School science education prepares learners to study science at a higher level, prepares them to follow a career in science and to become scientific literate citizens. It is the responsibility of the educator to ensure the learners’ conceptual framework is developed to the extent that secures success at higher level studies. The purpose of this study was to test the effectiveness of conceptual change teaching strategies on the conceptual development of grade 11 learners on Newton’s second law of motion. The two strategies employed were the cognitive conflict strategy and the development of ideas strategy. A sequential explanatory mixed-method research design was used during this study. The qualitative data were used to elucidate the quantitative findings. The quantitative research consisted of a quasi-experimental design consisting of a single-group pre-test–post-test method. During the qualitative part of the research a phenomenological research approach was utilised to gain a better understanding of participants’ learning experiences during the intervention. The quantitative research made use of an adapted version of the Force Concept Inventory (FCI). The data collected from the pre-test were used to inform the intervention. The intervention was videotaped and the video analysis or qualitative data analysis was done. After the intervention the post-test was written by the learners. Hake’s average normalised learning gain <g> from pre- to post-scores was analysed to establish the effectiveness of the intervention. The two sets of results (quantitative and qualitative) were integrated. Information from the qualitative data analysis was used to support and explain the quantitative data. The quantitative results indicate that there was an improvement in the students’ force conception from their initial alternative conceptions, such as that of an internal force. Especially the learners’ understanding of contact forces and Newton’s first law of motion yielded significant improvement. The qualitative data revealed that the understanding of Newton’s second law of motion by the learners who partook in this study did improve, since the learners immediately recognised the mistakes made when confronted with the anchor concept. The cognitive conflict teaching strategy was effective in establishing the anchor concept of force which proved to be useful as bridging concept in the development of ideas teaching strategy. The data from both datasets revealed that the cognitive conflict teaching strategy for the initial part of the intervention was effective. It was evident that for development of the idea teaching strategy the two data sets revealed mixed results. Recommendations were made for future research and implementation of conceptual development teaching strategies. / MEd (Natural Sciences Education), North-West University, Potchefstroom Campus, 2014

Not only is Quantum Mechanics (QM) conceptually rich, it is also a theory that physics students have found abstract and technically formidable. Nevertheless, compared to other classical topics of physics, university students’ understanding of QM has received minimal attention in the physics education literature. The principal purpose of this study was to characterize the variation in the ways that undergraduate physics students depict the basic concepts of QM and to extrapolate the results to scaffold possible changes to instructional practices at the university that provided the context for the study. In so doing, an adaptation of a developmental phenomenographic perspective was chosen. Empirically, the study was approached through in-depth interviews with 35 physics students from two Ethiopian governmental universities after they had been exposed to the traditional QM course for one-third of a semester. Interview responses were analyzed using phenomenographic approach where a picture of students’ depictions was established for each quantum concept by expounding the given responses. For each basic quantum concept addressed, the structure of the description categories was separately constructed, and overall, it was found that naive, quasi-classical ontology and/or variants of classical ways of visualization are dominant in students’ responses. For example, it was found that students’ depictions of the photon concept could be described with three distinct categories of description, which are (a) classical intuitive description, (b) mixed model description and (c) quasi-quantum model description. Similarly, the findings revealed that it is possible to establish three qualitatively different categories of description to picture students’ depictions of matter waves, namely, (a) classical and trajectory-based description, (b) an intricate blend of classical and quantum description and (c) incipient quantum model description. Likewise, it was found that students’ depictions of uncertainty principle can be described as: (a) uncertainty as classical ignorance, (b) uncertainty as measurement disturbance and (c) uncertainty as a quasi-quantum principle. With regard to learning QM, the categories of description made clear several issues: most students did not have enough knowledge to depict the basic concepts of QM properly; they were influenced by the perspective of classical physics and their perceptions in making explanations about QM; and they also applied mixed ideas, one based on their classical model and the other from newly introduced QM. These results are also supported by the findings of previous studies in similar domains. Findings from the study were used to guide the design of multiple representations-based instructions and interactive learning tutorials on the conceptual aspects of QM that has been shown to address specific difficulties identified in the study. Theoretical and practical implications of the study, as well as potential future considerations are drawn. / Mathematics, Science and Technology Education / D. Phil. (Mathematics, Science and Technology Education)

