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How can basic research on mathematics instruction contribute to instructional improvement? In our research on the practical rationality of geometry teaching we describe existing instruction and examine how existing instruction responds to perturbations. In this talk, I consider the proposal that geometry instruction could be improved by infusing it with activities where students use representations of figures to model their experiences with shape and space and I show how our basic research on high school geometry instruction informs the implementing and monitoring of such modeling perspective. I argue that for mathematics education research on instruction to contribute to improvements that teachers can use in their daily work our theories of teaching need to be mathematics-specific.

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... Matematiksel modelleme kavramının sıkça kullanıldığı ancak geometrik modelleme kavramına rastlamanın (en azından şu günlerde) mümkün olmamasından dolayı ispat ve kazanımlar özelinde çalışmalar yapan araştırmacılara (Đokić, 2018;Herbst, Fujita, Halverscheid, Weiss, 2017;Herbst, 2016) rastlanmaktadır. Herbst (2016), özellikle geometri öğretiminde modelleme perspektifi kavramını defaatle kullanmıştır. ...

... Matematiksel modelleme kavramının sıkça kullanıldığı ancak geometrik modelleme kavramına rastlamanın (en azından şu günlerde) mümkün olmamasından dolayı ispat ve kazanımlar özelinde çalışmalar yapan araştırmacılara (Đokić, 2018;Herbst, Fujita, Halverscheid, Weiss, 2017;Herbst, 2016) rastlanmaktadır. Herbst (2016), özellikle geometri öğretiminde modelleme perspektifi kavramını defaatle kullanmıştır. Geometri çalışmalarının organize edilmesinde spesifik bir yaklaşım olarak model ve modelleme eğitimi, geleneksel öğretimden ayıran en önemli özelliktir. ...

... Geometri çalışmalarının organize edilmesinde spesifik bir yaklaşım olarak model ve modelleme eğitimi, geleneksel öğretimden ayıran en önemli özelliktir. İlgili notasyonların (somut diyagramların araştırılması, çokgensel bölgelerin karşılaştırmak suretiyle alansal değerlendirilmesi, üç boyut modelleme yöntemlerinin teknolojide kullanılması gibi..) türetilmesi de yine modellemenin geometri özelinde kullanılması olarak ifade edilmektedir (Herbst, 2016). ...

