Article

How Theory-Building Research on Instruction can Support Instructional Improvement: Toward a Modeling Perspective in Secondary Geometry

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Abstract

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). ...
Thesis
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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.
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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.
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