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Matematik Öğretmenleri ile Öğretmen Adaylarının Öğretimsel Açıklamalarının Matematiksel Modeller Bağlamında İncelenmesi
Abstract
Bu araştırmanın amacı matematik öğretmenleri ile öğretmen adaylarının kesirlerle bölmeye yönelik öğretimsel açıklamalarının matematiksel modeller bağlamında incelenmesidir. Araştırma kapsamında durum çalışması tarama araştırması yönteminden yararlanılmıştır. Araştırmanın katılımcılarını aynı ilde yer alan farklı devlet okullarında görev yapmakta olan 2 matematik öğretmeni ile bir devlet üniversitesinin İlköğretim Matematik Öğretmenliği programında öğrenim görmekte olan 2 son sınıf öğrencisi oluşturmaktadır. Bu araştırmada veri toplama aracı olarak araştırmacılar tarafından oluşturulan sekiz adet açık uçlu soru ile yarı yapılandırılmış görüşmeler kullanılmıştır. Araştırma kapsamında yer alan katılımcıların öğretimsel açıklamalarında kullandıkları matematiksel modeller iki farklı boyutta değerlendirilmiştir. Bunlar matematiksel ve pedagojik boyutlardır. Matematiksel boyutta kullanılan modellerin özelliklerine, pedagojik boyutta ise kullanım düzeylerine yer verilmiştir. Araştırma sonucunda katılımcıların kullandıkları öğretimsel açıklama ve modellerin matematiksel olarak genelde doğru ve geçerli olmakla birlikte ilişkili oldukları matematiksel durumu tüm yönleriyle her zaman yansıtmadığı, pedagojik boyutta ise genel olarak kavramsal düzeye ve problem çözme düzeyine uygun olduğu görülmüştür. Pedagojik boyutta en düşük performans ise epistemik düzeye aittir. Araştırma sonucunda ayrıca öğretmenlerin kullandıkları öğretimsel açıklama ve modellerin, öğretmen adaylarına nazaran matematiksel ve pedagojik boyutlarda yer alan göstergelerle daha uyumlu olduğu sonucu elde edilmiştir. Elde edilen bulgular alan yazınla ilişkili olarak tartışılmış ve sonuçlar doğrultusunda bazı önerilerde bulunulmuştur.
The purpose of this study is to examine middle school mathematics teachers' preferences and performances in using mathematical models in situations involving different fraction schemes and fraction operations. The study, utilizing the case study survey method, involves fifteen mathematics teachers currently working in the Altınordu district of Ordu Province in Turkey. Purposeful sampling methods, including convenience sampling and criterion sampling, were employed in determining the participants of the study. Accordingly, the criteria for selecting teachers in the study were having a minimum of 10 years of professional experience, being stationed in the central district, and volunteering to participate in the study. In this study, the Questionnaire on Model Use Preferences, Open-Ended Questions on Model Use and semi-structured interviews developed by the researchers were used as data collection tools. According to the results of the study, it was observed that the participant teachers generally preferred to use the rectangle model-circle model-number line model and finally the set model when different fraction schemes were considered, and the rectangle model-number line model-circle model and finally the set model when fraction operations were considered. Teachers generally preferred continuous models and did not use discrete models. When the teachers' performances of using models were examined, it was seen that their performance levels were generally adequate except for the cases involving iterative fraction schemes. When the performances for fraction operations were analyzed, it was seen that the teachers generally performed adequately except for multiplication and division operations. In general, teachers used mathematical models not as a tool to support learning, but to complete the tasks assigned to them in the study process. In this context, it can be said that the models used by teachers do not fully include the conceptual meanings and differences related to the current situation in general.
This study is an examination of 79 elementary prospective teachers’ (PSTs’) capacity for recognizing the core ideas involved in modeling fraction addition problems and their difficulties in solving and presenting the process of fraction addition using area, length, and set models. PSTs completed a written task in which they represented the process of solving a set of fraction addition problems of varying complexity in terms of the sizes of the addends, the sums, and the denominators using the three models. The PSTs’ responses to the task were analyzed and clustered to identify the most salient and recurring patterns of erroneous approaches to modeling. While the PSTs recognized the core ideas involved in modeling fraction addition problems, their actual work samples demonstrated several areas for improvement. Some PSTs had invalid answers and representations that were indicative of their misunderstanding of key concepts. Some PSTs had correct answers, but their model did not support sense-making. Additionally, some PSTs’ representations did not clearly incorporate the salient properties of the area, length, and set models. Our analysis generated implications for mathematics teacher educators regarding what PSTs need to learn, develop further, unlearn, and refine in order to effectively teach uses of representation in elementary classroom.
