Matematik Öğretmenleri ile Adaylarının Tamsayılarla Dört İşlemi Sayma Pullarıyla Modelleme Başarıları

Bu araştırmanın amacı matematik öğretmen adaylarının ve öğretmenlerinin tam sayılarla dört işlemi sayma pulları ile modelleme başarılarını belirlemek ve sayma pullarının öğretimde kullanımına ilişkin görüşlerini incelemektir. Bu amaçla 14 öğretmen ve 49 öğretmen adayı ile çalışılmıştır. Araştırmanın verileri tam sayılarla dört işlemin sayma pullarıyla modellenmesinin istendiği açık uçlu klasik bir test ve görüşleri ortaya koymak üzere kullanılan 2 adet açık uçlu soru vasıtasıyla toplanmıştır. Veriler yüzde ve frekans değerleri ile temsil edildikten sonra modelleme sürecindeki başarılar arasında meydana gelen farkın anlamlılığı Mann Whitney-U ile test edilmiştir. Bulgulara göre hem öğretmenlerin hem de öğretmen adaylarının tam sayılarla toplama işleminin modellenmesinde diğer işlemlere göre daha başarılı oldukları çıkarma, çarpma ve bölme işlemlerini modellemede ise zorluk yaşadıkları görülmüştür. Modellemede yaşanılan zorluklarla birlikte, öğretmen adayları ve öğretmenlerin tam sayılarla dört işlemin öğretiminde sayma pullarına alternatif olarak kullanmayı düşündükleri materyaller açısından farklılaştıkları; öğretmenlerin öğretmen adayları ile kıyaslandığında daha fazla alternatif materyal öne sürebildikleri ve sayma pullarını olumlu ve olumsuz yönleri açısından daha iyi irdeleyebildikleri söylenebilir. Ortaya çıkan bu sonucun öğrencilerle ve öğretim materyali ile daha fazla zaman geçirmiş olma ve deneyimden kaynaklandığı düşünülebilir

The Success of Service and Preservice Mathematics Teachers’ on Modeling Integer Operations

Beginning from the first grade of secondary school (5-8 grades), the concepts and operations regarding integer numbers are important in mathematics teaching. For this age group where it is more appropriate to teach concrete materials and objects (Clements and McMillen, 1996) operations with integers have difficult concepts and processes. For this reason, many models and contexts are suggested to help teaching about integers in the literature (Ball, 1993; Javier, 1985; Kilpatrick, Swafford and Findell, 2001; Peled and Carraher, 2007; Peterson, 1972; Rabin, Fuller and Harel, 2013). Some of them are income and dept (Ball, 1993; Gregg and Gregg, 2007); elevator model (Ball, 1993); balloons and sand bags (Reeves and Webb, 2004); team scores in game (Linchevsky and Williams, 1999); witch model (Javier, 1985); distributive property (Rabin, Fuller and Harel, 2013); dancing couples (Dienes, 2000); electric charge (California Common Core Standards: Mathematics, 2013); happy and sad days (Whitacre, Bishop, Lamb, Philipp, Schappelle and Lewis, 2012); the people who get on and get off the bus (Streefland, 1996); reward and punishment (Shore, 2005); colored sticks (Flores, 2008; Lappan, Fey, Fitzgerald, Friel and Phillips, 2006); counter parts (counters, charge model) (Liebeck, 1990; Lestari, Hartono and Ilma, 2015); temperature (Altıparmak and Özdoğan, 2010); postman, water tank, walking, the graphs which are drawn on the Cartesian coordinate system (Peterson, 1972) and magic peanut model can be ordered (Ball, 1993). There are disadvantages as well as advantages of using these models and materials in teaching. For example, it is doubtful that models such as the magic peanut model would be useful for future use of modeling integers in realistic situations because it can create the image that mathematics contains some mysteries (Peled and Carraher, 2007). The number line which is one of the most frequently model used for teaching integers (Chilvers, 1985) is presented in a graphical form and has some limitations with respect to counters. As counters are more concrete they become more sensible and meaningful in context (Battista, 1983). Counters are also one of the materials recommended in the elementary mathematics curriculum for teaching four operations with integers, which is a subject perceived as difficult to learn by students. Proper and correct understanding of counters by the elementary mathematics teachers and pre-service teachers is of high importance for preventing many potential teaching problems. Therefore, this study aims to examine the opinions of service and pre-service elementary mathematics teachers about the functionality of the use of counters in modeling four operations with integers as well as revealing their skills in modeling. This study was conducted with 14 mathematics teachers working in different provinces in the spring term of the 2015-2016 academic year and 49 final grade students studying in the department of elementary mathematics education at a state university. The data were collected using a mathematics test and two interview questions asking to model all four operations with integers. This study is a phenomenological research and the data obtained were analyzed using frequencies and percentages. Some of the responses to the questions during the interview were given as direct quotations. The findings showed that both elementary mathematics teachers and pre-service teachers were most successful at addition in terms of modeling the operations by using counters. Subtraction, multiplication and division was followed this, respectively. Most of the participants among both teachers and pre-service teachers believed that counters were effective in addition and subtraction operations, but were not so in multiplication and division. The participants stated that the use of counters in teaching the operations with integers was advantageous in ensuring permanent learning, being easy to prepare and facilitating abstract operations, but was not useful as their use made it difficult and time-consuming to perform operations on large numbers. Some of the teachers indicated that they used counters after they finished teaching just because their use was an official requirement by the curriculum. At the end of the study, we observed that both elementary mathematics teachers and preservice teachers were not successful enough on the whole at modeling the operations with integers and did not believe the functionality of counters in teaching. This indicates that teachers prefer using the methods they could understand more easily, and they avoid from using certain materials while teaching the subjects they have trouble with. To prevent this and ensure diversity in teaching, workshops can be organized regarding the subjects that both teachers and pre-service teachers need. Moreover, further studies can be conducted with teachers with more experience and knowledge of mathematics teaching in order to obtain more healthy results as well as making relevant adjustment for making arrangements for mathematics teaching.

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