The aim of this study was to determine the effects of teacher training programs on pre-service mathematics teachers' beliefs about mathematics. The sample of the study consisted of 173 pre-service mathematics teachers from an education faculty in the Central Anatolia Region of Turkey. The data were obtained from an open-ended questionnaire, categorized in terms of content analysis and analyzed quantitatively. The results of the study showed that there was no statistical difference between both elementary and secondary pre-service mathematics teachers' beliefs from when they started the teaching program to when they graduated from the programs. Yet, the mean scores about mathematical beliefs in the senior class were higher than those of the freshman class.

The purpose of this study is to examine how students understand fractals depending on age. Students' understandings were examined in four dimensions: defining fractals, determining fractals, finding fractal patterns rules and mathematical operations with fractals. The study was conducted with 187 students (grades 8, 9, 10) by using a two-tier test consisting of nine questions prepared based on the literature and Turkish mathematics and geometry curriculums. The findings showed that in all grades, students may have misunderstandings and lack of knowledge about fractals. Moreover, students can identify and determine the fractals, but when the grade level increased, this success decreases. Although students were able to intuitively determine a shape as fractal or not, they had some problems in finding pattern rules and formulizing them.Keywords: Fractal Geometry. Students' Understandings. Cross-age Study. elaboradas por meio da literatura nos currículos turcos em matemática e geometria. Os resultados mostraram que em todos os graus, os alunos podem ter falta e conhecimentos equivocados sobre fractais. Além disso, os estudantes podem identificar e determinar os fractais, mas quando o nível de ensino aumenta, a identificação e a determinação dos fractais diminui. Embora, os alunos pudessem determinar intuitivamente uma forma como fractal ou não, eles tiveram alguns problemas em encontrar regras e padrões na formulação dos fractais. ResumoPalavras-chave: Geometria Fractal. Entendimentos dos Alunos. Faixa-etária do Estudo.

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