In the Teaching Principle (National Council of Teachers of Mathematics [NCTM], 2000), it explained that development and utilization of pedagogical content knowledge required teachers to continually increase their knowledges of mathematics content and pedagogy. This study researched the amalgamation of multi-faceted factors and inter-relatedness of pedagogical content knowledge (PCK), content knowledge for teaching mathematics (CKTM), and mentoring support perceptions throughout elementary and middle level student teachers’ and interns’ initial full-time teaching experiences. In the first article 13 elementary and seven middle grade student teachers’ are examined based on differences between pedagogical content knowledge and content knowledge for teaching mathematics. Standardized difference scores were calculated and compared using multivariate contrasts on certification level. Results showed statistically significant differences (p < .01) on all three CKTM domains but no statistically significant differences were found on any of the five Classroom Observation and Assessment for Teachers (COPAT) domains. Both groups had the highest mean difference in the CKTM number/concept domain. COPAT results showed middle grade level pre-service teachers primarily had all positive mean differences, in comparison to the elementary level pre-service teachers, which had all negative mean differences. In the second article the mathematics mentoring support perceptions of 11 first year teachers who participated in a year-long urban internship program were examined. Semi-structured interviews revealed that district and grade level campus mentors provided the most mathematics instruction and pedagogically-based support to both groups of interns. Middle school level interns relied more on their team of mathematics teachers and elementary level interns received more mathematical content support from their district mentor than did middle level interns. Pedagogical support was greatest in the areas of lesson design and implementation of classroom management strategies. In the third article 14 elementary and six middle level student teachers were observed and interviewed on general and content-specific pedagogical skills and perceptions. Results indicated both groups of student teachers perceived themselves as most competent in having lesson plans ready, routines evident, and utilizing studentcentered instruction. Conversely, both groups felt least competent in getting students on task quickly, using a variety of teaching strategies, using critical thinking skills, and handling inappropriate behavior effectively.

This doctoral research was concerned with the role of language and its implications for the learning, teaching and assessment of mathematics for children aged 4-9 years. Earlier research by the author had established language and assessment as bridges enabling learning although they had the potential to increase the divide between teacher and learner. Reflection raised the question on how children achieved in mathematics despite potential difficulties with language and assessment. Review of the literature concluded that resources and sociocultural norms were also bridges between learner and teacher. A model was established of the relationships and processes between all perceived variables that provided an external, theoretical structure to be evaluated against structuralist, pragmatic and integrational linguistic approaches and empirical outcomes. The overarching approaches adopted were institutional ethnography and case study. An appropriate methodology was devised whereby sophisticated ICT equipment captured all visual and speech events during classroom interactions. Frequency analysis at word level, content analysis at utterance level and discourse analysis at total speech level triangulated with content analysis of interviews and evaluation of documentation completed the empirical research. Data analysis revealed five registers of children’s talk. Evidence suggested that the peer-peer ‘conditioned talk’ used in focused group work was the most effective for learning as it enabled them to discern the small steps in the inferential leaps in discourse made by their teachers, work out problems together, inform their peers, share findings and reinforce each others’ learning. Learners’ language showed aspects of structural, pragmatic and integrational linguistics, confirming a conclusion of the literature review that the various linguistic approaches discussed should be used to support and not exclude each other. The contribution made to knowledge is the ethnomethodology provided by the model, ICT resource and the five registers of talk revealed by the linguistic approach to discourse analysis. Teachers would be able to understand nuances of language used by their pupils and acquire essential skills and tools to put into effect the personalised learning agenda. Peer-peer observation of teachers would be an appropriate platform for the observation of the different registers used by learners, the resources that generate those registers, and their most effective use to close the gap between natural language and the subject specific language of mathematics.