Bu çalışmanın amacı, ilköğretim matematik öğretmenlerinin bölme kavramı ile ilgili alan eğitimi bilgilerinin yapısını araştırmaktır. Çalışma, 3 farklı ilköğretim okulunda görev yapan 4, 12 ve 25 yıllık mesleki deneyime sahip 3 matematik öğretmeni ile yürütülmüştür. Öğretmenlerin alan eğitimi bilgilerine dair veriler iki aşamada toplanmıştır. İlk aşamada öğretmenlere bölme kavramıyla ile ilgili 3 senaryo tipi mülakat sorusu yöneltilmiş ve daha sonraki aşamada ise yarı yapılandırılmış gözlem formu yardımıyla sınıflarında gözlemler yapılmıştır. Bu çalışmayla, matematik öğretmenlerinin bölme kavramının farklı anlamlarına vurgu yapmadıkları ve tersine bölme kavramını kural ve işlem eksenli öğretmeye çalıştıkları saptanmıştır. Kural ve işlem eksenli öğretim yaklaşımının ise öğretmenlerin matematik öğretme yaklaşımlarını sınırlandırdığı anlaşılmıştır. ABSTRACT The aim of this study is to investigate the elementary mathematics teachers' pedagogical content knowledge structures concerning division concept. This study is carried on with 3 mathematics teachers. Data about pedagogical content knowledge of the teachers is collected in two phases. Interviews formed of 3 questions about teaching scenario based division concept were directed to the teachers in the first phase and then observations are done in classrooms with the help of semi-structured observation forms in the latter phase. As a result of this study, it is observed that mathematics teachers cannot relate applications of division concept in different contexts, and predominantly emphasize rules and procedures in their mathematics understandings and approach to teaching.
The aim of this study was to examine pre-service classroom teachers' knowledge of students in order to understand students' errors and their instructional written explanations to eliminate these errors. In this context, the participants of the study were composed of 69 pre-service classroom teachers who were studying in the Kazım Karabekir Faculty of Education at Atatürk University and were selected via the purposive sampling method from among the non-random sampling methods. Accordingly, the case study method, which is based on the qualitative research approach, was used in the study. The answers given by secondary school students to the open-ended questions on fractions in the study of Soylu and Soylu (2005) were used as the data collection tool in the study. Incorrect answers of the students were given to the pre-service teachers in written form and the pre-service classroom teachers were requested to find the errors in these questions and correct these incorrect answers. The data obtained from the answers of the pre-service classroom teachers were analysed using the content analysis technique. The results of the study showed that the pre-service classroom teachers generally did not experience much difficulty in identifying the students' errors related to the operations in fractions and, to a certain extent; they correctly detected the students' errors. Conversely, it was observed from the obtained data that the instructional explanations of the pre-service teachers for the question on how to correct students' errors were not at an adequate level.
Existing studies have quantitatively evidenced the relatedness between problem posing and problem solving, as well as the magnitude of this relationship. However, the nature and features of this relationship need further qualitative exploration. This paper focuses on exploring the interactions, i.e., mutual effects and supports, between problem posing and problem solving. More specifically, this paper analyzes the forms of interactions that happened between these two activities, the ways that those interactions supported prospective primary teachers’ conceptual understanding, and the difficulties that prospective teachers encountered while engaged in alternating problem-posing and problem-solving activities. The results indicate that problem posing contributes to problem-solving effectiveness while problem solving supports participants in posing more reasonable problems. Finally, multiple difficulties that demonstrate prospective primary teachers’ misunderstanding with fractions and their operations provide insight for teacher educators to design problem-posing tasks involving fractions.
Modelling can be considered as an effective way of ensuring permanent learning in the ‘Fractions’ topic, as well as the other topics of mathematics. When the effect of teacher’s knowledge on students’ learning is considered, it is important to investigate modelling skills of tomorrow’s teachers regarding fractions. Within this context in the present study, prospective middle school mathematics teachers’ modelling skills of multiplication and division in fractions were examined. The study was carried out with a total of 104 prospective middle school mathematics teachers who study at the Faculties of Education at two different universities in Turkey. The Modelling Fractions Test (MFT) with six items, three of which were on multiplication and three of which were on division in fractions, was used as a data collection tool. In analyzing the data, descriptive statistics was used and items were evaluated one by one. The findings indicated that the participants showed better performance on modelling with operations in multiplication than with the ones in division, created better models regarding operations requiring multiplication and division of a whole number and unit fraction, and also half of the participants answered questions in MFT as completely correct.
Experts' explanations have been shown to better enhance novices' transfer as compared with advanced students' explanations. Based on research on expertise and text comprehension, we investigated whether the abstractness or the cohesion of experts' and intermediates' explanations accounted for novices' learning. In Study 1, we showed that the superior cohesion of experts' explanations accounted for most of novices' transfer, whereas the degree of abstractness did not impact novices' transfer performance. In Study 2, we investigated novices' processing while learning with experts' and intermediates' explanations. We found that novices studying experts' explanations actively self-regulated their processing of the explanations, as they showed mainly deep-processing activities, whereas novices learning with intermediates' explanations were mainly engaged in shallow-processing activities by paraphrasing the explanations. Thus, we concluded that subject-matter expertise is a crucial prerequisite for instructors. Despite the abstract character of experts' explanations, their subject-matter expertise enables them to generate highly cohesive explanations that serve as a valuable scaffold for students' construction of flexible knowledge by engaging them in deep-level processing. (PsycINFO Database Record (c) 2014 APA, all rights reserved).