Ortaokul öğrencilerinin geometri kazanımlarından olan düzgün çokgenler konusundaki bilişsel modelleme yeterlik düzeyini tespit etmek ve uygulamalar neticesinde gelişimlerini inceleme amacıyla yapılan bu çalışmada, geometrik modelleme kavramı doğrudan kullanılamamıştır. Çünkü geometrik modelleme kavramının ampirik süreçler de dâhil olmak üzere kuramsallaşamamış olduğu görülmektedir. Net olarak kullanılamayan bu kavram yerine “geometri perspektifinde modelleme” kavramı kullanılmıştır.
Bu araştırmada, öğrencilerin geometri kazanımlarında modelleme becerilerinin işlemsel ve kavramsal olarak geliştirilmesi düşünüldüğünden dolayı; Bilişsel Perspektif Altında Modelleme Döngüsü gerek veri toplanması gerekse de analiz aşamasında kavramsal çerçeve olarak kullanılmıştır. Ortaokul 7. sınıf öğrencilerinin geometri kazanımlarından olan düzgün çokgenler konusunda bilişsel modelleme yeterlik düzeyini tespit etmek ve var olan durumlarının gelişimlerini inceleme amacıyla yapılan bu çalışmada eylem araştırması deseni kullanılmıştır. Gaziantep’te bulunan bir ortaokuldaki 12 öğrenci, çalışma grubu olarak belirlenmiştir. Öğrenci (Çalışma Kâğıtları, Öğrenme Günlükleri ve Video Transkriptleri) ve öğretmen (Gözlem Notları ve Araştırma Günlüğü) dokümanları kullanılarak veriler elde edilmiştir. Elde edilen verilerin çözümlenmesi esnasında betimsel analiz, doküman analizi gibi sistematik çoklu yöntemler kullanılmıştır.
Çalışmanın başlangıcında, öğrenciler yöneltilen soru esas alınarak rubrik kapsamında değerlendirilmiştir. Alınan puanlar doğrultusunda öğrenci grupları homojen bir şekilde oluşturulmuştur. İlk eylem planından başlamak suretiyle son plana kadar amaç, süreç ve zorluklar başlıklarında incelenmiştir. Yaşanan zorluklar doğrultusunda amaçlara bağlı kalınarak süreç ve müdahaleler şekillenmiştir. Yapılan çalışmalarda öğrencilerin özellikle ilk eylem planlarında zorlandıkları gözlenmiştir. Özellikle model oluşturma noktasında zorluklar yaşayan öğrencilerin, matematikselleştirme ve matematiksel olarak çalışma yeterliklerinde zorluk yaşadıkları görülmüştür. Süreçle birlikte varsayımlar oluşturabilen öğrenciler, bu basamaklarda başarı göstermişlerdir. Her eylem planının sonunda öğrencilerden alınan öğrenme günlükleri ve video transkiptlerinden elde edilen verilere göre, bir sonraki eylem planları için gerekli müdahaleler yapılmıştır. Öğrencilerin genellikle modelleri çalıştırdıkları ancak günlük hayat bağlamında yorumlamalarda ciddi zorluklar yaşadıkları görülmüştür. Doğrulama yeterliğinde ise ilk çalışmalarda hemen hemen hiç dikkat etmekleri söylenebilir. Yapılan doğrulama işlemleri ise ilk çalışmalarda işlem eksenli kalmıştır. Son çalışmalarda öğrencilerin daha rahat tavır sergiledikleri ve modeller oluşturdukları gözlenmiştir. Özellikle gerçekçi varsayımlara dayalı modelleri oluşturan öğrencilerin arttığı ve tüm yeterlikleri sağlandığı 6. eylem planı ile çalışma sonlandırılmıştır.
Genel olarak çalışma sonuçlarına göre, geometri özelinde öğrencilerin yeterliklerinde artış olduğu belirlenmiştir. Çalışmanın sürece yayılması, araştırmacının doğrudan katılması, çözümlerin öğrenciler tarafından yapılarak açıklanması, her öğrenciden ayrı ayrı geri dönütlerle sürecin kısmen tekrarlanması ve öğretmen müdahaleleri ile bahsi geçen yeterliklerin gelişiminde bu faktörlerin katkı sağladığı görülmüştür. Farklı kazanımlarla -özellikle de geometri kazanımlarında- farklı yaş gruplarına uygulanabilecek süreçlerle modelleme yeterliklerinin geliştirilebileceği düşünülmektedir. Ayrıca öğretim programlarının içeriğine entegre edilmesi neticesinde bu gelişim genele yayılması muhtemeldir. Sonraki yapılacak çalışmalarda, yapılan çalışma doğrultusunda yeni eylem planları ile farklı gruplara farklı kazanımlarda gelişimin incelenebileceği önerilmektedir.

We examined geometric calculation with number tasks used within a unit of geometry instruction in a Taiwanese classroom, identifying the source of each task used in classroom instruction and analyzing the cognitive complexity of each task with respect to distinct features: diagram complexity and problem-solving complexity. We found that instructional tasks were drawn from multiple sources, including textbooks, tests,supplemental materials, and the teacher. Our analysis of cognitive complexity indicated that the instructional tasks frequently involved both diagram complexity and problem-solving complexity. Moreover, the geometric calculation with number tasks from nontextbook sources tended to be more cognitively demanding than those found in the textbooks. Implications of task analysis on geometry domain and textbook analysis studies are discussed.

We elaborate on the notion of the instructional triangle, to address the question of how the nature of instructional activity can help justify actions in mathematics teaching. We propose a practical rationality of mathematics teaching composed of norms for the relationships between elements of the instructional system and obligations that a person in the position of the mathematics teacher needs to satisfy. We propose such constructs as articulations of a rationality that can help explain the instructional actions a teacher takes in promoting and recognizing learning, supporting work, and making decisions.

The authors present an analysis of portfolio entries submitted by candidates seeking certification by the National Board for Professional Teaching Standards in the area of Early Adolescence/Mathematics. Analyses of mathematical features revealed that the tasks used in instruction included a range of mathematics topics but were not consistently intellectually challenging. Analyses of key pedagogical features of the lesson materials showed that tasks involved hands-on activities or real-world contexts and technology but rarely required students to provide explanations or demonstrate mathematical reasoning. The findings suggest that, even in lessons that teachers selected for display as best practice examples of teaching for understanding, innovative pedagogical approaches were not systematically used in ways that supported students' engagement with cognitively demanding mathematical tasks.