Η μετάβαση των μαθητών από την Πρωτοβάθμια στη Δευτεροβάθμια Εκπαίδευση κατά γενική ομολογία, δημιουργεί πολλές δυσκολίες στους μαθητές, οι οποίες είναι ιδιαίτερα έντονες στο μάθημα των Μαθηματικών. Αυτό συμβαίνει γιατί οι παραπάνω έρχονται αντιμέτωποι με έναν καινούριο χώρο όπου έχει έναν διαφορετικό τρόπο σκέψης και γραφής από αυτόν που είχαν συναντήσει στην Αριθμητική κατά τη φοίτηση τους στις τάξεις του Δημοτικού. Ο χώρος αυτός δεν είναι άλλος από τον ‘όμορφο’ χώρο της Άλγεβρας. Οι μαθητές λοιπόν, στο μάθημα της Άλγεβρας έρχονται αντιμέτωποι με σωρεία καινούριων πληροφοριών παρατηρώντας πολλές διαφορές αλλά και αρκετές ομοιότητες με τις γνώσεις που είχαν λάβει από την Αριθμητική, γεγονός που πολλές φορές τους δημιουργεί σύγχυση. Ιδιαίτερα έντονες είναι οι δυσκολίες τους στην κατανόηση βασικών εννοιών όπως της μεταβλητής αλλά και στην επίλυση μιας πρωτοβάθμιας εξίσωσης που παρουσιάζονται στην Άλγεβρα. Η παρούσα διπλωματική εργασία αποτελείται από μια μελέτη της υπάρχουσας βιβλιογραφίας και των ερευνών που έχουν γίνει προς αυτή τη κατεύθυνση από πλήθος Ελλήνων και ξένων ερευνητών καθώς και παράθεση ερωτηματολογίου σε μαθητές Γ’ Γυμνασίου, εντοπισμός των βασικών λαθών τους και συνήθων συγχύσεων. Αναλυτικότερα, η δομή της εργασίας έχει ως εξής: • Στο κεφάλαιο 1 γίνεται μια ιστορική αναδρομή για το πώς φτάσαμε από τα πρώτα πρώιμα μαθηματικά, στην χρήση αγνώστων και στην επίλυση εξισώσεων έως τα σύγχρονα χρόνια καθώς και τη μετάβαση από την Αριθμητική στην Άλγεβρα. • Στο κεφάλαιο 2 επισημαίνεται η δυσκολία που συναντούν οι μαθητές από τη μετάβασή τους στη δευτεροβάθμια εκπαίδευση. Παραθέτονται ενδεικτικά οι έρευνες ομάδων ερευνητών Ελλήνων και μη και επισήμανση συγκεκριμένων κοινών χαρακτηριστικών. Στο σημείο αυτό περιγράφουμε το σκοπό της έρευνας μας καθώς και την επιλογή των μαθητών που παραθέτουμε το ερωτηματολόγιο, τη μέθοδο συλλογής και καταγραφής των αποτελεσμάτων. • Στο κεφάλαιο 3 παραθέτουμε το ερωτηματολόγιο που δώσαμε στους μαθητές χωρισμένο σε Α και Β ομάδα και εξηγούμε το στόχο κάθε ερώτησης που θέσαμε. Κατόπιν, αναλύουμε εκτενώς τις απαντήσεις των παιδιών, παραθέτοντας και αυτούσιους διαλόγους που είχαμε μαζί τους. Εντοπίζουμε έτσι τα πιθανά λάθη και παρερμηνεύσεις που αναμέναμε αλλά και ό,τι επιπλέον προέκυψε από τις απαντήσεις τους. • Τέλος, στο κεφάλαιο 4 καταστρώνουμε ένα διδακτικό σχέδιο βασισμένο στα λάθη των μαθητών και τις λοιπές παρατηρήσεις που είδαμε νωρίτερα. Δίνουμε ένα πλάνο διδασκαλίας της άλγεβρας στις μαθητικές αίθουσες, με ποιους τρόπους θα πρέπει να μεταλαμπαδεύουμε τις γνώσεις και την «όρεξη» μας για την άλγεβρα και με ποια τεχνολογικά μέσα. Έπειτα συνδέουμε την διδασκαλία αυτή με ένα ερωτηματολόγιο ελέγχου επιτυχίας της προηγούμενης διαδικασίας. Κλείνουμε με τα συμπεράσματα της έρευνάς μας. / -

The purpose of this study was to examine preservice teachers' beliefs and experiences in learning about teaching mathematics for social justice during a one-semester mathematics methods course at a large university in the Southwest. The study examined their beliefs on three levels: a) their beliefs about issues of equity, diversity, and social justice in general; b) their beliefs about how issues of equity, diversity, and social justice relate to teaching (i.e., if and how the preservice teachers perceive themselves as critical educators); and c) their beliefs about teaching mathematics for social justice. This study also analyzed the preservice teachers' experiences as they learned about and then discussed lessons and projects that could be used for teaching mathematics for social justice. The purpose of analyzing their experiences was to document what the pre-service teachers identified as positive and negative influences on their learning to teach mathematics for social justice.