Ratio (and associated topics such as fractions and proportion) is known to be an area of mathematics that students find difficult. Multiplicative thinking is necessary, and students benefit from a wide range of strategies and representations for interpreting ratio. This study examined aspects of teachers' pedagogical content knowledge for teaching ratio, and investigated their knowledge of a typical misconception together with the strategies that they would use for dealing with such a misconception. The nature of the numerical examples that they suggested might be useful in teaching was also examined. Most teachers were able to recognise the misconception, but not all were able to generate examples that might help students to deal with it. Teachers also appeared to have only a limited repertoire of strategies to assist students. Research into children's learning has long revealed that the topic of ratio—together with the allied topics of fractions, proportions, and percentages—is one that students find difficult (e.g., Hart, 1982). This is no surprise to teachers, whose experiences often lead them to list the topic as one of the more problematic in the curriculum. This small study examines the extent to which teachers can recognise a typical misconception associated with ratio understanding, and what strategies they have for addressing it. This implies an examination of their understanding of children's conceptions, their knowledge of models and explanations for teaching, and their capacity to modify examples in pedagogically useful ways, all of which lie in the domain of pedagogical content knowledge (PCK) (Shulman, 1986; see also Chick, Baker, Pham, & Cheng, 2006; Chick, 2007). The study also adds to the growing literature examining PCK using questionnaires with open-ended items that involve pedagogical situations; see, for example, the study of Watson, Beswick, and Brown (2006) which investigated teachers' knowledge about the teaching of fractions.
The data for the present paper was a part of a large research project conducted to assess preservice teachers' knowledge related to fractions and place value at a southwestern public university in 2007. The study utilized convenience sampling, consisting of 150 elementary preservice teachers who were enrolled in a mathematics methods course before their student teaching. The results demonstrated preservice teachers’ knowledge of teaching comparison, addition, subtraction, and multiplication of fractions was insufficient even though these should be basic knowledge. Teacher preparation programs should emphasize profound knowledge for teaching fractions using representations.
While teacher content knowledge is crucially important to the improvement of teaching and learning, attention to its development and study has been uneven. Historically, researchers have focused on many aspects of teaching, but more often than not scant attention has been given to how teachers need to understand the subjects they teach. Further, when researchers, educators and policy makers have turned attention to teacher subject matter knowledge the assumption has often been that advanced study in the subject is what matters. Debates have focused on how much preparation teachers need in the content strands rather than on what type of content they need to learn. In the mid-1980s, a major breakthrough initiated a new wave of interest in the conceptualization of teacher content knowledge. In his 1985 AERA presidential address, Lee Shulman identified a special domain of teacher knowledge, which he referred to as pedagogical content knowledge. He distinguished between content as it is studied and learned in disciplinary settings and the "special amalgam of content and pedagogy" needed for teaching the subject. These ideas had a major impact on the research community, immediately focusing attention on the foundational importance of content knowledge in teaching and on pedagogical content knowledge in particular. This paper provides a brief overview of research on content knowledge and pedagogical content knowledge, describes how we have approached the problem, and reports on our efforts to define the domain of mathematical knowledge for teaching and to refine its sub- domains.
Although explanations are a common means of instruction, research shows that they often do not contribute to learning. To unravel the factors giving rise to the ineffectiveness of instructional explanations, we propose a framework that brings together empirical work on instructional explanations from a variety of research fields, including classroom instruction, tutoring, cooperative learning, cognitive skill acquisition, learning from texts, computer-supported learning, and multimedia learning. In our framework, we identify the distinctive characteristics of instructional explanations, present general guidelines for designing instructional explanations, and describe factors influencing both the generation and use of instructional explanations. It is argued that future research should uncover in more detail the interrelations between the different aspects of providing and using instructional explanations and their specific effects on learning.
This study illuminates claims that teachers' mathematical knowledge plays an important role in their teaching of this subject matter. In particular, we focus on teachers' mathematical knowledge for teaching (MKT), which includes both the mathematical knowledge that is common to individuals working in diverse professions and the mathematical knowledge that is specialized to teaching. We use a series of five case studies and associated quantitative data to detail how MKT is associated with the mathematical quality of instruction. Although there is a significant, strong, and positive association between levels of MKT and the mathematical quality of instruction, we also find that there are a number of important factors that mediate this relationship, either supporting or hindering teachers' use of knowledge in practice.
