Didactic-methodical competencies of mathematics teachers

Raširenost izazova i zahtjeva nastave matematike u obrazovnim sustavima diljem svijeta dovela je do afirmacije novog interdisciplinarnog znanstvenog područja koje se u široj znanstvenoj zajednici navodi kao „Matematičko obrazovanje“ (eng. Mathematics Education). Pored sadržaja supstratne znanosti, izazovi suvremene nastave matematike predmet su interesa stručnjaka različitih profila (pedagoga, psihologa, sociologa, antropologa, komunikologa i sl.) što implicira kompleksnost zahtjeva koji predstoje predmetnim metodikama i didaktikama čiji položaj i klasifikacija u sustavno-znanstvenom sistematskom poretku nisu razjašnjeni u znanstvenoj zajednici. Specifičnost zahtjeva nastave matematike pokazuje se kao izazov visokog prioriteta suvremenih obrazovnih sustava. Glavni ciljevi tog područja su obrazovanje nastavnika matematike, unapređenje kvalitete nastave matematike te jačanje znanstveno-istraživačkog potencijala u području matematičkog obrazovanja. U nastojanjima da se učenicima pruži suvremena i kvalitetna nastava matematike, nužno je razmatrati zastupljenost pedagoških i didaktičko-metodičkih sadržaja u obrazovanju nastavnika te usklađenost obrazovnih standarda s nastavnim planovima i programima. U suvremenom društvu mnogobrojnih trendova kompetencije nastavnika matematike moraju se sagledavati u okviru šireg društvenog i obrazovno-političkog konteksta unutar kojeg promatramo i cjelokupno matematičko obrazovanje, a polazište u nastavi mora biti usmjereno na učenike, ali i čimbenike koji utječu na razvoj njihovih vještina i sposobnosti te odgojnih vrijednosti koje pritom stječu. Pritom treba koristiti suvremene didaktičke pristupe te ih prilagođavati s obzirom na specifičnosti učenja i poučavanja matematike te s obzirom na znanja, umijeća i sposobnosti nastavnika koja su u najiziravnijoj vezi s učenjem učenika. Razumijevanje pojedinačnih uvjerenja, stavova i postupaka nastavnika, koje čine učinkovitu nastavu matematike, suštinski su izazov i prioritet matematičkog obrazovanja te su stoga osnova istraživačkih interesa, kao i refleksije osobnih praktičnih iskustava i djelovanja nastavnika usmjerenih na unaprjeđivanje odgojno-obrazovne prakse.A quality upbringing and education can be attainable only with quality elementary and high school teachers, requiring the quality of their initial education and their ongoing training. Teachers’ competence is one of the key issues in the quality of educational system with a constant effort being made to determine a set of competencies of a successful teacher and the characteristics which distinguish them from the less successful ones. Even though recent research has indicated the need for a higher standard in the pedagogical knowledge of teachers, there are still disagreements on the competencies of math teachers, namely in the competence hierarchy relatable to the requirements of modern math teaching. The objective of this research is to analyse the beliefs, self-assessments, teaching practices and extra curricular activities of teachers that affect their performance. In order to conduct the research on the competence profile of math teachers and present the findings, an instrument was devised based on relevant research in the field of math education as well as pedagogy and psychology. The work is organized in five main chapters: introduction, theoretical segment, methodology of the empirical segment, findings, final deliberations and comments on research findings. In Introduction the author starts with the assumption that in theoretical and practical considerations on mathematical education social and affective aspects of recent challenges in teaching mathematics are often undermined, as well as the need for interdisciplinary approach in didactics and methodics of math teaching in order to offer solutions to a number of problems facing teaching in practice. The author asserts a critial approach in analysing the field of mathematical education, displaying how comprehensive and multilayered the challenges of modern math teaching and the suitable math teacher competence profile are. He describes the epistemology of mathematics and its relation to epistemological status of maths in schools, emphasizing social context as key distinction, presenting both a challenge as well as the potential for creating the requirements for a better quality math teaching. As essential factors in mathematical education he mentions stereotypes that have accompanied maths and science throughout history, along with educational policies and international research on education that have become a standard for successful education. The author concludes the introductory part with a reference to the current state of the educational system, taking into account teachers' position and competencies along with the quality of the teaching and learning process in relation to the contemporary educational policies. The following part, Theoretical segment comprises five sub-chapters introducing a more comprehensive outline of relevant aspects in mathematical education: competence approach to education; didactic and curricular approach to math education; students with special needs in math teaching; social and affective aspects of mathematical education; teachers' competencies. In the segment Competence approach to education the author displays a historical outline of the recent educational policies and the development of