This paper examines teachers’ classroom talk as teachers respond to, interact with and take forward learners’ contributions. Teachers’ responses to learner contributions provide a useful lens in understanding teachers’ hybrid practices as they take up aspects of reform practice. A set of codes for teacher moves is developed to describe teacher responses, which build on previous work in describing classroom talk and provide the beginnings of an elaborated language of description for changes in teaching practice. The codes illuminate the similarities and differences across four secondary school mathematics teachers as they shift their practices to take account of learners’ thinking.

Studies of teachers’ use of mathematics curriculum materials are particularly timely given the current availability of reform-inspired curriculum materials and the increasingly widespread practice of mandating the use of a single curriculum to regulate mathematics teaching. A review of the research on mathematics curriculum use over the last 25 years reveals significant variation in findings and in theoretical foundations. The aim of this review is to examine the ways that central constructs of this body of research—such as curriculum use, teaching, and curriculum materials—are conceptualized and to consider the impact of various conceptualizations on knowledge in the field. Drawing on the literature, the author offers a framework for characterizing and studying teachers’ interactions with curriculum materials.

Cet article développe les éléments d'un cadre théorique utilisé pour l'étude de l'enseignement de la géométrie élémentaire notamment en formation des enseignants. Les notions de paradigmes géométriques et d'espace de travail de la géométrie font l'objet d'une présentation détaillée. La géométrie élémentaire est ainsi vue comme éclatée en trois paradigmes géométriques différents. Ces derniers approfondissent la notion de cadre géométrique due à Douady. Ils sont ensuite mis en relation avec les niveaux de Van Hiele. La notion d'espace de travail permet d'étudier la spécificité de l'activité du géomètre qu'il soit expert ou apprenti. Une articulation avec l'approche cognitive de Duval est envisagée.

Four potential modes of interaction with diagrams in geometry are introduced. These are used to discuss how interaction with
diagrams has supported the customary work of ‘doing proofs’ in American geometry classes and what interaction with diagrams
might support the work of building reasoned conjectures. The extent to which the latter kind of interaction may induce tensions
on the work of a teacher as she manages students’ mathematical work is illustrated.
Vier mögliche Formen der Interaktion mit geometrischen Darstellungen werden aufgezeigt. Diese Formen werden thematisiert um
deutlich zu machen, wie visuelle Darbietungen im am erikanischen Geometrieunterricht das alltägliche Geschäft des Beweisens,
unterstützen. Dadurch soll auch gezeigt werden, welche Art der Interaktion mit geometrischen Darstellungen es erlaubt, das
Herstellen begründeter Vermutungen zu unterstützen. Zugleich wird das Ausmaß illustriert, mit welchem die letztere Art von
Interaktion Spannungen innerhalb der unterrichtlichen Arbeit, der Lehrerin hervorruft, die sich darum bemüht, die mathematischen
Beiträge, d.h. die mathematische Arbeit, der Schülerinnen und Schüler zu organisieren.
ZDM-ClassificationC63-C73-D43-E53-G43

We outline a theory of instructional exchanges and characterize a handful of instructional situations in high school geometry that frame some of these exchanges. In each of those instructional situations we inspect the possible role of reasoning and proof, drawing from data collected in intact classrooms as well as in instructional interventions. This manuscript is part of the final report of the NSF grant CAREER 0133619 “Reasoning in high school geometry classrooms: Understanding the practical logic underlying the teacher’s work” to the first author.All opinions are those of the authors and do not represent the views of the National Science Foundation.

IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching) is an exciting new series of texts for teacher education which aims to advance the learning and teaching of mathematics by integrating mathematics content with the broader research and theoretical base of mathematics education. The Learning and Teaching of Geometry in Secondary Schools reviews past and present research on the teaching and learning of geometry in secondary schools and proposes an approach for design research on secondary geometry instruction. Areas covered include: teaching and learning secondary geometry through history; the representations of geometric figures; students' cognition in geometry; teacher knowledge, practice and, beliefs; teaching strategies, instructional improvement, and classroom interventions; research designs and problems for secondary geometry. Drawing on a team of international authors, this new text will be essential reading for experienced teachers of mathematics, graduate students, curriculum developers, researchers, and all those interested in exploring students' study of geometry in secondary schools. © 2017 Patricio Herbst, Taro Fujita, Stefan Halverscheid, and Michael Weiss. All rights reserved.

