It is no unknown fact that South African learners underachieve in mathematics. Due to the fact that the quality of mathematics teaching is one possible factor that has an influence on learners' mathematics achievement, there are valid reasons questioning the conceptual mathematical knowledge of mathematics teachers. In order to facilitate conceptual understanding teachers themselves must possess profound mathematical knowledge. Apart from the influence of a teacher's knowledge for teaching, teachers' attitudes and beliefs play a meaningful role in the way they teach mathematics. The deficient nature of prospective and practising teachers' knowledge of school mathematics, as well as their attitudes and beliefs towards mathematics has serious implications for the training of prospective mathematics teachers. Literature reveals that a technologically enhanced environment can improve the conceptual learning of prospective mathematics teachers. The purpose of this study was to determine the influence of a dynamic technologically enhanced environment on the attitudes and beliefs, as well as the conceptualisation of prospective mathematics teachers regarding the function concept. In this study, prospective teachers were exposed to Geometer's Sketchpad®, a dynamic software programme providing a powerful learning context that promotes the investigation of algebraic relationships. In order to answer the research question, an explanatory mixed method design was used. In the quantitative part of the study, the Study Orientation Questionnaire in Mathematics and a function test were administered to prospective teachers. In the latter, conceptualisation of the function concept was measured in terms of four competence components, namely interpretation, modelling, translation and reification. In the qualitative part of the investigation semi-structured and task-based interviews were held with a group of prospective teachers. Analysis of the results revealed that the dynamic technologically enhanced environment did not contribute to an improvement of the prospective teachers' attitudes and beliefs. In fact, these decreased visibly. Regarding their conceptualisation, only the reification component showed a practically significant improvement. It therefore appears as if prospective teachers are not being prepared to benefit from the dynamic technologically enhanced environment. A model is proposed for the effective use of such a learning environment. The model involves that diagnostic assessment be made of prospective teachers' basic knowledge of the function concept, their study habits, their attitudes and beliefs with respect to mathematics, as well as their mathematics anxiety. The second component comprises recommendations made to prospective teachers as a result of the diagnostic assessment, as well as continuous support being offered as an integrated part of the mathematics module. Support is offered with respect to cognitive and meta-cognitive skills, affective factors and the creation of an advantageous technologically enhanced learning environment. Despite the restricted value of generalisation of the findings from this study, I still recommend the expansion, refining and implementation of the model so that prospective mathematics teachers can effectively benefit from a technologically enhanced environment. Key words for indexing: mathematics education, mathematics teacher education, teacher knowledge, prospective mathematics teachers, function concept, conceptual learning, tertiary education. / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2009.

School and learning readiness are continually being placed under the spotlight in an attempt to protect the grade 1 learner from possible learning failures, since unsuccessful learning experiences have negative effects even on a person's adult life. A factor that plays a significant role in the current teaching situation in the RSA is the fact that many grade 1 leamers are not instructed in their mother tongue and that they are not familiar with the concepts which are used in the language of learning and teaching. Another important factor is the fact that since 2004 learners may enrol as grade 1 learners at primary schools if they are five and a half years old and did not attend a grade R class. In this study the problems experienced by grade 1 leamers in primary schools since the implementation of Outcomes-based Education in the RSA in 1998, were determined by means of a pilot study. It was followed up by a literature study on the brain and factors that influence the working of the brain among young learners, as well as learning and factors that cannot only promote learning among this age group, but also disadvantage it. The literature was also consulted to look at stimulation of the child's development in the early development stage (0-9 years). As cross control for the study, the influence of the implementation of the RNCS was determined by means of a Likert-type questionnaire that was sent to grade1 teachers. As a result of factors that were identified from the above-named literature study and from the information acquired from the pilot project and Likert-type questionnaire, a conceptual stimulation programme for multicultural grade 1 classes was developed to provide in the shortcomings that currently occur among grade 1 learners. The Aptitude Test for School beginners (ASB), a standarised school readiness test of the HSRC, was used as a measuring instrument to acquire a differentiated image in a scientific way of the grade 1 learners (n = 39) for the purpose of this study. Thereafter, the learners were divided into an experimental and a control group. The conceptual stimulation programme was presented to the experimental group for ten consecutive weeks. The AS6 test was then conducted again with all the grade 1 learners involved. Both the first and final tests were marked according to the prescribed marking keys, interpreted according to the norm tables and statistically processed. The pre and post test scores of the two groups were compared and the experimental group improved in all 8 sections of the test battery, namely perception, spatial, reasoning, numerical, Gestalt, co-ordination , memory and verbal comprehension. In each of these categories there was an increase of d = 0.5 and more which according to Cohen's d-values, can be regarded as practically meaningful. The above-mentioned results indicate that the effective presentation of a conceptual stimulation programme to a group of multicultural grade 1 learners provides positive results and that the didactical methods used during the presentation can be recommended to grade 1 teachers in the current multicultural schools in the RSA. The programme can serve as a prototype to adress the shortcomings identified in the practise (questionnaires) and the literature study. Keywords for indexing are: stimulation programme, schoolreadiness, schoolreadiness programme, cognitive development, learning theories, conceptual learning, brain development. / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2006.