As part of a teacher profiling instrument, 42 middle school teachers were presented with a mathematics problem dealing with fractions and wholes and asked to suggest solutions that would be given by their students. Further they were asked how they would address inappropriate responses in the classroom. The students in their classes were presented with the same question as part of a larger survey of mathematical concepts important in the middle years. This study compares the expectations of teachers and their suggested remedial actions with their years of teaching, their previous mathematics study, and the performance of students. Results suggest explicit questioning of teachers is an effective way to explore teacher knowledge for teaching mathematics. The issue of teacher knowledge in relation to teaching mathematics and students' understanding of mathematics has long been a vexing question in mathematics education. What kinds of knowledge do teachers need in order to ensure learning for their students? Shulman (1987a, 1987b) began addressing issues of teachers' knowledge by suggesting seven types of knowledge that were required for teachers: content knowledge, general pedagogical knowledge, curriculum knowledge, pedagogical content knowledge, knowledge of learners and their characteristics, knowledge of education contexts, and knowledge of education ends, purposes, and values. These seven types of knowledge were featured and assessed in a teacher profiling instrument developed by Watson (2001) and used in relation to the chance and data part of the mathematics curriculum. She felt that the most important aspects in terms of student outcomes were associated with content knowledge, pedagogical content knowledge, and knowledge of students as learners. Among other aspects of the profiling instrument these three were addressed in questions that presented teachers with problems previously used in student surveys. Teachers were asked what appropriate and inappropriate responses students would give to these problems and how the teachers would use the student responses to devise remedial activities in the classroom. The issue of delving into these three types of teacher knowledge has often been considered delicate, in that teachers may feel threatened in particular by explicit questions about their mathematical knowledge. Often measures of teacher knowledge have been based on the number of mathematics courses completed, years of teaching, or self-report of confidence (e.g., Mewborn, 2003; Schoen, Cebulla, Finn, & Fi, 2003). These can be less reliable measures than asking teachers to produce mathematical responses for themselves in the three areas of interest. Recent work in this area by Hill, Rowan, and Ball (2005) has focussed on "Teachers' knowledge for teaching mathematics" as an extension of the work of Shulman. By this they mean the mathematical knowledge used to carry out the work of teaching mathematics. Examples of this "work of teaching" include explaining terms and concepts to students, interpreting students' statements and solutions, judging and correcting textbook treatments of particular topics, using representations accurately in the classroom and providing students with examples of mathematical concepts, algorithms, or proofs. (p. 373)
The expertise reversal effect occurs when learner’s expertise moderates design principles derived from cognitive load theory. Although this effect is supported by numerous empirical studies, indicating an overall large effect size, the effect was never tested by inducing expertise experimentally and using instructional explanations in a computer-based environment. The present experiment used an illustrated introductory text and a computer program about statistical data analyses with 93 students. Retention and transfer tests were employed as dependent measures. Each learner was randomly assigned to one condition of a 2 × 2 between subjects factorial design with the two factors expertise (novices vs. ‘experts’) and explanations (with vs. without). Expertise was induced by adding expository examples and illustrations to the introductory text to enhance text coherence and facilitate text comprehension. The expertise reversal effect was replicated for the dependent measure transfer, but not for retention. Results and implications for adaptive learning environments are discussed.
It should be noted that the authors in this volume represent neither of the continuum extremes presented in the previous section.
However, each author started with Shulman’s model and, based on their interpretation, shaped the model in unique ways that
fit their perceptions of the data on teacher cognition. Hopefully this book will enhance the reader’s understanding of PCK
through an analysis of both historic and current conceptions, an overview of the research literature, and a presentation of
the practical implications derived from this model. Does the construct of PCK help or constrain our pursuit of excellence
in teacher preparation? The answer to this question is left to the reader. An anticipated result of such contemplation will
lead to individual and community exploration, development, and evaluation of alternative models used to study teacher cognition.
As with PCK, future models will need to address the following questions: What knowledge do teachers need to possess in order
to be effective? What model of teacher knowledge best explains the data that exists and stimulates future attempts to reconcile,
synthesize, and expand our knowledge?
Regardless of future evaluation, the explication of PCK as a construct and a model has reintroduced the importance of content
knowledge into the teaching equation, promoted renewed vigor in the subject-specific teaching areas such as science education,
and highlighted the need for integration of the various domains of knowledge in research, teaching, and teacher preparation.
Using these criteria, PCK has proven to be an especially fruitful model.
The purpose of this study was to rethink the conceptualization of pedagogical content knowledge based on our descriptive research
findings and to show how this new conceptualization helps us to understand teachers as professionals. This study was a multiple
case study grounded in a social constructivist framework. Data were collected from multiple sources and analysed using three
approaches: (a) constant comparative method, (b) enumerative approach, and (c) in-depth analysis of explicit PCK. The results
indicated that (a) PCK was developed through reflection-in-action and reflection-on-action within given instructional contexts,
(b) teacher efficacy emerged as an affective affiliate of PCK, (c) students had an important impact on PCK development, (d)
students’ misconceptions played a significant role in shaping PCK, and (e) PCK was idiosyncratic in some aspects of its enactment.
Discussion centres on how these five aspects are related to teacher professionalism.