Not only is Quantum Mechanics (QM) conceptually rich, it is also a theory that physics students have found abstract and technically formidable. Nevertheless, compared to other classical topics of physics, university students’ understanding of QM has received minimal attention in the physics education literature. The principal purpose of this study was to characterize the variation in the ways that undergraduate physics students depict the basic concepts of QM and to extrapolate the results to scaffold possible changes to instructional practices at the university that provided the context for the study. In so doing, an adaptation of a developmental phenomenographic perspective was chosen. Empirically, the study was approached through in-depth interviews with 35 physics students from two Ethiopian governmental universities after they had been exposed to the traditional QM course for one-third of a semester. Interview responses were analyzed using phenomenographic approach where a picture of students’ depictions was established for each quantum concept by expounding the given responses. For each basic quantum concept addressed, the structure of the description categories was separately constructed, and overall, it was found that naive, quasi-classical ontology and/or variants of classical ways of visualization are dominant in students’ responses. For example, it was found that students’ depictions of the photon concept could be described with three distinct categories of description, which are (a) classical intuitive description, (b) mixed model description and (c) quasi-quantum model description. Similarly, the findings revealed that it is possible to establish three qualitatively different categories of description to picture students’ depictions of matter waves, namely, (a) classical and trajectory-based description, (b) an intricate blend of classical and quantum description and (c) incipient quantum model description. Likewise, it was found that students’ depictions of uncertainty principle can be described as: (a) uncertainty as classical ignorance, (b) uncertainty as measurement disturbance and (c) uncertainty as a quasi-quantum principle. With regard to learning QM, the categories of description made clear several issues: most students did not have enough knowledge to depict the basic concepts of QM properly; they were influenced by the perspective of classical physics and their perceptions in making explanations about QM; and they also applied mixed ideas, one based on their classical model and the other from newly introduced QM. These results are also supported by the findings of previous studies in similar domains. Findings from the study were used to guide the design of multiple representations-based instructions and interactive learning tutorials on the conceptual aspects of QM that has been shown to address specific difficulties identified in the study. Theoretical and practical implications of the study, as well as potential future considerations are drawn. / Mathematics, Science and Technology Education / D. Phil. (Mathematics, Science and Technology Education)

In 2005 the opportunity to apply the New Zealand ‘Numeracy’ approach to teaching Mathematics was extended into the secondary school sector. The goal was to alter teachers’ pedagogy so that ‘sense making’ rather than ‘instruction’ was the core objective of their lessons. Ultimately it is hoped that along with a familiarity and comprehension of Number will come a relatively seamless acquisition of the fundamentals of Algebra. This paper will present details of this approach for teaching Number, the status of Number in the secondary school curriculum, the focus and ramifications of teaching for understanding, as opposed to assimilating and learning to apply algorithms, and will also consider evidence of the effectiveness of the initiative.

The study investigated errors made by Grade 9 learners when simplifying algebraic expressions. Eighty-two (82) Grade 9 learners from a rural secondary school in Limpopo Province, South Africa participated in the study. The sequential explanatory design method which uses both quantitative and qualitative approaches was used to analyse errors in basic algebra. In the quantitative phase, a 20-item test was administered to the 82 participants. Learners’ common errors were identified and grouped according to error type. The qualitative phase involved interviews with selected participants. The interviews focused on each identified common error in order to establish the reasons why learners made the identified errors. The study identified six (6) common errors in relation to simplifying algebraic expressions. The causes of these errors were attributed to poor arithmetic background; interference from new learning; failure to deal with direction and operation signs; problems with algebraic notation and misapplication of rules. / Mathematics Education / M. Ed. (Mathematics Education)