In this dissertation, the researcher examined teachers' understandings and attitudes about the teaching of argument and how professional development (PD) affected those understandings and attitudes. The teaching of argument is important because argument promotes critical thinking skills, authentic learning opportunities, increased conceptual development, and meaningful discussions about topics that are important to students. The researcher gathered data about 14 teachers’ understandings and attitudes and looked at them as a qualitative case study bounded by their common experience in a PD opportunity during the summer with return visits to PD in the fall and the following spring. The findings include: (1) teachers had little to no previous experience with argument in their own learning or teaching, (2) the teaching of a new strategy, like argument, requires ongoing, systematic PD, and (3) after PD the teachers developed more complex, nuanced understanding of and value for the teaching of argument. These findings were significant in that preservice teacher programs, district PD opportunities, and teacher-led PD learning communities must strategically and intentionally address argument for its power in the 21st century classroom.

The ability to make sense of a wide array of stimuli presupposes the human tendency to organize information in a meaningful way. Efforts to assess the degree to which students organize information meaningfully have been hampered by several factors including the idiosyncratic way in which individuals represent their knowledge either with words or visually. Concept maps have been used as tools by researchers and educators alike to assist students in understanding the conceptual interrelationships within a subject domain. One concept-map assessment in particular known as the construct-a-map task has shown great promise in facilitating reliable and valid inferences from student concept-map ratings. With all of its promise, however, the construct-a-map task is burdened with several rating difficulties. One challenge in particular is that no published rubric has been developed that accounts for the degree to which individual propositions are important to an understanding of the overall topic or theme of the map. This study represents an attempt to examine the psychometric properties of two construct-a-map tasks designed to overcome in part this rating difficulty. The reliability of the concept-map ratings was calculated using a person-by-rater-by-occasion fully crossed design. This design made it possible to use generalizability theory to identify and estimate the variance in the ratings contributed by the three factors mentioned, the interaction effects, and unexplained error. The criterion validity of the concept-map ratings was examined by computing Pearson correlations between concept-map and essay ratings and concept-map and interview transcript ratings. The generalizability coefficients for student mean ratings were moderate to very high: .73 and .94 for the first concept-mapping task and .74 and .87 for the second concept-mapping task. A relatively large percentage of the rating variability was contributed by the object of measurement. Both tasks correlated highly with essay and interview ratings: .62 to .81.

The purpose of this study was to examine how second graders use writing and language when they are learning to add multidigit numbers in mathematics class. Second grade students were taught addition conceptually with a focus on sharing their strategies and thought processes with each other during the mathematics lesson. Two social norms were established with students so that sharing information and asking for clarity when they did not understand would be natural and expected. Students kept a daily mathematics journal to answer the class's Problems of the Day. Patterns found in student journals indicated three stages of multidigit learning. In Stage One, students used little or no words to explain their solution, illustrations show students using counting by ones strategies. Stage Two represents students using appropriate mathematics strategies and vocabulary to explain their solutions in detail. Lastly, Stage Three consists of students solving multidigit problems with little or no word explaining their solution process and illustrations are few. Results of the study indicated that students' oral explanations of solutions to addition problems included more detail compared to students' written justification of similar problems.