This paper draws on videotapes of mathematics lessons prepared and conducted by pre-service elementary teachers towards the
end of their initial training at one university. The aim was to locate ways in which they drew on their knowledge of mathematics
and mathematics pedagogy in their teaching. A grounded approach to data analysis led to the identification of a ‘knowledge
quartet’, with four broad dimensions, or ‘units’, through which mathematics-related knowledge of these beginning teachers
could be observed in practice. We term the four units: foundation, transformation, connection and contingency. This paper
describes how each of these units is characterised and analyses one of the videotaped lessons, showing how each dimension
of the quartet can be identified in the lesson. We claim that the quartet can be used as a framework for lesson observation
and for mathematics teaching development.
Bu araştırmanın amacı ortaokul matematik öğretmenlerinin farklı kesir şemaları bağlamında model kullanmaya yönelik pedagojik tercihlerinin incelenmesidir. Çalışma sonuçları ile alana katkı sağlanacağı düşünülmektedir.
Bu araştırmada durum çalışması tarama araştırması deseni kullanılmıştır. Durum çalışması tarama araştırması, küçük bir örneklem veya örneklem grubuna, grupta yer alan bireylerin bir yönünü veya özelliğini tanımlamak için bir anketin uygulandığı araştırma tasarımı olarak tanımlanmaktadır. Araştırmacılar, popülasyondaki bireylere görüş, davranış, yetenek, inanç veya bilgiyle ilgili kişisel ifadelerini incelemek için sorular sorar. Elde edilen yanıtlar, grubun eğilimlerini tanımlamak veya soruları veya hipotezleri test etmek için analiz edilir (Mills, Durepos, & Wiebe, 2010).
Araştırmanın çalışma grubunu Ordu ili Altınordu ilçesinde yer alan devlet okullarında görev yapmakta olan 15 matematik öğretmeni oluşturmaktadır. Katılımcı öğretmenlerin seçilmesinde araştırmayı yürüten araştırmacının kolay ulaşabileceği öğretmenler seçilerek araştırmaya hız ve pratiklik kazandırması amaçlanmış ve bu bağlamda uygun örnekleme tekniğinden (Yıldırım ve Şimşek, 2008) yararlanılmıştır.
Çalışmada veri toplama aracı olarak araştırmacı tarafından geliştirilen i) likert tipi anket, ii) açık uçlu sorular ve iii) yarı yapılandırılmış görüşmeler kullanılmıştır. Veri toplama araçlarının geliştirilmesinde Olive ve Steffe (2010) tarafından ortaya konulan farklı kesir şemalarından yararlanılmıştır. Buna göre hazırlanan anket öğretmenlerin farklı kesir şemalarını içeren durumlarda tercih ettikleri matematiksel modellerin tespitine yönelik olarak hazırlanmıştır ve toplamda 13 adet 5’li likert tipi soruyu içermektedir. Açık uçlu sorular ve yarı yapılandırılmış görüşmelerde ise öğretmenlerin farklı kesir şemalarını içeren durumlarda farklı kesir modellerini kullanma durumlarının gerekçeleri ile analiz edilerek söz konusu modelleri kullanma konusunda öğretmen performanslarını incelemeyi amaçlamaktadır.
Çalışmadan elde edilen verilerin analizinde tümevarımsal kodlamaya gidilerek, süreç sonunda araştırmacılar tarafından oluşturulan analiz ve kategorilerden yararlanılmıştır. Çalışma sonucunda öğretmenlerin farklı kesir şemaları ve kesir işlemlerinde genel olarak dikdörtgen modelini tercih ettiği, daha sonra sırasıyla daire ve sayı doğrusu modelini tercih ettikleri, küme nesne modelini ise nadiren tercih ettikleri ya da hiç tercih etmedikleri görülmüştür. Çalışmada yer alan öğretmenlerin farklı kesir şemalarında farklı matematiksel modelleri kullanma durumlarına yönelik veri analiz süreci devam etmekte olup, süreç sonunda elde edilen tüm sonuçlar alt problemler ışığında sunulacak ve yorumlanacaktır.