the concept of lifelong learning, along with competence approach to education. In doing so, he points out the discrepancies between economic trends and pedagogical aspects of education as one of the most significant challenges of modern education. He also illustrates the position of mathematical competence within the contemporary education and upbringing as well as the features and nature of mathematical language, script and thinking, linking them to the main objectives of mathematics in schools. In the following sub-chapter Didactic and curricular approach to mathematical education the author presents an outline of theories in didactics, analyses them and relates to the specifics of math teaching. In doing so, he describes fundamental didactic models, placing them in the context of math teaching and analysing them from the scientific pedagogical aspect. He states the existence of a discontinuity between the initial education of teachers and the requirements posed by practice and points to the predominance of core science in the methodics of math teaching. In the segment entitled Educational and upbringing standards in math teaching the author asserts a critical standpoint against the aim-oriented contemporary curricula and highlights numerous challenges from practice which thus remain unaddressed, together with problems on a theoretical and content level of the curriculum which have not even been taken into consideration within math teaching. Furthermore, he presents conceptual and organizational components in teaching along with the general didactic principles, focusing on specific usage in math teaching. In practical examples, he illustrates the potential usage of certain methods in math teaching. He views evaluation as one of the crucial elements and points out that evaluation of educational aims within the existing currricula and lesson plans and programs should consider the position of students and their educational and upbringing needs. In the sub-chapter Students with special needs in math teaching he addresses students with dyscalculia and the gifted ones. In that sense, he especially emphasizes insufficient representation of dyscalculia in theoretical, research as well as practical terms in comparison to other learning disorders and the need for a pedagogical approach and counseling of both children and their parents. This is followed by a concise outline of dominant theoretical approaches to studying giftedness. As a significant challenge to teaching and learning mathematics he presents the problem of recognizing the gifted students, where the concept of giftedness is not seen in binary terms following the concept of creativity, but instead considers the (mathematical) development of an individual as a process subjected to change and direction by well-trained experts. Along with the issue of giftedness he indicates the presence of stereotypes and the perception of mathematical education forced upon by society and individuals, emphasizing the necessity for math teacher training in theoretical aspects of stereotyping related to math teaching, as well as the practical competence in a form of pedagogical counselling and development of students' educational aspirations. In sub-chapter Social and affective aspects of mathematical education he presents the notion that math teaching is constructed within certain social frames and that subsequent success in teaching and learning math is under a significant influence of social and affective factors, which, therefore, have to be incorporated into subject curriculum planning as its integral part. As one of the crucial moments in math teaching the author claims that students have positive or neutral attitude and emotional reactions towards math in the beginning of their education, whereas, with years they become more negative. He presents a comprehensive outline of scientific research related to sudent beliefs and emotional reactions, by which he implies a critical attitude towards outcomes-set priorities in math teaching. He refers to one of the most significant challenges in considerations on teachers' competencies: the relationship between a teacher's competence and student achievement, and claims that studying teachers' competencies has to encompass a wide spectrum of their beliefs and attitudes in order to gain a perspective on their approach to students and teaching. This claim is supported with research showing that the role and actions of teachers, perception of the importance of mathematics, the ability to grasp it, perception of math as a social activity and as the field of success shape the beliefs and students' attitudes towards mathematics (Eynde and De Corte, 2003). He is guided by the notion that teachers have an influence on educational aspirations and academic success of their students (Marsh,1987; Cambria et al., 2017), emphasizing the complexity of each class dynamics, which, using the average student achievement variable determines the relationship between students in the context of the perception of their academic self-efficiency (Marsh, 1987). Theoretical segment concludes with an outline of relevant literature on teachers' competencies, their beliefs and self-assessment on the quality of performance and teaching patterns as well as expert training (sub-chapter Teachers' competencies). Within various fields of teacher competence, the author highlights didactic and methodical competence as the operative