The following study aims to examine the special mathematical knowledge needed in order to teach mathematics. Furthermore, the study attempts to explore how teachers’ views on the knowledge needed in order to teach mathematics affects their student’s opportunities to develop their conceptual understanding. Qualitative and quantitative empirical data was attained by observations and complementary interviews. A total of three teachers, all working at the same school, was observed and interviewed. The study used Ball, Thames &amp; Phelps (2008) practice-based theory of mathematical knowledge for teaching, MKT, as its theoretical framework when analyzing the empirical data. The result of the observations displays that math teachers tend to use common content knowledge far more than specialized content knowledge during their lessons. The outcome of this also study reveals that there is a tendency among teachers to interfuse mathematical concepts with terminology. Conceptual understanding is equated with the use of correct terminology. The students are not exposed to the underlying ideas of the mathematical concepts. The study also concludes that there seems to be a sectioning between the mathematical content taught in grade 4-6 from the rest of the content being taught in elementary school, with a low number of connections being made between mathematical topics and concepts included in the curriculum.

In 2005 the opportunity to apply the New Zealand ‘Numeracy’ approach to teaching Mathematics was extended into the secondary school sector. The goal was to alter teachers’ pedagogy so that ‘sense making’ rather than ‘instruction’ was the core objective of their lessons. Ultimately it is hoped that along with a familiarity and comprehension of Number will come a relatively seamless acquisition of the fundamentals of Algebra. This paper will present details of this approach for teaching Number, the status of Number in the secondary school curriculum, the focus and ramifications of teaching for understanding, as opposed to assimilating and learning to apply algorithms, and will also consider evidence of the effectiveness of the initiative.

The study aimed to explore misconceptions that Grade 9 learners at a school in Soweto had concerning the topic of fractions. The study was based on the ideas of constructivism in a bid to understand how learners build on existing knowledge as they venture deeper into the development of advanced constructions in the concept of fractions. A case study approach (qualitative) was employed to explore how Grade 9 learners describe the concept of fractions. The approach offered a platform to investigate how Grade 9 learners solve problems involving fractions, thereby enabling the researcher to discover the misconceptions that learners have/display when dealing with fractions. The research allowed the researcher to explore the root causes of the misconceptions held by learners concerning the concept of fractions. Forty Grade 9 participants from a township school were subjected to a written test from which eight were purposefully selected for an interview. The selection was based on learners’ responses to the written test. The researcher was looking for a learner script that showed application of similar but incorrect procedures under specific sections of operations of fractions, for example, multiplication of fractions. Both performance extremes were also considered, the good and the worst performers overall. The written test and the interviews were the primary sources of data in this study. The study revealed that learners have misconceptions about fractions. The learners’ definitions of what a fraction is were neither complete nor precise. For example, the equality of parts was not emphasised in their definitions. The gaps brought about by the learner conception of fractions were evident in the way problems on fractions were manipulated. The learners did not treat a fraction as signifying a specific point on the number system. Due to this, learners could not place fractions correctly on the number line. Components of the fraction were separated and manipulated as stand-alone whole numbers. Consequently, whole number knowledge was applied to work with fractions. A lack of conceptual understanding of equivalent fractions was evident as the common denominator principle was not applied. In the multiplication of fractions, procedural manipulations were evident. In mixed number operations, whole numbers were multiplied separately from the fractional parts of the mixed number. Fractional parts were also multiplied separately, and the two answers combined to yield the final solution. In the division of fractions, the learners displayed a lack of conceptual knowledge of division of fractions. Operations were made across the division sign numerators separate from the denominators. This reveals that a fraction was not taken as an outright number on its own by learners but viewed as one number put on top of the other which can be separated. Dividing across, learners rendered division commutative. A procedural attempt to apply the invert and multiply procedure was also evident in this study. Learners made procedural errors as they showed a lack of conceptual understanding of the keep-change-flip division algorithm. The study revealed that misconceptions in the concept of fraction were due to prior knowledge, over-generalisation and presentation of fractions during instruction. Constructivism values prior knowledge as the basis for the development of new knowledge. In this study, learners revealed that informal knowledge they possess may impact negatively on the development of the concept of fractions. For example, division by one-half was interpreted as dividing in half by learners. The prior elaboration on the part of a whole sub-construct also proved a barrier to finding solutions to problems that sought knowledge of fractions as other sub-constructs, namely, quotient, measure, ratio and fraction as an operator. Over generalisation by learners in this study led to misconceptions in which a procedure valid in a particular concept is used in another concept where it does not apply. Knowledge on whole numbers was used in manipulating fractions. For example, for whole numbers generally, multiplication makes bigger and division makes smaller. The presentation of fractions during instruction played a role in some misconceptions revealed by this study. Bias towards the part of a whole sub-construct might have limited conceptualisation in other sub-constructs. Preference for the procedural approach above the conceptual one by educators may limit the proper development of the fraction concept as it promotes the use of algorithms without understanding. The researcher recommends the use of manipulatives to promote the understanding of the fraction concept before inductively guiding learners to come up with the algorithm. Imposing the algorithm promotes the procedural approach, thereby depriving learners of an opportunity for conceptual understanding. Not all correct answers result from the correct line of thinking. Educators, therefore, should have a closer look at learners’ work, including those with correct solutions, as there may be concealed misconceptions. Educators should not take for granted what was covered before learners conceptualised fractions as it might be a source of misconceptions. It is therefore recommended to check prior knowledge before proceeding with new instruction. / Mathematics Education / M. Ed. (Mathematics Education)

