Mehmet Fatih Ozmantar

A Dialectical Approach to the Formation of Mathematical Abstractions

This paper is structured in two sections. The first examines views of mathematical abstraction in two broad categories: empiricist and dialectical accounts. It documents the difficulties involved in and explores the potentialities of both accounts. Then it outlines a recent model which takes a dialectical materialist approach to abstraction in context. This model constitutes the basis of the second section where we describe an empirical study designed to investigate mathematical abstraction in socially rich (e.g., peer-interacted and tutor-assisted) environments. We then present data on two students working with the help of a tutor on tasks concerned with graphs of absolute value functions. On the basis of these data, we discuss four particular themes which are relevant to the purpose of this special issue and are important in the discussion of mathematical abstraction: human and artefact mediation, tutor interventions in assisting the formation of mathematical abstractions, implications of a dialectical view on student development, and the things that are abstracted.

Are Mathematical Abstractions Situated

In this chapter we address the question: are mathematical abstractions situated? We first consider empiricist accounts of abstraction which see abstraction as a development process from the concrete to the abstract achieved through the recognition of commonalties isolated in a large number of instances. We discuss difficulties involved in empiricist accounts and propose an alternative approach which we call a dialectical account of abstraction. In this approach, an undeveloped initial idea develops through the use of mediational means and social interaction. This development is not from the concrete to the abstract but, rather, a dialectical to and fro between the concrete and the abstract. Unlike empiricist views, our approach regards context, in the formation of mathematical abstractions, as paramount. Although the construct ‘context’ is difficult to delineate precisely, we focus on the importance of students’ personal mathematical histories, the tools and knowledge artefacts they work with, the people they work with and the tasks they work on. We exemplify the importance of these contextual factors through a study where two teenage girls worked collaboratively, with an interviewer assisting them, in completing tasks designed to generate abstractions in the field of graphs of linear absolute value functions.

Factors shaping mathematics lecturers’ service teaching in different departments

In this article we focus on university lecturers’ approaches to the service teaching and factors that influence their approaches. We present data obtained from the interviews with 19 mathematics and three physics lecturers along with the observations of two mathematics lecturers’ calculus courses. The findings show that lecturers’ approaches to teaching the same topic vary across departments; that is, they consciously privilege different aspects of mathematics, set different questions on examinations and follow different textbooks while teaching in different departments. We discuss factors influencing lecturers’ decision of what (mathematics) to teach in different departments and offer educational implications for service mathematics teaching in terms of students’ mathematical needs and the role of mathematics for client students.

Use of multiple representations in technology rich environments

In this study we present part of a research project that aims to develop prospective mathematics teachers’ Technological Pedagogical Content Knowledge (TPCK) (Mishra and Koehler 2006). The project considers various TPCK components (see Ozmantar et al. 2010). This report focuses on a particular component, namely the knowledge of using multiple representations (MRs) with technology. A course was designed using the TPCK framework, and forty prospective mathematics teachers took part in the course. They were enrolled in a teacher preparation programme in a state university in Istanbul, Turkey. As part of this course, prospective teachers participated in workshops during which they were introduced to the TPCK framework, which was illustrated using function and derivative concepts. The workshops included examples of MRs, how to make links among them, and the way in which technology could contribute to achieving the intended linkages. Following the introductory workshops, the prospective teachers themselves prepared and carried out similar workshops for their peers with different topics including limit, continuity, probability, radian and integral. They were grouped into fours, assigned one of these topics and asked to prepare the workshops in collaboration. The participants also prepared lesson plans and teaching notes and performed micro-teaching (during which they taught lessons to their peers as if they were in a real classroom environment). Following the completion of the course, we attempted to see the participants’ development with regard to using multiple representations (MRs) with technology. For this purpose, we found the functional taxonomy of MRs proposed by Ainsworth (2006) useful in that it allowed us to see development of the participants through our course activities, as well as to perform comparative analyses of participants’ developmental trajectories. Ainsworth’s (2006) taxonomy differentiates three main functions that MRs serve in learning situations: to complement (different representations involve distinct yet complementary information), to constrain (confine inferences and provide a focus) and to construct (by providing cognitive linking of representations, which might eventually lead students to achieve a deeper understanding). Using this framework, we examined the extent to which two prospective teachers (Gamze and Mutlu) used different functions of MRs during instruction. These two

Matematik öğretmenlerinin ders kitabı dışında kaynak kullanımları: nedenler, sonuçlar ve çıkarımlar

Bu calismada, matematik ogretmenlerinin ders kitabi disinda kaynak kullanim durumlarinin belirlenmesi ve ogretmenleri kaynak kitap kullanimina iten nedenlerin incelenmesi amaclanmistir. Bu amac kapsaminda sekillenen calisma tarama modelinde betimsel bir arastirma olarak desenlenmistir. Calismaya 100 ortaokul matematik ogretmeni katilmistir. Katilimcilara acik uclu sorulardan olusan bir anket uygulanmistir. Anketten elde edilen veriler icerik analizine tabi tutularak cozumlenmistir. Yapilan analizler sonucunda katilimcilarin %80 gibi buyuk bir cogunlugunun ucretsiz olarak dagitilan ders kitabi disinda kaynak kitap kullandiklari belirlenmistir. Milli Egitim Bakanliginin kaynak kitap kullanilmamasi yonundeki duzenlemeleri, en azindan calismamiza katilan ogretmenler ozelinde, ders kitabi disinda kaynak kullanimini engelleyemedigi tespit edilmistir. Katilimci ogretmenler kendilerini ekstra kaynak kullanmaya iten sebepler bulundugunu; ders kitaplarinin hem soru bakimindan, hem konu anlatimi hem de ogrenci ozelligine uygunluk bakimindan beklentilerini karsilamadigini belirtmektedirler. Elde edilen bulgular ve ulasilan sonuclar, ders kitaplarinin hazirlanmasi ve secimi, ucretsiz dagitimi, kitaplarin kullanimi ve surecte ogretmenlerin ihtiyac duyacagi destek ve kitaplarin kullanimiyla birlikte ortaya cikan degisimin yonetimi konulari ekseninde tartisilmis ve bir takim onerilerde bulunulmustur.

Voices and values in shaping the subjectivity of pedagogical content knowledge

Abstract Pedagogical content knowledge involves subjective decisions on the parts of teachers in making the content comprehensible to learners. This paper is concerned with the formation of this subjectivity and asks: how do (pre-service) teachers come to know and decide upon the best approach to making the content instructional for learners? In answering this question, this study draws upon data obtained from one pre-service mathematics teacher’s microteaching and retrospective interviews. The data were examined through the lenses of the Bakhtinian notion of voice. The findings suggest that the pre-service teacher’s understanding of pedagogy is formed and transformed, to an important extent, by the voices of others who are not necessarily physically present within the immediate instructional setting and who might be distant in space and time. Assimilation of those voices enacts particular value judgments, which in turn shapes and hence forms the subjectivity of the pre-service teacher’s pedagogical practices.