This study investigates the ways in which parents' and teachers' experiences and representations mediate their child's mathematics learning as they make the transition between home and school to either a multiethnic or mainly white school. In particular, it examines if the forms of mediation they adopt can shed light on the academic success of the child in school mathematics. The focus on mathematics learning has been chosen for the study because of its relative neglect, until recent times, to be seen as a subject influenced by cultural representations. Furthermore, there are significant implications in the relative neglect of understanding the achievement of ethnic minority pupils in mathematics. The research was framed by Vygotskian sociocultural theory and Wenger's (1998) communities of practice to explore the construction of meaning, identity and representations of practice. The amalgam of Wenger's communities of practice with sociocultural theory provided three key theoretical facets: (i) multiple levels of understanding in the form of meaning, practice and identity, (ii) the scope to explore the social and cultural worlds of the learner and (iii) understanding the ways that past experiences impact on current practice. Three different forms of qualitative data collection were used within the context of an ethnographic approach: (i) investigations in the form of classroom observations, (ii) in-depth semi-structured interviews and (iii) a child identity task. Twenty-two parents, eight teachers and fifty-eight children took part in the interviews, which form the main part ofthe data analysis. Out ofthese fifty-eight children, twenty-seven undertook the child identity task. The research took place in three schools with different ethnic make-up: a multicultural school, a mainly white school and a predominantly South Asian school. Two year groups were chosen, year 2 (ages 6/7 years) and year 6 (10/11 years), balancing high and low achievers. This study has provided data, which suggests that the way parents and teachers mediate the child's learning involves more than representations of mathematics. In making meaning of the mathematical, they draw on wider representations of the educational world, which include aspects like child development, notions of achievement, past experiences and the child's projected futures. This complex picture emerged from studying the highly interwoven aspects ofthe construction of meaning, identity and representations of practice. Representations of learning can be borrowed from both communities, providing the ethnic minority pupil with the potential to create hybrid representations of learning as they make the transition between home and school, which may be attributed a cultural status within the home. Each social actor has the potential to borrow from the home or school community to a greater or lesser degree. lfthe gap between the shared representations of the home and school are large, then this increases the likelihood of difficulties for the child in transition. However, the data suggests that even if the cultural representations of the home are very different from the school, the identification of high achievement and the engagement in mathematical activity at home can still provide success in learning. From the school community perspective, classrooms were represented by the teacher informants as 'cultureless' in both the multi ethnic and mainly white school. For example, in the multicultural school the teachers felt that there were so many ethnicities that differences were not visible. In the mainly white school, there were so few ethnic minority children that teachers also struggled to identify issues of culture. In the predominantly South Asian school, issues surrounding culture were brought to the forefront of the teacher discourse. However, in many ethnic minority homes, parents described how culture was influential in mediating representations ofleaming. This has implications in the educational arena with respect to the teachers' understanding of the transitional process that ethnic minority children undergo and the levels of visibility that culture and ethnicity is given in the school community.

Τα βασικά θέματα για την παρούσα Διπλωματική Εργασία είναι δύο: αφενός σκοπός μας είναι να εντοπίσουμε και να παρουσιάσουμε τους γεωμετρικούς κανόνες που χρησιμοποιούσαν, κατά την απόδοση των θεμάτων τους, οι καλλιτέχνες της Αναγέννησης, αφετέρου θέλουμε να αξιοποιήσουμε τη σύνδεση αυτή, μεταξύ τέχνης (κυρίως της ζωγραφικής) και γεωμετρίας, ώστε η διδασκαλία των γεωμετρικών εννοιών να αποκτήσει ένα πιο φιλικό χαρακτήρα για τους μαθητές. Ουσιαστικά η επιλογή του θέματος πραγματοποιήθηκε, εστιάζοντας στο διδακτικό στόχο που είχαμε κατά νου, ο οποίος αφορά τον τρόπο με τον οποίο μπορούν να συνδυαστούν δυο διαφορετικοί τομείς μεταξύ τους, είτε μοιάζουν να συνδέονται άμεσα ή έμμεσα, η ζωγραφική και η γεωμετρία, με τη βοήθεια της τεχνολογίας, ώστε να ενισχυθεί η κατανόηση, η κριτική σκέψη και ο δημιουργικός χαρακτήρας της μάθησης, κατά τη διαδικασία της διδασκαλίας της γεωμετρίας. / The main themes for this thesis is two fold: first goal is to identify and present the geometrical rules used in the performance of their subjects, the artists of the Renaissance, other want to make the connection between art (mainly painting) and geometry, that the teaching of geometric concepts to gain a more friendly nature for students. Essentially the choice of subject was focusing on the learning objectives we had in mind, which is the way in which they can combine two different sectors together, seem to be associated either directly or indirectly, painting and geometry, using technology to enhance understanding, critical thinking and creative nature of learning in the process of teaching geometry.