This study employed a qualitative approach to investigate the use of cooperative learning to enhance conceptual understanding of trigonometry in a Grade 11 mathematics classroom, conducted at a high school in Moletlane Circuit, Capricorn District in Limpopo Province, South Africa. A single case study was used as a research design to get an in-depth analysis and collect detailed data using semi-interviews and lesson observation of the cooperative learning of trigonometry in Grade 11 from the learners and the teacher. Participants were purposely chosen and consisted of (n=30) Grade 11 mathematics learners and their mathematics teacher. Data from the participants were collected through semi-structured interviews and observation, with the aid of observation guide (Appendix C) for three weeks. The salient findings from the study showed how cooperative learning was used, research questions, the approaches, the teacher did not adequately highlight the importance of trigonometry to students without integrating the topic to real-life situations. Some students said that the teacher did not teach trigonometry in a manner that they understood, which made trigonometry challenging for them. Concerning cooperative learning, the study found that many learners were passively engaged, listened to or watched the teacher. Mainly, the study recommends teacher-training institutions to host practical workshops to help teachers integrate theoretical training and practical cooperative learning experience. While this study was qualitative in nature, future researchers could conduct quantitative data collection. This would allow for the collection of numerical findings through survey questionnaires. / Mathematics Education / M. Ed. (Mathematics Education)