aspect of the superimposed pedagogical one. In third chapter Methodology of empirical research he highlights the emphasis the traditional pedagogy attributes to the styles of learning (Marsh, 1994), and not so much the content of learning. In doing so, he views the concept of didactic and methodical competence of math teacher as operative teacher competence, i.e. as a form of manifestation of overall teacher competence profile. As the main objective of the research the author sets determining predictive value of individual teacher characteristics, deriving from their attitudes, beliefs, self-assessment and teaching practice and the amount in which they contribute to the quality of teaching, starting with the assumption that the perception of academic self-efficiency, teacher system of values and beliefs reflect teacher competence profile, determine their teaching and upbringing role and are the foundation for acquiring values and achieving educational outcomes of students. A research tool was devised on that premises comprising four parts: 1. Socio-demographic data; 2. Scale of teacher beliefs about teaching and education in general, including specific and problematic areas of math education; 3. Scale of teacher self-efficiency; 4. Scale of occurence of teaching practices, including participation in profesional training and enhancement. Based on the analysis of the research on teacher competence, the author claims that the research where the given aspects ( beliefs, sels-assessment and teaching practice) are mainly observed separately outnumber the ones that incorporate them and studies their inter-relationship. He thus claims that beliefs, self-assessment and teaching practice of the participants in the research were firstly analysed as separate, and subsequently in their inter-relationship, which provides a framework in methodological approach to the empirical segment of this study. Participants in the research were elementary and high school math teachers. Total of 348 teachers from all Croatian counties took part, making up a representative sample according to Cohran formula for assessing sample size (1963). Fourth part Research findings encompasses the majority of the paper and is organized according to the sequence of data analysis and research phases. This refers to as follows: descriptive analysis, factor analysis and co-relation analysis of three research scales. Among a number of specific data included, also taken into account in direct or indirect manner of questioning were teachers' beliefs and attitudes on: the quality of teaching, their expectations, pedagogical optimism, class environment, education and various forms of teacher training, curriculum, insight into student's individuality, perception of giftedness, competence to identify disabilities such as dyscalculia, etc. Taken into account was also teacher self-assesment in the field of: social competence, professional competence, emotional competence and insight into curricular documents and specific teaching practice with students with math learning disabilities. Related to classroom and outside of classroom activities taken into account were: student-oriented teaching activities, methods and actions of the participants in the research, involvement in mobility projects and action research, reflective aspects of teaching practice, following innovations in science, and command of ICT. Following the descriptive analysis and assessing differences among the participants considering socio-demographic factors and professional teacher profile, the author confirms the existence of various methodological restrictions in determining potential predictive aspects of certain variables, which leads to a conclusion of correlation analysis with a number of regression analyses. This is clear from the existing redundancies in results where for a number of variables it is dificult to determine whether they indicate or even explain the potential latent constructs in specific test conclusions in descriptive and inferential methods and actions. In regression analysis models are therefore included among the primary independent variables (gender, work experience, position, initial education, further education and training) also beliefs and participant self-assessment, which are separated by factor analyses of specific scales, and are as such included into predictive variables in the process of synthesis and filtrating of the results obtained by previous analyses. Regarding the insight into the specific needs of math students, the only significant predictor was proven to be the gender of the teacher, where female teachers showed a much higher level of self-assessment than their male colleagues. When considering professional competence, significant predictors were work experience and further education and training, where the more experienced teachers assessed their expertise on a much higher level compared to their colleagues with 5 years of work experience or less. In self-assessing social competence, two significant predictors were present, out of which a more relevant variable is workplace, whereas gender plays a less relevant role. In working with students with certain disabilities teacher’s workplace has proven to be the single and fairly significant predictor, which was confirmed in previous analyses, indicating the higher level of competence self-assessment among elementary school teachers, compared to the high school