This study investigated the use of an advanced technological development (free GeoGebra software) within the secondary educational setting in four relatively under-resourced schools in the Gauteng Province of South Africa. This advancement is viewed as having the potential to promote the teaching and learning of complex ideas in mathematics, even within traditionally deprived communities. The focus in this study was on the teaching and learning of transformation geometry at Grade 9 and attainment was reflected in terms of the van Hieles' levels of geometrical thinking. A mixed methods approach was followed, where data was collected through lesson observations, written tests and semi-structured interviews. Four Grade 9 teachers from four schools were purposively selected, while twenty-four mathematics learners (six from each school) in the Tshwane metropolitan region were randomly selected. The teachers' lesson observations and interview outcomes were coded and categorised into themes, and the learners' test scripts were marked and captured. The analysis of test scores was structured according to the van Hieles' levels of geometric thought development. As far as the use of GeoGebra is concerned, it was found that teachers used the program in preparation for, as well as during lessons; learners who had access to computers or android technology, used GeoGebra to help them with practice and exercises. As far as the effect of the use of GeoGebra is concerned, improved performance in transformation geometry was demonstrated. / Dissertation (MEd)--University of Pretoria, 2016. / Science, Mathematics and Technology Education / MEd / Unrestricted

The purpose of the research study is to investigate the effect of peer instruction method on the 8th grade students&rsquo / mathematics achievement and mathematics attitudes in transformation geometry (fractals, rotation, reflection, translation) in crowded classrooms (more than 50 students). Besides, in this study it was aimed to investigate the gender differences regarding mathematics achievement and mathematics attitude. The study was conducted during the academic year 2009-2010. The sample was consisted of 112 eighth grade students from a public elementary school in K&uuml / &ccedil / &uuml / k&ccedil / ekmece district in Istanbul. Two classes, instructed by the researcher, were randomly assigned as experimental and control groups. The experimental group students were taught the subject transformation geometry through peer instruction method, while the control group students were taught the subject transformation geometry conventionally. Mathematics Achievement Test (MAT) and Attitude towards Mathematics Scale (ATMS) were administered to students as measuring instruments. The two-way ANCOVA and two-way ANOVA statistical techniques were performed in order to answer to the research questions. Results indicated that the peer instruction method has significant positive effects on students&rsquo / mathematics achievement and attitudes towards mathematics. Also, it was shown that there is not a significant difference between the female and male students&rsquo / mathematics achievement and mathematics attitudes.

Dem Erwerb geometrischer Kompetenzen wird in der Primarstufe im Allgemeinen eine wichtige Bedeutung zugemessen. Geometrische Inhalte finden sich in Deutschland sowohl in den nationalen Bildungsstandards als auch in den Lehrplänen der Bundesländer wieder. An einer strukturierten empirischen Aufarbeitung der Kompetenzentwicklung, der Fähigkeiten sowie der Schwierigkeiten der Schülerinnen und Schüler in der Auseinandersetzung mit diesen Inhalten mangelt es dennoch in einigen dieser Bereiche. Damit einhergehend fehlen auch erforderliche Grundlagen, die es Lehrkräften ermöglichen, Unterricht planen zu können und somit ein anschlussfähiges Verständnis bei den Schülerinnen und Schülern aufzubauen. Aus der Lehrerprofessionsforschung liegen Modelle vor, welche zentrale Wissensfacetten, die eine Rolle bei der unterrichtlichen Vermittlung fachlicher Inhalte spielen, beschreiben (u.a. Weinert, Schrader & Helmke 1990; Shulman 1986; Bromme 1992, 1997; Baumert & Kunter 2006, 2011; Ball, Thames & Phelps 2008). Insbesondere dem fachlichen sowie dem fachdidaktischen Wissen der Lehrperson wird eine entscheidende Rolle bezüglich des Lernerfolgs der Schülerinnen und Schüler zugeschrieben (u.a. Kunter et al. 2013). Ein geometrischer Inhaltsbereich, in dem bereits Wissen vorliegt, auf das Lehrerinnen und Lehrer zurückgreifen können, ist die Achsenspiegelung und -symmetrie. Dieses Wissen bezieht sich jedoch überwiegend auf die Sekundarstufe und ist wenig systematisch angelegt bzw. lückenhaft. Diese Arbeit verfolgt daher das Ziel, systematisch Wissen herauszuarbeiten, auf das Lehrkräfte zurückgreifen können, um im Geometrieunterricht der Grundschule fachlich fundierte Entscheidungen treffen zu können und im Bereich der Achsenspiegelung und Achsensymmetrie mit Bezug auf die individuellen Fähigkeiten der Schülerinnen und Schüler ein anschlussfähiges und tragfähiges Verständnis aufzubauen. Basierend auf einer integrativen Betrachtung der Facetten professionellen Wissens werden inhaltsspezifisch relevante Facetten des Wissens von Lehrkräften herausgearbeitet. Diese Perspektiven der Lehr-Lern-Forschung werden aufgegriffen und bieten eine Rahmung zur fachlichen und fachdidaktischen Einordnung der einzelnen Forschungsarbeiten dieser Dissertation. Die Forschungsschwerpunkte dieser Arbeit werden daraufhin ebenfalls in der Rahmung verankert, anschließend vorgestellt und ihre Ergebnisse zusammengefasst. • Der erste Forschungsschwerpunkt setzt sich mit wichtigen Grundlagen zu spezifischen Anforderungen von Aufgabenstellungen sowie typischen schwierigkeitsgenerierenden Merkmalen bei der Achsenspiegelung und Achsensymmetrie auseinander (vgl. Götz & Gasteiger 2019; Götz et al. 2020; Götz 2018). Dabei führt der Abgleich psychometrisch erfasster Aufgabenschwierigkeiten mit inhaltlich fachdidaktischen Analysen der Fehler und theoretischen Erkenntnissen zur Achsenspiegelung zu einem Kategorienschema mit unterschiedlichen schwierigkeitsgenerierenden Merkmalen bei Aufgaben zur Achsenspiegelung und deren Ausprägungen. • Der zweite Forschungsschwerpunkt nimmt das Wissen über das mathematische Denken von Schülerinnen und Schülern bei der Achsenspiegelung in den Fokus. Um dieser Frage nachzugehen, wurden in dieser Studie Äußerungen der Schülerinnen und Schüler während ihrer Lösungsprozesse bei Aufgaben zur Achsenspiegelung beobachtet und analysiert (vgl. Götz & Gasteiger 2022). Der Mehrwert dieser Arbeit liegt insbesondere in den Erkenntnissen zu fachdidaktischem Wissen bezüglich des Unterrichtens der Achsensymmetrie und Achsenspiegelung in der Grundschule. Ein Kategorienschema zu den schwierigkeitsgenerierenden Merkmalen bei der Achsenspiegelung, ein Überblick über Fehlerarten bei Aufgaben zur Achsensymmetrie sowie Zusammenstellungen, die Rückschlüsse auf das Verständnis von Schülerinnen und Schülern bei der Achsenspiegelung ermöglichen, bieten eine wertvolle Wissensbasis für Lehrkräfte in der Grundschule. Für die Unterrichtspraxis sind die Ergebnisse dieser Arbeit an unterschiedlichen Stellen relevant. Zum einen stellen die Ergebnisse dieser Arbeit ein Handlungsgerüst zur Erstellung adäquater Aufgaben, zum anderen liefern sie eine wertvolle Unterstützung bei der Kompetenzdiagnostik. Gleichzeitig geht aus den Ergebnissen dieser Arbeit die hohe Bedeutung adäquater Unterrichtsmaterialien hervor – wobei diesbezüglich wertvolle Hinweise zur Auswahl geeigneter Arbeitsmittel gegeben werden.