Problem Durumu: Öğretmenin öğretimdeki rolü göz önüne alındığında eğitimin niteliğinin doğrudan öğretmenin niteliğine bağlı olduğunu söylemek mümkündür. Öğretmen niteliğinin sahip olduğu bu önemden dolayı dünyanın birçok ülkesinde eğitim reformları gerçekleştirilmiş ve öğretmenlerde bulunması gereken niteliklere ilişkin yapılan çalışmalarda bir artış görülmüştür (Bolat ve Sözen, 2009). İyi bir öğretmende olması gereken yeterlilikler dikkate alındığında alan bilgisinin son derece önemli olduğu ifade edilmektedir (Tanışlı, 2013). Alan bilgisi Shulman (1987) tarafından öğretmenin alanındaki kavramların doğru ve yanlışlığını, geçerlik ve geçersizliğini saptamada kullanılan yöntemler hakkındaki bilgisi olarak tanımlanmaktadır. Öğretim için uygun etkinliklerin seçimi, öğrencilerin ne öğrendiklerini değerlendirme gibi birçok öğretim faaliyeti öğretmenlerin sahip oldukları alan bilgisiyle ilişkilidir (Ball ve McDiarmid, 1990). Ball (1990) tarafından öğretmen eğitiminde alan bilgisi gelişimine önem verilmesi gerektiği ifade edilmiştir. Kesirler konusu müfredattaki başka öğrenme alanları ile olan ilişkisinden dolayı öğrenilmesi ve öğretilmesi önemli bir konu olarak karşımıza çıkmaktadır. Kesirler konusu cebir, olasılık, oran ve orantı, yüzdeler gibi öğrenme alanları için bir temel oluşturmaktadır (Van de Walle, Karp ve Bay-Williams, 2013). Kesirler bu kadar önemli olmasına rağmen öğrencilerin en çok zorlandıkları, birçok kavram yanılgısı ve hataya sahip oldukları konulardandır (Işıksal, 2006; Ma, 1999). Kesir kavramı ve kesirlerle işlemler konusunda sahip olunan zorluklar sadece öğrencilerle sınırlı değildir. Yapılan araştırmalar öğrenciler dışında öğretmen ve öğretmen adaylarının da kesir kavramı ve özellikle kesirlerde bölme işlemine yönelik güçlüklere sahip olduğunu göstermektedir (Ball, 1990; Simon, 1993). Kesir konusunun iyi öğrenilmesi ileri düzeydeki matematik konularının daha rahat anlaşılabilmesi adına önem taşıdığından (Alacaci, 2014) kesirlerin öğretilmesi konusunda
The national assessment of Educational Progress (NAEP) is used by the federal government and by states to gauge achievement in several subject areas, including mathematics. The results of the NAEP tests in mathematics at the eighth grade are used here to help us explore students' mathematics achievement over the decade from 1990 to 2000. In particular, we use these data to counteract the media portrayal of students' achievement in mathematics as steadily declining.
The goal of this study was to gain a better understanding of the intuitive activity that precedes the construction of quantity, in particular, to make inferences concerning developmental trends and needs in children's unitizing processes. Using a cross-sectional design, the study analyzed the partitioning strategies of 346 children from grades four through eight in terms of a framework that translated economy in the number or size of pieces and the use of perceptual cues into sophistication in unitizing. At each grade level, a greater percentage of students used economical partitioning strategies than used less economical cut-and-distribute strategies. As grade level increased, the percentage of students using economical strategies increased, indicating a shift away from the distribution of singleton units toward the use of more composite units. Strategies were heavily influenced by social practice related to the commodity being shared and, to a lesser degree, by the numerical portion of the given extensive quantities.
This article reports an analysis of 19 prospective elementary and secondary teachers' understanding of division. Interview questions probed the prospective teachers' understanding of division in three contexts. Although many of the teacher candidates could produce correct answers, several could not, and few were able to give mathematical explanations for the underlying principles and meanings. The prospective teachers' knowledge was generally fragmented, and each case of division was held as a separate bit of knowledge.
This study explored the strategies used by 13 prospective secondary school mathematics teachers to develop and validate functions as mathematical models of real-world situations. The students, enrolled in an elective mathematics course, had continuous access to curve fitters, graphing utilities, and other computing tools. The modeling approaches fell under 4 general categories of technology use, distinguished by the extent and nature of curve-fitter use and the relative dominance of mathematics versus reality affecting the development and evaluation of models. Data suggested that strategy choice was influenced by task characteristics and interactions with other student modelers. A grounded hypothesis on strategy selection and use was formulated.
This article analyzes from several vantage points a classroom lesson in which a student teacher was unsuccessful in providing a conceptually based justification for the standard division-of-fractions algorithm. We attempt to understand why the lesson failed, what it reveals about learning to teach, and what the implications are for mathematics teacher education. We focus on (a) the student teacher's beliefs about good mathematics teaching, her knowledge related to division of fractions, and her beliefs about learning to teach; and (b) the treatment of division of fractions in the mathematics methods course she took. The student teacher's conception of good mathematics teaching included components compatible with current views of effective mathematics teaching. However, these beliefs are difficult to achieve without a stronger conceptual knowledge base and a greater commitment to use available resources and to engage in hard thinking than she possessed. Further, the mathematics methods course did not require the student teacher to reconsider her knowledge base, to confront the contradictions between her knowledge base and at least some of her beliefs, or to reassess her beliefs about how she would learn to teach. These findings suggest that mathematics teacher education programs should reconsider how they provide subject matter knowledge and opportunities to teach it, and whether and how they challenge student teachers' existing beliefs.
Since its emergence over two decades ago, the construct of pedagogical content knowledge (PCK) has significantly impacted preservice and inservice teacher education, educational policy, and educational research. PCK has served to re-focus educators' attention on the important role of subject matter in educational practice and away from the more generic approach to teacher education that dominated the field prior to 1975.
This ambitious text is the first of its kind to summarize the theory, research, and practice related to pedagogical content knowledge. The audience is provided with a functional understanding of the basic tenets of the construct as well as its applications to research on science teacher education and the development of science teacher education programs. The authors are prominent educators representing a variety of subject matter areas and K-12 grade levels. Although the focus of the text is science education, it should provide valuable reading for any individuals with interests in professional teacher education.
