Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma

Abstract

Farklı disiplinlerde etkili olmasına rağmen özellikle geometri alanında daha etkili olduğu düşünülen ve zihinsel yeteneğin bir parçası olarak kabul edilen görsel uzamsal akıl yürütme becerisi birçok araştırmacının üzerinde durduğu bir konudur. Bu bağlamda geleceğin bilim insanlarının öğretim için gerekli görsel akıl yürütme becerilerini sağlayabilmede ve geliştirebilmede matematik öğretmenlerine sorumluluk düşmektedir. Bu çalışma ile matematik öğretmenlerinin görsel teoremleri ispatlama bağlamında, görsel akıl yürütme becerileri ile geometrik düşünme düzeyleri ve uzamsal görselleştirme becerileri arasındaki ilişkiyi nitel olarak özel durum yöntemiyle inceleme amaçlanmıştır. Çalışma grubunu on lise matematik öğretmeni oluşturmaktadır. Çalışmadan elde edilen veriler iki farklı görüşme sürecinde elde edilmiştir. İlk görüşmede öğretmenlerden “van Hiele Geometri Düzeyleri Testini ve Uzamsal Görselleştirme Beceri Testini” doldurmaları istenmiş, ikinci görüşmede ise öğretmenlerle üç farklı görsel teorem üzerinden klinik mülakatlar yürütülmüştür. Bu iki süreçten elde edilen veriler betimsel olarak analiz edilmiştir. Sonuç olarak,  matematik öğretmenlerinin görsel akıl yürütme becerileri ile geometrik düşünme düzeyleri ve uzamsal görselleştirme becerileri arasında bir ilişki olduğu,  daha yüksek geometrik düşünme düzeyine ulaşan öğretmenlerin, görsel teoremleri tanımada, onlar üzerine akıl yürütmede ve ispatlamada daha yetenekli olduğu tespit edilmiştir.

Keywords

• geometri,görsel akıl yürütme,matematik öğretmenleri

A Qualitative Study with Mathematics Teachers on Visual Theorems

Abstract

The geometry that develops the aesthetic sensation of individuals and event that allows them to think in many ways, at the same time helping individuals to better understand the world they live in and to relate mathematical concepts and events in life. Despite being effective in different disciplines, visual (diagrammatic) reasoning skill that is thought to be more effective, especially in the field of geometry, and which is considered to be part of mental ability, it is a topic that many researchers have pointed out. Because visual reasoning is an important skill that affects students especially to prove or solve geometric problems. Many researchers have suggested that an individual working at a higher level of geometric thinking should have stronger visual reasoning skills and that visual reasoning can also be improved by geometric teaching. However, despite the considerable importance given to geometry in recent years, it has been shown in many studies that the level of comprehension of the geometry of students is not expected and desired. Such results indicate that the objectives of the curriculum are not reached, such as training individuals with geometric and spatial thinking skills. In this sense, mathematics teachers need to possess the visual reasoning skills necessary for teaching in the training of the individuals (mathematicians, scientists, engineers, doctors, graphic designers, etc.) who will form the human power of the future. The aim of this study is to qualitatively examine the relationships between math teachers' visual reasoning skills and their level of geometric thinking and spatial visualization skills in the context of proving visual theorems.

This research is a descriptive study conducted using the case study method. The study group of the study is composed of ten mathematics teachers with a postgraduate degree in a university located in the Eastern Black Sea Region. In the selection of the study group, the maximum diversity method was chosen from the purposive sampling methods. As a result of two different interviews with the teachers, the data of the study were obtained. During the first interview, it was asked to answer open-ended questions prepared by the researchers with the help of literature to determine their geometric backgrounds. Then the teachers were asked to answer van Hiele Geometry Thinking Level Test and Spatial Visualization Skill Test. The data obtained from these two different tests were presented using descriptive statistics.  In this context, in order to determine the teachers' level of geometric thinking, a criterion was used in which teachers responded "at least 4 of 5 questions correctly" to each level. In the spatial visualization skill test, each teacher was given a test score of 36 points by giving (1) points to the questions that the teachers answered correctly, (0) points that they answered incorrectly or left empty. In the second interview, in order to reveal the problem solving or proving behaviours, visual reasoning skills, geometrical information about visual expressions and validation situations, clinical interviews were conducted on four different visual theorems. Data collected with clinical interviews were analysed using descriptive analysis technique.

In general, teachers with more geometric background scores have been found to have higher geometric thinking levels and better spatial visualization skills. In particular, teachers who are graduates of the Faculty of Education and who work as permanent staff were found to have achieved a level 4 in the van Hiele geometry thinking level test and higher scores than the spatial visualization skill test. In addition, teachers who attain a higher level of geometric thinking and have better spatial visualization skill scores, have been found to be more capable in recognizing visual theorems, in reasoning about them, and in verifying relationships. Another result is that there is a relationship between the visual reasoning that teachers have conducted on visual theorems and the way of thinking or behaviour attributed to van Hiele levels. However, it was found that most of the teachers who participated in the study had lack of knowledge about geometry and geometrical background, geometric thinking level test and spatial visualization skill test scores. In particular, it has been determined that teachers who have graduated from two faculties other than the Faculty of Education have a lack of information to be gained in geometry and these deficits affect the process of proving their visual theorems. The lack of information in the teachers has a negative effect on the learning of the students. Because there is a positive relationship between teacher knowledge and student achievement. In short, teachers have to know and understand the geometry they teach in order to make successful teaching. This work underscores some of the deficiencies in teacher education in the context of geometry and the need to re-examine the academic needs and curriculum changes needed to address these deficiencies.

Keywords

• geometry,visual reasoning,mathematics teachers

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