This chapter conceptualizes and illustrates StoryCircles, a form of professional education that builds on the knowledge of practitioners and engages them in collective, iterative scripting, visualization of, and argumentation about mathematics lessons using multimedia. The drive to invent and study new forms of professional education for mathematics teachers, such as StoryCircles, is predicated on the need to improve mathematics instruction. While many such efforts aim to support teachers to make broad sweeping changes, few take into account the actual predicaments of practice that make such changes difficult. StoryCircles aims to support teachers in making incremental improvements to practice by eliciting teachers’ practical wisdom and enabling participants to use each other’s knowledge and experience as resources for professional learning. In this chapter we outline critical characteristics of the StoryCircles interaction and illustrate how they are connected to seminal anchors in the professional development literature. We also illustrate those features with examples from various instantiations of StoryCircles. We close by providing some considerations for the affordances we see for the model both for the profession and for individual groups of teachers.

Edited and translated by Nicolas Balacheff, Martin Cooper, Rosamund Sutherland and Virginia Warfield. Excerpts available on Google Books (link below). For more information, go to publisher's website :http://www.springer.com.gate6.inist.fr/education+&+language/mathematics+education/book/978-0-7923-4526-8

In this article, Heather C. Hill and Pam Grossman discuss the current focus on using teacher observation instruments as part of new teacher evaluation systems being considered and implemented by states and districts. They argue that if these teacher observation instruments are to achieve the goal of supporting teachers in improving instructional practice, they must be subject-specific, involve content experts in the process of observation, and provide information that is both accurate and useful for teachers. They discuss the instruments themselves, raters and system design, and timing of and feedback from the observations. They conclude by outlining the challenges that policy makers face in designing observation systems that will work to improve instructional practice at scale.

This article focuses on mathematical tasks as important vehicles for building student capacity for mathematical thinking and reasoning. A stratified random sample of 144 mathematical tasks used during reform-oriented instruction was analyzed in terms of (a) task features (number of solution strategies, number and kind of representations, and communication requirements) and (b) cognitive demands (e.g., memorization, the use of procedures with [and without] connections to concepts, the “doing of mathematics”). The findings suggest that teachers were selecting and setting up the kinds of tasks that reformers argue should lead to the development of students’ thinking capacities. During task implementation, the task features tended to remain consistent with how they were set up, but the cognitive demands of high-level tasks had a tendency to decline. The ways in which high-level tasks declined as well as factors associated with task changes from the set-up to implementation phase were explored.

This paper analyzes the tension between the traditional foundation of efficacy in teaching mathematics and current reform efforts in mathematics education. Drawing substantially on their experiences in learning mathematics, many teachers are disposed to teach mathematics by "telling": by stating facts and demonstrating procedures to their students. Clear and accurate telling provides a foundation for teachers' sense of efficacy--the belief that they can affect student learning--because the direct demonstration of mathematics is taken to be necessary for student learning. A strong sense of efficacy supports teachers' efforts to face difficult challenges and persist in the face of adversity. But current reforms that de-emphasize telling and focus on enabling students' mathematical activity undermine this basis of efficacy. For the current reform to generate deep and lasting changes, teachers must find new foundations for building durable efficacy beliefs that are consistent with reform-based teaching practices. Although productive new "moorings" for efficacy exist, research has not examined how practicing teachers' sense of efficacy shifts as they attempt to align their practice with reform principles. Suggestions for research to chart the development of, and change in, mathematics teachers' sense of efficacy are presented.

In this paper, we argue that to prepare pre-service teachers for doing complex work of teaching like leading classroom mathematics discussions requires an implementation of different pedagogies of teacher education in deliberate ways. In supporting our argument, we use two frameworks: one curricular and one pedagogical. The curricular framework is based on the work of Hammerness et al. (Preparing teachers for a changing world. What teachers should learn and be able to do. San Francisco, Jossey-Bass Educational Series, pp 358–388, 2005) outlining four main goals of teacher learning: a vision of practice, knowledge of students and content, dispositions for using this knowledge, and a repertoire of practices and tools. The pedagogical framework is based on the work of Grossman et al. (Teach Teach Theory Pract 15(2):273–289, 2009a; Teach Coll Record 111(9):2055–2100, 2009b) outlining three pedagogies of practice: representations, decompositions, and approximations of practice. We use the curricular framework to examine the opportunities for teacher learning that were afforded by these three different pedagogies of practice in a unit on leading classroom mathematics discussion in a secondary mathematics methods course. We use evidence from our analysis to show how the coordination of those pedagogies of practice is better than any one of them in addressing important goals for teacher learning about classroom discussions.