ones. The same has been established for adjusting to and helping students with learning disabilities that have negative attitude towards learning math, to whom elementary school teachers dedicated a substantially more amount of time. Regarding recognizing students with dyscalculia, elementary school teachers also assess their competence to detect children having this disability much higher than those working in high schools, however more thorough analyses have proven self-assessment in recognizing dyscalculia not to be consistent in any of the groups according to workplace with contemporary definition of dyscalculia. Teaching directed towards the student has three substantial predictors: workplace, gender and the acquired additional education. Teacher workplace has proven to be the best predictor for student-oriented teaching, demonstrating high level of adjustment to students among elementary school teachers, compared to high school teachers. Regarding the metacognitive aspect, it has been determined that gender and further education equally contribute to the metacognitive aspect of teacher performance. Female teachers have demonstrated a higher level of student support and the accompanying regressive model has established gender as the single statistically relevant factor in this dependant variable. Professional development is also influenced by gender, showing female teachers as more prone to professional training. However, further acquired education based on this aspect of teaching has turned out to be a better predictor. Fermale teachers also more frequently evaluate their work performance by taking into account student feedback on their teaching and are more prone to expert training through mobility projects, peer evaluation and action research. In the factor analysis of the teacher self-assessment scale it has been determined that the differences in the self-assessment of the professional competence are most prominent considering teacher work experience, where the more experienced ones consider themselves significantly more competent than the younger ones with up to five years work experience. Beliefs on student mathematical capabilities are a negative predictor, meaning that the teachers displaying a more optimistic attitude towards students and their capabilities have a slightly lower assessment of their professional competence, in connection to the group of elementary school teachers who graduated from the Faculty of Teacher Education. As the most prominent predictor of teachers’ professional competence have proven to be beliefs about curriculum, showing higher level of math competence assessment among teachers who consider curriculum aspects of modern education relevant. Workplace is the best predictor for self-assessment of social competence, demonstrating the connection between elementary school workplace and teacher self-assessment in working with children with specific disabilities in learning math and the notion of student-oriented teaching in general. These analyses have shown that elementary school teachers tend to offer help to students with learning difficulties more often than high school teachers, link tasks with everyday situations, help students that have negative attitude towards learning maths. In the frequency of certain actions and practice in the metacognitive aspect of teacher performance, the greatest connection is with beliefs about teacher social skills and self-assessment of awareness of student individuality in math teaching, which are strongly connected with elements of teaching oriented towards supporting students. Work experience is negatively connected with teaching methods related to student support, meaning that younger teachers are more inclined to emphasize the importance of persistence and work habits in learning maths and more frequently encourage attitudes towards learning among less successful students and believe and advocate in their teaching that every student can be successful at maths. Teachers with 30 years of work experience and more feel most competent in working with gifted students. Significant differences have been established among the most experienced teachers and both groups with up to 15 years work experience, showing teachers to be gradually more prone to working with gifted students with increasing work experience. Predictors of professional development and teacher training through professional meetings are attitudes and beliefs on curriculum and, to a lesser extent selfassessment of professional competence. Reflexive aspect of teacher performance is mostly determined by awareness on student individuality in math teaching, showing female teachers as more successful. Teachers that consider themselves more competent in the field of expertise more often participate in professional training through mobility projects, peer evaluation and action research. Male teachers consider themselves more competent in the field of expertise, whereas in the assessment of the methodical aspect of teaching, they consider themselves less successful than female teachers. The results show that female teachers follow guidelines from documents encompassed by National Frame Curriculum in preparation, performance and teaching evaluation significantly more frequently than male teachers. In the final combined regression analyses the author determines the relationship between socio-demographic factors and professional profile of teachers and