It is concluded that tracking teacher learning through two complementary discourses of the teacher’s instructional practice enabled a unification of individual and collective levels of the teacher learning by self-inquiry of own practice. Moreover, through the continuum of discourses, this study provided insights about the generation of new meanings through transformation of old meanings in the teacher learning.

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.

10th Congress of the European-Society-for-Research-in-Mathematics-Education (CERME) -- FEB 01-05, 2017 -- Dublin, IRELAND, WOS: 000467053304043, The purpose of this study was to investigate middle school mathematics teachers' knowledge for teaching algebra. The participants of the study were 48 mathematics teachers from various middle schools. A questionnaire was conducted in order to collect data about the teachers' knowledge related to teaching of algebra. The results showed that the participating teachers were competent in making transitions among different algebraic representations. However, they had difficulties in explaining the conceptual bases of some of the algebraic concepts and procedures. In addition, results indicated that some of the teachers had difficulties and misconceptions similar to those of students as depicted in the scenarios provided in the questionnaire., European Soc Res Math Educ

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.

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.

The purpose of this study was to investigate the problem solving strategies, problem solving episodes, and metacognitions of five Turkish high school freshmen and explore the interplay of these on their problem solving success in mathematics. Participants worked on ten problems selected among the released mathematical literacy items used in Programme for International Student Assessment (PISA) 2003. The research data had been collected through clinical interviews and a self-monitoring questionnaire filled by the participants. The data were coded and analysed according to a framework adopted from various researchers in the field. Our results have confirmed that the problem solving success is too complex to be clarified by a unique property or behaviour of the solver. It requires overcoming various obstacles to reach a successful result. Therefore, not only the students should have the required mathematical knowledge and a good repertoire of different problem solving strategies, but also they should know when and how to use those strategies, along with monitoring and regulating their problem-solving processes using their metacognitive skills.

WOS: 000352197500015, The purpose of this study was to investigate three second-year graduate students' awareness and understanding of the relationships among the "big ideas" that underlie the concept of derivative through modeling tasks and Skemp's distinction between relational and instrumental understanding. The modeling tasks consisting of warm-up, model-eliciting, and model-exploration activities were used to stimulate participants to reflect and construct ideas about the concept of change. The data indicated that the participants' understanding of derivative was rather instrumental. Their explanations couldn't reveal the role of the big ideas regarding the concept of derivative with respect to what they mean, why and how they are related to the derivative and to each other. The results highlighted the fact that even if one of big ideas is ignored, the concept of derivative may not be fully understood relationally due to the compartmentalization of these big ideas in students' conceptual systems. If this happens to be the case, even though students can solve differentiation tasks/problems correctly, which implies procedural understanding, they may not be actually making sense of what the concept of derivative conceptually means., Scientific and Technological Research Council of Turkey (TUBITAK) [110K250]; Turkish Academy of Sciences through the Young Scientist Award Program [A.K.E./TUBA-GEBIP/2012-11], Work reported here is based upon a research project supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under grant number 110K250. Opinions expressed are those of the authors and do not necessarily represent those of TUBITAK. Ayhan Kursat ERBAS is supported by the Turkish Academy of Sciences through the Young Scientist Award Program (A.K.E./TUBA-GEBIP/2012-11).

(ProQuest: ... denotes formulae omitted)Quadratic equations have been a fundamental topic, not only in secondary mathematics curricula around the world but also in the historical development of algebra. Various approaches for solving quadratic equations were used at different stages in this histor-ical development, through representations including arithmetic or numerical, algebraic or symbolic, and visual or geometric (Katz & Barton, 2007). From a contemporary perspective, quadratic equations are considered important in school mathematics curricula because they serve as a bridge between mathematical topics such as linear equations, functions, and polynomials (Saglam & Alacaci, 2012). Furthermore, like linear equations, quadratic equations are powerful representations used in other disciplines, such as physics, engineering, and design, due to their usefulness in solving many kinds of word problems and for modeling realistic or real-life situations.Student Performance in Solving Quadratic EquationsVarious researchers (e.g., Vaiyavutjamai & Clements, 2006) have illustrated that very little attention has been paid to quadratic equations in mathematics education literature, and there is scarce research regarding the teaching and learning of quadratic equations. A limited number of research studies focusing on quadratic equations have documented the techniques students engage in while solving quadratic equations (Bosse & Nandakumar, 2005), geometric approaches used by students for solving quadratic equations (Allaire & Bradley, 2001), students' understanding of and difficulties with solving quadratic equations (Kotsopoulos, 2007; Lima, 2008; Tall, Lima, & Healy, 2014; Vaiyavutjamai, Ellerton, & Clements, 2005; Zakaria & Maat, 2010), the teaching and learning of quadratic equations in classrooms (Olteanu & Holmqvist, 2012; Vaiyavutjamai & Clements, 2006), comparing how quadratic equations are handled in mathematics textbooks in different countries (Saglam & Alacaci, 2012), and the application of the history of quadratic equations in teacher preparation programs to highlight prospective teachers' knowledge (Clark, 2012).In general, for most students, quadratic equations create challenges in various ways such as difficulties in algebraic procedures, (particularly in factoring quadratic equations), and an inability to apply meaning to the quadratics. Kotsopoulos (2007) suggests that recalling main multiplication facts directly influences a student's ability while engaged in factoring quadratics. Furthermore, since solving the quadratic equations by factorization requires students to find factors rapidly, factoring simple quadratics becomes quite a challenge, while non-simple ones (i.e., ax2 + bx + c where a ^ 1) become harder still. Factoring quadratics can be considerably complicated when the leading coefficient or the constant term has many pairs of factors (Bosse & Nandakumar, 2005). Lima (2008) and Tall et al. (2014) suggest that students' lack of understanding on the procedures of linear equations, and their understanding based on "procedural embodiments," affects students' work on quadratic equations. Students tend to allocate meaning to equations and solving methods, however, the given meaning is related to the movement of the symbols rather than the mathematical concept. They also documented that students perceive quadratic equations as mere calculations, without paying attention to the unknown as a fundamental characteristic of an equation. Students mostly focus on the symbolic world to perform operations with symbols. For example, students used procedural embodiment associated with the exponent of the unknown, and solved the equation by transforming it into m = 1/9 to solve m2 = 9. In this case, students' use of the procedural embodiments "switching power to roots" (p. 15) resulted in failing to recognize the other root (i.e., m = -3). Moreover, they reported that students attempted to transform quadratic equations into linear equations. …