The study explores Mathematics educators’ use of students’ out-of-school experiences in the teaching of Transformation Geometry. This thesis focuses on an analysis of the extent to which students’ out-of-school experiences are reflected in the actual teaching, textbook tasks and national examination items set and other resources used. Teachers’ teaching practices are expected to support students’ learning of concepts in mathematics. Freudenthal (1991) argues that students develop their mathematical understanding by working from contexts that make sense to them, contexts that are grounded in realistic settings. ZIMSEC Examiners Reports (2010; 2011) reveal a low student performance in the topic of Transformation Geometry in Zimbabwe, yet, the topic has a close relationship with the environment in which students live (Purpura, Baroody & Lonigan, 2013). Thus, the main purpose of the study is to explore Mathematics teachers’ use of students’ out-of-school experiences in the teaching of Transformation Geometry at secondary school level. The investigation encompassed; (a) teacher perceptions about transformation geometry concepts that have a close link with students’ out-of-school experiences, (b) how teachers are teaching transformation geometry in Zimbabwe’s rural secondary schools, (c) the extent to which students’ out-of-school experiences are incorporated in Transformation Geometry tasks, and (d) the extent to which transformation geometry, as reflected in the official textbooks and suggested teaching models, is linked to students’ out-of-school experiences. Consistent with the interpretive qualitative research paradigm the transcendental phenomenology was used as the research design. Semi-structured interviews, Lesson observations, document analysis and a test were used as data gathering instruments. Data analysis, mainly for qualitative data, involved coding and categorising emerging themes from the different data sources. The key epistemological assumption was derived from the notion that knowing reality is through understanding the experiences of others found in a phenomenon of interest (Yuksel & Yildirim, 2015). In this study, the phenomenon of interest was the teaching of Transformation Geometry in rural secondary schools. In the same light, it meant observing teachers teaching the topic of Transformation Geometry, listening to their perceptions about the topic during interviews, and considering how they plan for their teaching as well as how students are assessed in transformation geometry. The research site included 3 selected rural secondary schools; one Mission boarding high school, a Council run secondary school and a Government rural day secondary school. Purposive sampling technique was used carefully to come up with 3 different types of schools in a typical rural Zimbabwe. Purposive sampling technique was also used to choose the teacher participants, whereas learners who sat for the test were randomly selected from the ordinary level classes. The main criterion for including teacher participants was if they were currently teaching an Ordinary Level Mathematics class and had gained more experience in teaching Transformation Geometry. In total, six teachers and forty-five students were selected to participate in the study. Results from the study reveal that some teachers have limited knowledge on transformation geometry concepts embedded in students’ out-of-school experience. Using Freudenthal’s (1968) RME Model to judge their effectiveness in teaching, the implication is teaching and learning would fail to utilise contexts familiar with the students and hence can hardly promote mastery of transformation geometry concepts. Data results also reveal some disconnect between teaching practices as espoused in curriculum documents and actual teaching practice. Although policy stipulates that concepts must be developed starting from concrete situations and moving to the abstract concepts, teachers seem to prefer starting with the formal Mathematics, giving students definitions and procedures for carrying out the different geometric transformations. On the other hand, tasks in Transformation Geometry both at school level and the national examinations focus on testing learner’s ability to define and use procedures for performing specific transformations at the expense of testing for real understanding of concepts. In view of these findings the study recommends the revision of the school Mathematics curriculum emphasising pre-service programmes for teacher professional knowledge to be built on features of contemporary learning theory, such as RME theory. Such as a revision can include the need to plan instruction so that students build models and representations rather than apply already developed ones. / Curriculum and Instructional Studies / D. Ed. (Curriculum Studies)