abstract: The research indicated effective mathematics teaching to be more complex than assuming the best predictor of student achievement in mathematics is the mathematical content knowledge of a teacher. This dissertation took a novel approach to addressing the idea of what it means to examine how a teacher's knowledge of mathematics impacts student achievement in elementary schools. Using a multiple case study design, the researcher investigated teacher knowledge as a function of the Mathematics Teaching Cycle (NCTM, 2007). Three cases (of two teachers each) were selected using a compilation of Learning Mathematics for Teaching (LMT) measures (LMT, 2006) and Developing Mathematical Ideas (DMI) measures (Higgins, Bell, Wilson, McCoach, & Oh, 2007; Bell, Wilson, Higgins, & McCoach, 2010) and student scores on the Arizona Assessment Collaborative (AzAC). The cases included teachers with: a) high knowledge & low student achievement v low knowledge & high student achievement, b) high knowledge & average achievement v low knowledge & average achievement, c) average knowledge & high achievement v average knowledge & low achievement, d) two teachers with average achievement & very high student achievement. In the end, my data suggested that MKT was only partially utilized across the contrasting teacher cases during the planning process, the delivery of mathematics instruction, and subsequent reflection. Mathematical Knowledge for Teaching was utilized differently by teachers with high student gains than those with low student gains. Because of this insight, I also found that MKT was not uniformly predictive of student gains across my cases, nor was it predictive of the quality of instruction provided to students in these classrooms. / Dissertation/Thesis / Ph.D. Curriculum and Instruction 2013

Σκοπός της παρούσης έρευνας είναι η μελέτη της συνεισφοράς που μπορεί να έχει η διδασκαλία μέσω επίλυσης προβλήματος στην κατανόηση των ανισώσεων και στην ανάπτυξη της ικανότητας μοντελοποίησης από μαθητές της Β΄ Γυμνασίου. Σχεδιάστηκε ένα οιονεί πείραμα που αφορούσε τη διαφοροποιημένη διδασκαλία του κεφαλαίου των ανισώσεων σε δύο τμήματα 17 μαθητών (πειραματική ομάδα και ομάδα ελέγχου). Αξιολογήθηκαν η κατάκτηση του γνωστικού αντικειμένου και η ικανότητα μοντελοποίησης-επίλυσης μιας κατάστασης-προβλήματος μέσω γραπτής δοκιμασίας, ενώ διενεργήθηκαν και συνεντεύξεις. Παράλληλα σκοπός μας ήταν η διερεύνηση της ικανότητας μοντελοποίησης-επίλυσης μιας κατάστασης-προβλήματος ενός ευρύτερου δείγματος μαθητών Β΄ Γυμνασίου, σχολείων αγροτικής, ημιαστικής και αστικής περιοχής. Πραγματοποιήθηκε επισκόπηση σε ένα δείγμα 39, 48 και 53 μαθητών αντίστοιχα, οι οποίοι κλήθηκαν να αντιμετωπίσουν γραπτώς μια κατάσταση-πρόβλημα, ενώ επίσης διενεργήθηκαν συνεντεύξεις. Από την ποσοτική και ποιοτική ανάλυση των αποτελεσμάτων προκύπτει ότι οι μαθητές μεσαίας επίδοσης είναι αυτοί που κυρίως επωφελήθηκαν από την διδασκαλία μέσω επίλυσης προβλήματος. Επιβεβαιώθηκε η διάκριση τεσσάρων επιπέδων ανάπτυξης στην ικανότητα δόμησης και χρήσης μαθηματικών μοντέλων από μέρους των μαθητών, ενώ κατέστησαν εμφανείς οι μεγάλες δυσκολίες που αντιμετωπίζουν οι τελευταίοι στην ανωτέρω διαδικασία. / The purpose of this research is to study the contribution of teaching through problem solving, in understanding inequalities and in the development of modeling capacity by students of the 2nd high school. A quasi-experiment was designed on differentiated instruction of inequalities in two classes of 17 students (experimental and control group). The achievement of the knowledge object and the ability to resolve a problem situation through mathematical modeling, were assessed by means of a written test and interviews. At the same time, our aim was to investigate the modeling capacity of a larger sample of 2nd high school students, of rural, suburban and urban schools. A survey was carried out in a sample of 39, 48 and 53 students respectively, who were invited to address a problem situation in writing, while interviews were also conducted. The quantitative and qualitative analysis of the results shows that medium performance students were the ones who largely benefited from the “teaching through problem solving” approach. The identification of four levels in the development of constructing and using mathematical models was confirmed, while became apparent major problems faced by the students in the above process.