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.

Mathematical modeling and its role in mathematics education have been receiving increasing attention in Turkey, as in many other countries. The growing body of literature on this topic reveals a variety of approaches to mathematical modeling and related concepts, along with differing perspectives on the use of mathematical modeling in teaching and learning mathematics in terms of definitions of models and modeling, the theoretical backgrounds of modeling, and the nature of questions used in teaching modeling. This study focuses on two issues. The first section attempts to develop a unified perspective about mathematical modeling. The second section analyzes and discusses two approaches to the use of modeling in mathematics education, namely modeling as a means of teaching mathematics and modeling as an aim of teaching mathematics.

As teachers prepare to teach the Common Core State Standards for Mathematics (CCSSM), students’ exploration of patterns and relationships between numbers has gained more importance. Specifically, students’ conceptual understanding of numerical patterns is critical in middle school, as it lays a groundwork for fostering mathematical thinking at all levels. Educational technologies can enhance student’s explorations of patterns by providing opportunities to represent patterns, test conjectures, and make generalizations. In this chapter, the authors illustrate how spreadsheets can support students’ explorations of both arithmetic and geometric patterns in the middle grades.

WOS: 000295860100025, The purpose of this study was to investigate the effects of using a dynamic geometry environment (DGE) together with inquiry-based explorations on the sixth grade students' achievements in polygons and congruency and similarity of polygons. Two groups of sixth grade students were selected for this study: an experimental group composed of 66 students (34 boys and 32 girls); and a control group composed of 68 students (35 boys and 33 girls). The students in the experimental group taught with a DGE, while the students in the control group received textbook-based direct instruction. An achievement test was administered as pre-test, post-test, and delayed post-test in both groups. Qualitative data were collected through videotaped classroom observations. The results showed that the DGE together with open-ended explorations significantly improved students' performances in polygons and congruency and similarity of polygons. Furthermore, students in the experimental group showed greater interest and motivation toward learning geometry compared to those in the control group whom often showed lack of interest and curiosity. Also, students' comments and interpretations during lessons and tests were more accurate and advanced in the experimental group as they engage more in the DGE. Moreover, qualitative data suggested that boys showed greater interest in the computer-based learning environment than girls in the experimental groups although no significant gender effect on achievement was found. (C) 2011 Elsevier Ltd. All rights reserved.

The study investigates the effects of certain school characteristics on students' mathematics performances in Turkey in the PISA 2006 while controlling for family background and demographic characteristics. Three models of Hierarchical Linear Modeling (HLM) are constructed. The results reveal that 55% of the variance is attributable to between-schools and the remaining 45% to individual student characteristics. About two-thirds of the 55% is explained by selectivity in admissions, time to study mathematics and students' SES, gender and the geographical region. The findings are interpreted to explain why Turkish schools differed greatly in average student performance in PISA 2006 by using the conceptual efforts on school quality factors and family background characteristics. © 2009 Elsevier Ltd. All rights reserved.

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

The purpose of the study was to investigate students' perceptions of the concept of functions in vocational high schools on industry. The participants were tenth grade students (n = 11) studying in the department of Computer Technology and Electrics and Electronics. A 10-item diagnostic test was designed to assess students' understanding of the concept of function. After analyzing students' written works for the test, three students were selected to be interviewed to reveal more about their ways of thinking in deciding whether a representation is a function or not. Results have revealed that the students had very weak perceptions of the function concept. © 2010 Elsevier Ltd. All rights reserved.