Es ist eine weit verbreitete Wahrnehmung, dass der Übergang zwischen der Mathematik der gymnasialen Oberstufe und der Mathematik an der Universität für Studierende problematisch sein kann. Besondere Verständnisschwierigkeiten in Bereich der lineare Algebra (lA) bereiten den Studierenden die verschiedenen Herangehensweisen auf diesen beiden Ebenen. Dies lässt sich auf die strukturell-axiomatischer Herangehensweisen an die lA an der Universität, im Gegensatz zu ihrer arithmetisch-geometrischen Darstellung in der Schule, zurückführen. Dies bedingt ebenfalls Unterschiede im prozeduralen und konzeptuellen Verständnis. Ziel dieser Arbeit ist es, zu untersuchen, wie Schüler konzeptuelles Verständnis, Bezug nehmend auf die Theorien von concept definition/image in Verbindung mit multiplen Modi der Beschreibung und des Denkens von Konzepten wie Bilinearität z.B. Skalarprodukt und Multilinearität z.B. Determinanten gewinnen können. Um dies zu erreichen wurde eine substanzielle Lehr-Lernumgebung unter Verwendung einer dynamischen Geometriesoftware (DGS) entwickelt. Die Lerneinheit wurde an einem Berliner Gymnasium eingesetzt und dabei ein vollständiger design-based research Zyklus durchlaufen und eine multiple-level Datenanalyse durchgeführt. Die Ergebnisse der Untersuchung zeigen nicht nur, dass eine Erweiterung der Vorstellungen der Schüler, eine Entwicklung multipler Denkmodi und ein Gewinn tieferen konzeptuellen Verständnisses in der lA erfolgreich vermittelt werden können, sondern geben auch Einblicke in ein mögliches theoretisches Modell, mit dessen Hilfe sich diese Prozesse weiter untersuchen lassen. Weiterhin werden die interaktiven Lehr-Lernmaterialien für die weitere Verwendung im Rahmen von Lehre und Forschung zur Verfügung gestellt. Es öffnen sich neue Forschungsfragen hinsichtlich lokalen Axiomatisierens in der lA der gymnasialen Oberstufe, welches auf einer Integration geometrischer, algebraischer und axiomatischer Denkmodi, unterstützt durch DGS, basieren könnte. / A common perception among researchers in mathematics education is that the transition between secondary- and tertiary level of mathematics may be problematic for the students. In particular, the exact and abstract nature of the theory of Linear algebra versus its arithmetic-geometric presentation in school appears to be difficult for the novice students. The application of properties for defining concepts at university in contrast to their usage for describing concepts in school points out a possible occurrence of obstacles for learning and discrepancies in procedural and conceptual understanding. The aim of this study is to examine how could upper-high school students develop a conceptual understanding based on concept definition and concept image in connection to multiple modes of description and thinking about concepts such as bi-linearity exemplified by the dot product of vectors and multi-linearity exemplified by determinants. In order to achieve this, I have created a specific teaching/ learning sequence in a dynamic geometry environment (DGE), then implemented it and evaluated it in a high school in Berlin, following a complete cycle of design-based research and conducting a multiple-level data analysis. The findings of the study show not only that widening students' concept images, developing multiple modes of thinking and gaining deeper conceptual understanding can successfully be mediated by dynamic geometries, but also give insights into an eventual theoretical model of how can they be further examined. Moreover, the study promotes authorized open-source interactive teaching/ learning materials for further sustainable practice and research. It opens new research questions about revisiting axiomatic approaches on local levels in upper high-school Linear algebra which may base on the integration of all three modes of description and thinking geometric, algebraic and abstract possibly facilitated by DGE. / Честа перцепција кај многумина истражувачи во областа на математичкото образование е дека транзицијата помеѓу средното и високото образование по математика може да биде проблематична за студентите. Егзакноста и апстрактноста на теоријата по Линеарна алгебра наспроти нејзината аритметичко-геометриска презентација во средното гимназиско образование се покажува како особено тешка за студентите. Примена на својствата на математичките поими за нивно дефинирање на универзитетско ниво наспроти нивното употреба за опишување на претходно дефинирани поими на училишно ниво, укажува на можна појава на тешкотии при нивното изучување и несовпаѓање на процедуралното и концептуалното разбирање на истите. Целта на оваа студија е да истражи како средношколците би можеле да развијат концептуално разбирање на поимите врз основа на концепт дефиниција и концепт слика во врска со мулти-моди на мислење, конкретно за поими како билинеарност, пр. скаларен производ на вектори, и мултилинеарност, пр. детерминанти. За да ја постигнам оваа цел, креирав наставна содржина поддржана од еден динамичен геометриски систем (ДГС) и следејќи целосен циклус на т.н. design-based research и спрoведувајќи мулти-анализа на податоци, истата ја имплементирав и евалуирав во едно средно училиште во Берлин. Резултатите од студијата укажуваат не само на фактот дека проширувањето на концепт сликите на учениците, развојот на мулти-моди на мислење и стекнувањето на длабоко концептуално разбирање на поимите можат да бидат успешно посредувани од ДГС туку овозможија и увид во еден теоретски модел за тоа коко тие можат понатаму да се истражуваат. Уште повеќе, студијата промовира авторизирани open-source интерактивни материјали за предавање и учење на содржините кои може да служат за понатамошни одржливи истражувања и развој. Студијата отвора нови истражувачки прашања за средношколската Линеарна алгебра која може да се базира на интеграција на сите три моди на мислење, геометриски, алгебарски и апстрактен, поддржан од ДГС.