The purpose of the study was to determine potential errors, which might be experienced by seventh grade middle school students related to the problems posed by them about the subtraction operation with fractions. The study was conducted with 143 students studying at six middle schools existing in Erzurum. Problem posing test composed of four items about the subtraction operation with fractions was used as a data collection tool. The answers given by the students were classified in accordance with the categories of problem, not a problem and empty. Then, the errors in the answers within the category of problem were analyzed. Twelve errors were determined in the problems posed by the students. Moreover, students made more errors in posing problems about the subtraction operations with fractions which the minuend and subtrahend fractions are mixed fractions.
Describes pedagogical content knowing (PCKg) based on a constructivist view of teaching and learning, emphasizing knowing and understanding as active processes. PCKg requires teachers to understand students' learning and the environmental context in which teaching and learning occur. The paper applies the model of PCKg to teacher education curriculum. (SM)
In this article I present and discuss an attempt to promote development of prospective elementary teachers' own subject-matter knowledge of division of fractions as well as their awareness of the nature and the likely sources of related common misconceptions held by children. My data indicate that before the mathematics methods course described here most participants knew how to divide fractions but could not explain the procedure. The prospective teachers were unaware of major sources of students' incorrect responses in this domain. One conclusion is that teacher education programs should attempt to familiarize prospective teachers with common, sometimes erroneous, cognitive processes used by students in dividing fractions and the effects of use of such processes.
Building on the work of Ball and McDiarmid, this study provides an equivalent at the secondary level to the work of Liping Ma at the elementary level in that it provides a better understanding of the conceptual knowledge of school mathematics held by prospective secondary teachers, along with examples of the sorts of knowledge needed to teach for understanding within the domain of integer subtraction. Part of an eight-year longitudinal study of secondary teacher candidates' conceptions of instructional explanations, this analysis of interaction in the author's methods course and its discussion of epistemological obstacles and changes combines subject-matter and interactionist perspectives. The author concludes that secondary teacher candidates can deepen their relational knowledge of secondary mathematics within a methods course by focusing on instructional explanations.
Prospective elementary teachers must understand fraction division deeply in order to meaningfully teach this topic to their future students. This paper explores the nature of the subject content knowledge of fraction division possessed by a group of Taiwanese prospective elementary teachers at the beginning of their mathematics methods course. The findings provide preliminary evidence that many prospective Taiwanese elementary teachers have developed the knowledge package of fraction division as described by Ma (Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum Associates, Mahwah, 1999). The nature of various strategies used by these teachers provides further illustration of a secure common content knowledge that can serve as a benchmark for the development of mathematics courses for prospective teachers. However, the findings also show that the tasks of representing fraction division, through either word problems or pictorial diagrams, are challenging even for those highly proficient in elementary and middle school mathematics. The broader implications of this research for the international community are discussed, and recommendations for elementary teacher education programs are presented.
This article presents a survey of post‐war development and present states and trends in the role of applications and modelling in mathematics curricula at all levels. Characteristic phases in the development of elementary and post‐elementary curricula respectively are identified. The present situation is described as a period of movement, investigation and experimentation, with the following predominant features. (a) Applications and modelling are being secured a footing in more and more curricula, but considerable variety exists. There are differences between countries with central and countries with local curriculum authorities; (b) the distance between ‘the front’ and ‘the main‐stream’ of applications and modelling in mathematics curricula is great. The article finishes by addressing a few key problems. There is a need for a more balanced view of applications and modelling than is usually seen. What are the barriers to applications and modelling obtaining a reasonable position in curricula, and how may these barriers be overcome?.† The present paper is the slightly revised manuscript of a survey lecture given by the author at ICME V (Adelaide) in a summary session of the Theme Group 6, ‘Applications and Modelling’, for which the author was one of the Chief Organizers.
In this article, the authors examine two distinct but closely related fields, research on teaching and research on teacher education. Despite its roots in research on teaching, research in teacher education has developed in isolation both from mainstream research on teaching and from research on higher education and professional education. A stronger connection to research on teaching could inform the content of teacher education, while a stronger relationship to research on organizations and policy implementation could focus attention on the organizational contexts in which the work takes shape. The authors argue that for research in teacher education to move forward, it must reconnect with these fields to address the complexity of both teaching as a practice and the preparation of teachers.
This article presents a summary of considerations for teachers when assessing a student's understanding of Division concepts. As with all mathematical concepts, children develop a basic idea of Division through interactions with others in daily life, beginning from the preschool years. As they move into elementary school, Division is introduced as an arithmetic operation, building on their prior knowledge of addition, subtraction, and multiplication. As children advance to middle school, various uses of fractional concepts are introduced, along with algorithms involving rational numbers. Although there is no clear demarcation as to the grade at which a more advanced level of understanding may be taught, for the purpose of discussion, this article is organized into three broad levels: Preschool, Elementary, and Middle School. Within each level, the assessment of (a) student products, (b) student procedures and strategies, and (c) student concepts and explanations, is presented. Examples of assessment activities and examples of possible student misconceptions are provided. The article concludes with a discussion of the assessment of teaching practices using Division as a model for other areas of mathematics.