This paper documents efforts to develop an instrument to measure mathematical knowledge for teaching high school geometry (MKT-G). We report on the process of developing and piloting questions that purported to measure various domains of MKT-G. Scores on a piloted set of items had no statistical relationship with total years of experience teaching, but all domain scores were found to have statistically significant correlations with years of experience teaching high school geometry. Other interesting relationships regarding teachers??? MKT-G scores are also reported. We use these results to propose a way of conceptualizing how instruction specific considerations might matter in the design of MKT items. In particular, we propose that the instructional situations that are customary to a course of studies, can be seen as units that organize much of the mathematical knowledge for teaching such course.

To solve two enduring problems in education—unacceptably large variation in learning opportunities for students across classrooms and little continuing improvement in the quality of instruction—the authors propose a system that centers on the creation of shared instructional products that guide classroom teaching. By examining systems outside and inside education that build useful knowledge products for improving the performance of their members, the authors induce three features that support a work culture for creating such products: All members of the system share the same problems for which the products offer solutions; improvements to existing products are usually small and are assessed with just enough data; and the products are jointly constructed and continuously improved with contributions from everyone in the system.

Novel (as opposed to familiar) tasks can be contexts for students’ development of new knowledge. But managing such development is a complex activity for a teacher. The actions that a teacher took in managing the development of the mathematical concept of area in the context of a task comparing cardstock triangles are examined. The observation is made that some of the teacher’s actions shaped the mathematics at play in ways that seemed to counter the goals of the task. This article seeks to explain a possible rationality behind those contradictory actions. The hypothesis is presented that in managing task completion and knowledge development, a teacher has to cope with three subject-specific tensions related to direction of activity, representation of mathematical objects, and elicitation of students’ conceptual actions.

This article presents a comparison of the first 2 years of an experienced middle school mathematics teacher's efforts to change her classroom practice as a result of an intervening professional development program. The teacher's intention was for her teaching to better reflect her vision of reform-based mathematics instruction. We compared events from the 1st and 2nd year's whole class discussions within a multilevel framework that considered the flow of information and the nature of peer- and teacher-directed scaffolding. Discourse analyses of classroom videos served both as an analytic tool for our study of whole classroom interactions, as well as a resource for promoting discussion and reflection during professional development meetings. The results show that there was little change in the teacher's specific goals and beliefs in light of a self-evaluation of her Year 1 practices, but substantial changes in how she set out to enact those goals. In Year 2, the teacher maintained a central, social scaffolding role, but removed herself as the analytic center to invite greater student participation. Consequently, student-led discussion increased manifold, but lacked the mathematical precision offered previously by the teacher. The analyses lead to insights about how classroom interactions can be shaped by a teacher's beliefs and interpretations of educational reform recommendations.

Editors' preface Acknowledgments Author's introduction 1. A problem and a conjecture 2. A proof 3. Criticism of the proof by counterexamples which are local but not global 4. Criticism of the conjecture by global counterexamples 5. Criticism of the proof-analysis by counterexamples which are global but not local: the problem of rigour 6. Return to criticism of the proof by counterexamples which are local but not global: the problem of content 7. The problem of content revisited 8. Concept-formation 9. How criticism may turn mathematical truth into logical truth Appendices Bibliography Index of names Index of subjects.

This investigation analyzes the structure and process of multidigit multiplication. It includes a review of recent theories of mathematical knowledge and a description of several fourth-grade math lessons conducted in a regular classroom setting. Four types of mathematical knowledge are identified: intuitive, concrete, computational, and principled knowledge. The author considers each type in terms of its relation to instructional issues and suggests that instruction should focus on strengthening the connections among the four types. Illustrations from instructional sessions show children generating and testing hypotheses when salient connections are made between concrete materials and principled, computational practices. Implications for teaching are discussed along with suggestions for future research.

This study explores interactions with diagrams that are involved in geometrical reasoning; more specifically, how students publicly make and justify conjectures through multimodal representations of diagrams. We describe how students interact with diagrams using both gestural and verbal modalities, and examine how such multimodal interactions with diagrams reveal their reasoning. We argue that when limited information is given in a diagram, students make use of gestural and verbal expressions to compensate for those limitations as they engage in making and proving conjectures. The constraints of a diagram, gestures and linguistic systems are semiotic resources that students may use to engage in geometrical reasoning.

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