Matematik är en stor del av en människas liv och behövs för att kunna verka i samhället. Det betyder att matematikundervisningen bör utformas för att bidra till en förståelse för matematikens olika delar, så att alla elever får möjlighet att utvecklas för att just kunna fungera i samhället. Elever har emellertid olika förutsättningar och behov, vilka behöver tas i beaktande när läraren genomför undervisningen. Olika anpassningar görs i undervisningen på olika vis för att stötta och vägleda alla elevers utveckling och lärande. Syftet med studien är att, med utgångspunkt i Shulmans syn på PCK, undersöka hur lärares olika kunskaper påverkar mötet med elever i svårigheter inom matematikområdet bråk. Studien undersöker lärares resonemang om vilka svårigheter som finns inom matematikområdet bråk, vilka kunskaper lärarna anser sig behöva för att möta och stötta elever i svårigheter samt hur lärares arbete och olika anpassningar kan se ut. Studien grundar sig dessutom i det sociokulturella perspektivet, med fokus på lärares påverkan och syn på elevers lärande. Studiens resultat visar att lärare anser att elever är i svårigheter gällande del av antal, storleksordning av bråk samt förståelsen för vad som ska delas upp. Lärares resonemang gällande vilka elever det är som är i svårigheter är dock spridda. Vissa lärare anser att bråk generellt sett är ett svårt område inom matematiken, medan andra lärare anser att det enbart är vissa delar i området bråk som är generellt svårt för eleverna. Ytterligare åsikter är att elever som är i svårigheter med andra områden inom matematiken, även är i svårigheter gällande området bråk. Resultatet av studien visar dessutom att lärare anser att deras erfarenhet, kunskap och förståelse för eleverna samt kunskap om hur undervisningen ska läggas upp på bästa sätt, är de kunskaper som lärare behöver för att stötta och vägleda elever som är i svårigheter med matematikområdet bråk. Studiens resultat visar utöver ovanstående att lärarna i studien anser att det är viktigt med varierad och nivåanpassad undervisning för att stötta alla elever i deras lärande och då i synnerhet elever i svårigheter. Slutsatsen i studien är att lärarnas olika kunskaper om svårigheter, elever och undervisning, har stor påverkan på hur stöttning och vägledning för elever i svårigheter ser ut. / Mathematics is a major part of a person's life and is necessary for us to work in society. This means that mathematics education should be designed to contribute to an understanding of the different parts of mathematics, so that all students will be able to develop to function in society. However, students have different capability and needs, which need to be considered when the teacher teach. Various adjustments are made in the teaching in various ways to support and guide all students' development and learning. The purpose of the study is, based on Shulman's view of PCK, to study how teachers' various knowledge affects the encounter with students in difficulty with fraction in mathematics. The study examines the teachers' reasoning about the difficulties with fraction, which skills the teachers consider necessary to meet and support students in difficulty as well as how teachers' work and how different adaptations can be formed. In order to achieve the purpose of the study, semi structured interviews were used to examine teachers' perceptions of mathematical difficulties in the area fraction. The study is also based on the socio-cultural perspective, with focus on teachers 'influence and views on students' learning. The study result show that the teachers believe that students are in difficulty regarding fraction of number, order of fractional numbers and understanding of what to divide. Teachers' reasoning about which students are there are difficulties, however, are spread. Some of the teachers believe that fraction is generally a difficult area within mathematics, while other teachers consider that there are only certain parts in the area of fraction that are generally difficult for the students. Further opinions are that students in difficulty with other areas of mathematics are also in difficulty in the area of fraction. The results of the study also show that teachers consider that their experience, knowledge and understanding of the students, as well as knowledge of how the teaching is best designed, are the skills that teachers need to support and guide students in difficulty with the mathematic area fraction. The study's results show, in addition to the above, that the teachers in the study believe that it is important to provide varied and level-based education to support all students in their learning, especially students in difficulty. The conclusion of the study is that the teachers' different knowledge of difficulties, students and teaching has a major influence on the support and guidance for students in difficulty.

The findings of this investigation support the following conclusions. 1. The review of published research reports yielded conflicting conclusions as to the value of using minicalculators in teaching mathematics to students. 2. The use of minicalculators does not improve a student's mathematical achievement as measured by the Comprehensive Tests of Basic Skills. 3. The use of minicalculators does not improve a student's attitude as measured by the Likert-Tye Attitude Scale. 4. The use of minicalculators does not improve a student's self-esteem as measured by the Self-Esteem Inventory. 5. There is no significant difference in attitude, selfesteem or mathematical achievement between boys and girls in the area of mathematics.