The purpose of this study was to compare 6th grade Turkish, Singapore, and American mathematics textbooks in terms of certain features of textbook design. Textbooks were compared based on their visual design, text density, internal organization, weights of curriculum strands, topics covered, and content presentation. The results revealed varied assumptions for student learning and choices of design. Singapore books reflected simple features of text density and enriched use of visual elements, fewer number of topics, and an easier inner organization to follow. American books were mainly designed as reference books. Turkish books reflected a measured middle way between the two and reflected a design that valued active student learning. However, Turkish books could use ideas to improve visual design and presentation of certain topics.

Teaching efficacy beliefs have attracted researchers' attention in recent decades because of its close association with and potential impact on the implementation of new ideas and skills in education. In the present study, we have explored the psychometric properties and construct validity of the Turkish adaptation of the Mathematics Teacher Efficacy Belief Instrument developed by Enochs, Smith, & Huinker (2000) for in-service mathematics teachers. The instrument distinguishes between two dimensions of efficacy beliefs for mathematics teachers: personal mathematics teaching efficacy and mathematics teaching outcome expectancy. The sample consisted of 1355 in-service elementary school teachers and middle school mathematics teachers from 368 schools. Exploratory and confirmatory factor analysis revealed a two-factor structure similar to that found in other studies. Also, scores from the two subscales indicated acceptable internal consistency. Keywords: self-efficacy, mathematics education, teachers, exploratory factor analysis, confirmatory factor analysis. En decadas recientes, las creencias sobre la eficiencia de la ensenanza han atraido la atencion de los investigadores debido a su cercana relacion y potencial impacto en la implementacion de nuevas ideas y estrategias de educacion. En el presente studio hemos explorado las propiedades psicometricas y la validez de constructo de la adaptacion turca del Instrumento de Creencias de la Eficacia del Profesor de Matematicas desarrollado por Enochs, Smith, & Huinker (2000) para profesores de matematicas en activo. El instrumento distingue entre dos dimenciones de creencias de eficacia para profesores de matematicas: eficacia personal en la ensenanza de las matematicas y resultados de las expectativas en la ensenanza de las matematicas. La muestra consistio en 1355 profesores de matematicas de educacion elemental y de escuela media en activo de 368 escuelas. Los analisis exploratorios y los analisis de factor confirmatorio revelaron una estructura de dos factores similar a la encontrada en otros estudios. Asimismo, los datos de las dos subescalas tuvieron valores de consistencia interna aceptables. Palabras clave: auto eficacia, educacion en matematicas, profesores, analisis factorial exploratorio, analisis factorial confirmatorio. The assessment of teaching efficacy beliefs has become an important concern for researchers because of its critical role on various educational outcomes, particularly on the adaptation of educational innovations (De Mesquita & Drake, 1994, Ghaith & Yaghi, 1997; Guskey & Passaro, 1994; Tschannen-Moran, Hoy, & Hoy, 1998). Research studies reveal that teachers with high teaching efficacy beliefs tend to be flexible in their teaching approaches, open to new ideas and skills, and inclined to change their teaching practices by adopting new educational ideas (Czerniak & Lumpe, 1996; Enochs & Riggs, 1990; Ghaith & Yaghi, 1997; Guskey, 1988). Although several models and instruments have been developed to measure teaching efficacy beliefs, there is a need for cross-culturally validated teaching efficacy belief scales (Brouwers, Tomic, & Stijnen, 2002; Henson, Kogan, & Vacha-Haase, 2001). The primary focus of this study was to contribute to the work on factor structure and psychometric properties of the Mathematics Teaching Efficacy Belief Instrument (MTEBI, Enochs, Smith, & Huinker, 2000) by translating it into Turkish and evaluating its factor structure and reliability through in-service elementary and middle school mathematics teachers in a Turkish sample. The secondary purpose of this study was to examine how the mathematics teachers' teaching experience, gender, and grade-level taught interacted with mathematics teaching efficacy beliefs. The MTEBI was selected for this study because of three reasons. The first reason is that this scale specifically focuses on mathematics teaching efficacy. …

The purpose of this study was to explore the effect of a technology-supported learning environment utilizing an interactive whiteboard (IWB) and NuCalc graphing software compared to a traditional direct instruction-based environment on student achievement in graphs of quadratic functions and attitudes towards mathematics and technology. Sixty-five high school graduates attending cram schools (called dershane in Turkish) to study for the university entrance examination participated in the study. The significant interaction effect between time of testing and groups indicated that student performance across time of testing was dependent upon the presence or absence of the treatment. Comparisons suggest that although both experimental and control group students' performances increased from pre-test to post-test and then decreased from post-test to retention test, the rate of decrease was about the same and the rate of increase was different, with students showing a greater rate of increase in the experimental group. The results also revealed that the treatment had positively affected students' attitudes towards technology and mathematics. Moreover, students' reasoning and interpretation skills regarding graphs of quadratic functions were better in the experimental group as compared to those in the control group.

The purpose of this study was to investigate student difficulties, common errors and misconceptions in solving simple linear algebraic equations within a group of students (n = 217) from four different schools. The results suggest that the students in lower achievement groups have conceptual problems and mal-rules, some of which might be considered as misconceptions. In general, students in the middle and higher achievement levels did arithmetical and computational errors.