This paper follows the ways in which publications in TATE, that focus on teacher knowledge, provide insights into the development and growth of scholarly understanding of teacher knowledge. Relevant questions are: How is teacher knowledge defined? What modes of inquiry are adopted by the researchers? What are conceived as the implications of teacher knowledge for schooling? In order to answer these questions, nine papers were chosen from TATE according to the following criteria:1.distributed over a period of 20 years from 1988 to 20092.representing an international group of scholars3.reflecting modes of inquiry4.focusing on a variety of themes related to teacher knowledgeThese papers were analyzed according to the following aspects:–Definition of teacher knowledge–Mode of inquiry–Emphasis on one or more of the commonplaces of education – subject matter, learner, teacher, milieu (Schwab, 1964)–Emphasis on one or more of the kinds of teacher knowledge suggested byShulman (1986)The analysis of each paper is presented followed by a discussion.Several tendencies in the development of the concept of teacher knowledge are noted. There is the extension of the term to include societal issues. As well, one finds a growing focus on the personal aspects of knowledge. The role of context in shaping teacher knowledge plays a crucial role in the analyzed papers, reflecting changes in the milieu of schooling. The main mode of inquiry in the analyzed papers is qualitative, interpretative. The authors of the various papers were interested in the concrete experiences and views of student teachers, and teachers, concerning their knowledge and its acquisition. This approach yields important insights but leaves open several questions. First, the curricular question: what concrete opportunities for gaining knowledge are offered to student teachers? Another question concerns the modes of teachers’ use of their professional knowledge. This question requires detailed observations and documentation of teachers’ actions in classrooms, trying to link their knowledge and practice.The papers analyzed in this review share a common scholarly language and are based in Western culture. It is important to see, as well, studies conducted in other cultures, which might have a different view of teacher knowledge.
In this research, I examine some of the classroom processes that may be responsible for the stellar mathematical performance among Asian children compared to U.S. children. The study documents differences in the frequency and type of mathematical explanations during lessons observed in 80 U.S., 40 Chinese, and 40 Japanese 1st- and 5th-grade classrooms. Explanations occurred more frequently in the Japanese and Chinese classrooms than in U.S. classrooms. Furthermore, typical explanations in the Asian classrooms were more substantive than those in the U.S. lessons, and Japanese children were learning about more complex topics than their peers in Taiwan or the United States.
Basic reading comprehension and summary tend to be the focus in social studies and history classrooms, if reading and writing are included at all. But such a focus inhibits a conception of history as an interpretive discipline grounded in evidence that is analyzed, not simply accepted. Understanding the past is impossible without such historical reasoning, as is advanced literacy. This study examines the discipline-specific literacy instruction of one history teacher and the simultaneous growth in his students' historical reasoning and writing. Student data included pre- and post-instruction writing samples as well as regularly assigned essays, interviews, and annotations of readings. Teacher data included observations, interviews, and artifacts such as assignments and feedback from one term of a required 11th-grade U.S. history course. Analysis included developing codes based on patterns, testing propositions, and searching for alternative explanations. Through a focus on historical evidence use, perspective, and interpretation students learned to construct more accurate, grounded interpretations of the past. Three teaching strategies emphasized these aspects of historical thinking: annotating primary source readings; regular informal writing prompts that focused on historical perspectives followed by writing prompts that called for a synthesis of major issues; and feedback focused on evidence use and accuracy of interpretation. This study suggests that discipline-specific ways of reading and writing can help students understand history and learn to think historically while developing advanced literacy skills.
This study contributes to an emerging body of research on how to develop prospective teachers’ pedagogical content knowledge (PCK) in teacher education. The main focus is on the knowledge transformation process and on the cognitive strategy used to shift prospective teachers’ instructional explanations within the domain of integer subtraction from an instrumental to a relational understanding of mathematics. The strategy rests on the observation that transformation is not a unidirectional process from subject-matter knowledge to pedagogical content knowledge as the literature suggests. The adequacy of prospective teachers’ instructional explanations, and the depth of their subject-matter understanding, is analyzed using a modification of Perkins and Simmons’ levels of subject-matter understanding framework. The study concludes with observations about the interface between student achievement and prospective teachers’ subject-matter and pedagogical content knowledge.
The purpose of this paper is to share some results from a year-long teaching experiment in which fourth grade students were given the opportunity to understand fraction concepts prior to the introduction of algorithmic instruction. In particular, this paper focuses on the means by which children solved problems involving division of fractions. Children were given a task-based activity specifically designed to promote solutions that would be grounded in conceptual understanding. Three distinct solution methods, all related to counting, emerged. When the activity was replicated as part of regular classroom practice seven and a half years later, the same solution methods were observed.