This diploma thesis briefly introduces the concept of inquiry based teaching mathematics. In inquiry based teaching, emphasis is placed primarily on the active activity of the pupil, the aim of which is to discover a certain reality. This activity mainly involves solving problems and finding the right paths to achieve the right goal. We can call this path to a goal as a research. The inquiry process involves observing, formulating questions, identifying information, designing possible processes, and verifying them. In this educational method, the role of the teacher is not to pass the facts to the pupils, but to target them and to supervise the correctness of their practices. We can understand the teacher here as a guide, adviser or assistant on the path to the goal (discovery). In the thesis a few examples of mathematics of elementary and secondary schools are given, in which the inquiry based approach is applied. Mathematics in the given examples is not complicated, emphasis is put on the practical use. In the examples interdisciplinary relationships are developed as well, which is an important part of the inquiry based teaching. In each example, additional questions are provided.

CARVALHO, Alanna Oliveira Pereira. A avaliação diagnóstica como subsídio às práticas docentes no ensino da matemática: uma análise dos resultados das avaliações dos alunos do 2º ano do ensino fundamental do Estado da Bahia. 2014. 206f. – Dissertação (Mestrado) – Universidade Federal do Ceará, Programa de Pós-graduação em Educação Brasileira, Fortaleza (CE), 2014. / Submitted by Márcia Araújo (marcia_m_bezerra@yahoo.com.br) on 2014-04-28T11:50:20Z No. of bitstreams: 1 2014-DIS-AOPCARVALHO.pdf: 3392902 bytes, checksum: eecd9ab312ade4c528e9317fb8704733 (MD5) / Approved for entry into archive by Márcia Araújo(marcia_m_bezerra@yahoo.com.br) on 2014-04-28T14:30:53Z (GMT) No. of bitstreams: 1 2014-DIS-AOPCARVALHO.pdf: 3392902 bytes, checksum: eecd9ab312ade4c528e9317fb8704733 (MD5) / Made available in DSpace on 2014-04-28T14:30:53Z (GMT). No. of bitstreams: 1 2014-DIS-AOPCARVALHO.pdf: 3392902 bytes, checksum: eecd9ab312ade4c528e9317fb8704733 (MD5) Previous issue date: 2014 / Esta pesquisa tem como principal objetivo analisar os resultados da Provinha Brasil aplicada aos alunos do 2º ano do Ensino Fundamental da rede pública do Estado da Bahia dos municípios participantes do Programa Avalie Alfa, no ano de 2012. Após a divulgação dos resultados desta avaliação, ao final do primeiro semestre letivo, os professores são orientados a realizar atividades de intervenção pedagógica em sala de aula visando desenvolver as aprendizagens das crianças apontadas pela avaliação. Compreender a avaliação externa e em larga escala como propulsora de subsídios à pratica docente é um novo desafio para gestores e professores da Educação Básica. Para auxiliar a tecedura teórica sobre avaliação educacional, os estudos de autores contemporâneos como Luckesi, Perrenoud, Guerra, Vianna e Desprebiteris foram mencionados, a fim de fundamentar as análises realizadas. Para a construção da arquitetura teórica sobre a prática docente no ensino da Matemática nos anos inicias do Ensino Fundamental foram utilizadas as ideias de Piaget, Vygostsky, Vergnaud, Kami, Valente, Bicudo e Lorenzato. De cunho bibliográfico e documental, este estudo realizou, além da discussão dos resultados quantitativos da Provinha Brasil de Matemática 2012, uma investigação de cunho qualitativo, a fim de identificar como os resultados podem auxiliar os professores nas intervenções em sala de aula por meio de um trabalho pedagógico para o aprofundamento e a consolidação das competências e habilidades que os educandos demonstram ter nessa fase dos processos de ensino e de aprendizagem. De acordo com os resultados desse estudo, a Provinha Brasil aplicada e analisada nos moldes do Programa Avalie Alfa Bahia pode contribuir significativamente para o aprimoramento da prática docente no Ensino da Matemática e, por conseguinte, na aprendizagem dos estudantes, uma vez que possibilita por parte do professor a compreensão acerca das competências e habilidades que devem ser consolidadas pelos estudantes nessa etapa de escolarização. Dessa forma, o presente estudo apresenta perspectivas de análise e de intervenção para auxiliar a ação docenteno trabalho pedagógico na área da Matemática.