Procedural and conceptual learning are two types of learning, related to two types of knowledge, which are often referred to in mathematics education. Procedural learning involves only memorizing operations with no understanding of underlying meanings. Conceptual learning involves understanding and interpreting concepts and the relations between concepts. The relationship between these learning types has been discussed for a long time. For some researchers, procedural knowledge forms the basis for conceptual knowledge, while for others the relationship is reversed. The aims of the study reported here were first to explore the nature of students' learning in traditional differential equations (DEs) courses and second to clarify the relationship between procedural and conceptual learning. To address these aims an achievement test with 13 open-ended questions, probing procedural and conceptual learning with regard to DEs, was administered to 77 candidate mathematics teachers, enrolled in a traditional DEs course. The analysis of students' responses to the test items showed that 85% of candidate teachers gave correct responses on procedural questions whilst only 30% of them gave correct responses to the conceptual questions. These findings suggest that the candidate teachers' learning was primarily procedural in the context of traditional instruction and content and that this did not lead them to develop the conceptual knowledge needed to interpret new situations properly and to produce new ideas beyond the ones they had memorized. In addition, based upon the student levels in both procedural and conceptual learning, it was concluded that conceptual learning supports and generates procedural learning but procedural learning does not support conceptual learning.

No abstract

Öz: Bu makale, Öğretmenlik Uygulaması derslerinde uygulanan ders imecesi modelinin sınıf öğretmeni adaylarının alanı öğretme bilgilerine etkisini izlemek amacıyla yapılan bir araştırmanın bir bölümünü yansıtmaktadır. Makalede alanı öğretme bilgisinin alt bileşenlerinden olan öğrenme-öğretme sürecini planlama boyutuna odaklanılarak 'Ders imecesi uygulaması, sınıf öğretmeni adaylarının öğrenme öğretme sürecini planlama bilgilerinin gelişimini nasıl etkilemektedir?' sorusunun yanıtı aranmıştır. Çalışma 6'sı daha önce ders imecesi çalışmasına katılmış, 6'sı katılmamış olmak üzere 12 sınıf öğretmeni adayıyla, Öğretmenlik Uygulaması dersi kapsamında gerçekleştirilmiştir. Veri toplama aracı olarak video kayıtları, gözlem, alan notları, mülakat, ders planları kullanılmıştır. Ders imecesi grubunun bir dersin planlamasında nelere dikkat edilmesi gerektiği konusunda kendilerini geliştirdikleri ve bu adayların, etkinliklerin sayısını ayarlama, öğrenme-öğretme sürecini bir kazanım dâhilinde tamamlama ve etkinlikleri uygun sırada sıralama konularında diğer grubun adaylarına göre daha iyi oldukları belirlenmiştir.Abstract: This article reflects a special part of a research conducted to examine the effect of lesson study on prospective classroom teachers' mathematical pedagogical content knowledge (MPCK). In this article, the special part consists of prospective teachers' knowledge of lesson planning including a mastery of planning an affective lesson taking into account student's current knowledge, understanding and difficulties within mathematics. Therefore, the research question is how lesson study practices affect prospective classroom teachers' knowledge of lesson planning as a sub component of MPCK. The research is conducted with 12 prospective classroom teachers, six of them have already assisted to lesson study and the others have not. Data collection tools consist of video records, class observations, field notes, interviews and lesson plans prepared and used by prospective teachers participated in lesson study. Findings indicated that the prospective classroom teachers who participated in lesson study improved their knowledge in terms of planning an affective lesson taking student's current knowledge and understanding into consideration. They appeared to be aware of selecting and ordering appropriate activities related to the actual objectives of the mathematical topics. They also appeared to be better in lesson organization and lesson presentation comparing to the other group of prospective teachers who did not participated in lesson study.

The aim of this study was to determine preschool teachers' beliefs about teaching mathematics to young learners. In this context, we compared preschool teachers' beliefs with mathematical learning, talent-development-age appropriateness for mathematical learning, the nature of mathematics, the curriculum, teacher efficacy, and the teacher's role by taking into account the preschool teachers' graduation programs and their experiences. Data was collected from 139 preschool teachers. The teachers were selected based on their teaching experiences and their graduation programs. A 32-item Likert-type "Belief Scale towards Mathematics Learning and Teaching" was used to determine their views about teaching, and the nature of the preschool mathematics teachers learning mathematics. The results of an ANOVA test to determine whether there was a statistically significant difference between the score averages the of their beliefs and teaching practice time and graduated program, revealed a significant difference among preschool teachers about mathematics and the role of teachers on teaching. Furthermore, the test revealed that the more experienced teachers have a better understating of the curriculum and the children and indicated a positive change in their opinions in the scale's related section.

No abstract