Studiens syfte var att undersöka hur lärare som undervisar i matematik i årskurs 7-9 använder matematiskt språk i relation till ett vardagligt samt i vilken utsträckning de använder en kombination av de två språken. Studien var kvalitativ och metoderna som använts är observation med efterföljande samtal. Syftet med studien medför att ett lärarperspektiv var givet. De observerade lektionerna och samtalen resulterade i olika kategorier av begrepp vilka sedan resulterade i en teori kring hur lärare använder de olika språken vid genomgångar. Samtliga lärare använde både matematiskt och vardagligt språk vid genomgång medan endast ett fåtal använde språken i kombination. Studien visade att lärare använder sig av både matematiskt och vardagligt språk vid genomgångar med elever, men i varierande grad. / The purpose of the study was to investigate how teachers teaching mathematics in grades 7-9 use mathematical language in relation to a everyday language as well as the extent to which they use a combination of the two. The study was qualitative and the methods used are observation with subsequent conversations. The purpose of the study implies that a teacher's perspective was necessary. The observed lessons and the conversations resulted in different categories of concepts which then resulted in a theory of how teachers use the different languages during the lecture. All the teachers used both mathematical and everyday language during the lecture while only a few used the two languages in combination. The study showed that teachers use both mathematical and everyday language during lectures with students, but in varying extent.

Studien bygger på en interventionsforskning med syfte att tydliggöra att praktiska övningar bidrar till begreppsinlärning hos elever i lågstadiet. Studiens utformning bestod av en undervisningssekvens innehållande en förkunskapskoll, en praktisk övning, en genomgång med diskussion, följt av ytterligare en praktisk övning. Avslutningsvis fick eleverna utvärdera lektionen. Vid genomförandet av interventionen var eleverna indelade i halvklass. Undervisningssekvensen utfördes därför två gånger och totalt medverkade 22 elever från årskurs 3. För att väcka elevernas intresse och engagemang var de praktiska övningarna som utfördes konstruerade på ett lekfullt och lustfyllt sätt. Eleverna fick testa att dra en stol utan och med en person på, samt glida över golvet med och utan strumpor för att tydliggöra fenomenet friktion. De komplement som eleverna fick till de praktiska övningarna var ett protokoll där de skulle skriva i vad de tror skulle hända, vad som hände och varför det hände. Det gjordes en observation under genomförandet av interventionen där en av oss förde löpande protokoll på dator och den andra av oss ansvarade för lektionen. Under de praktiska övningarna gick båda runt i klassrummet och observerade eleverna. Resultatet visar att eleverna kan utöka sin begreppsinlärning med hjälp av praktiska övningar samt vid diskussion, men att fenomenet är komplext och det behövs både mer tid att arbeta med det samt träning för att befästa djupare kunskaper. / The study is based on an intervention research with the aim of clarifying that practical exercises contribute to concept learning for students in primary school. The design of the study consisted of a teaching sequence containing a prior knowledge check, a practical exercise, a review with discussion, followed by another practical exercise. Finally, the students had to evaluate the lesson. At the time of the intervention, the students were divides into half classes. The teaching sequence was therefore performed twice and a total of 22 students from year 3 participated. To arouse the students´ interest and commitment, the practical exercises that were performed were constructed in a playful and fun way. The students had to try to pull a chair without and with a person on, and slide over the floor with and without socks to clarify the phenomenon of friction. The complement that the students received to the practical exercises was a protocol where they would write in what they thought would happen, what happened and why it happened. An observation was made during the implementation of the intervention where one of us kept notes on the computer and the other of us was responsible for the lesson. During the practical exercises, both walked around the classroom and observed the students. The results show that students can expand their concept learning with the help of practical exercises and in discussion, but that the phenomenon is complex, and it takes both more time to work with it and training to consolidate deeper knowledge.

The purpose of this study was to examine the effect of using Lakatos’ heuristic method to teach the surface area of the cone (SAC) on students’ learning. The Lakatos (1976) heuristic framework and the Oh (2010) model of “the enhanced-conflict map” were employed as framework for the study. The first research question examined the impact of the Lakatosian heuristic method on students’ learning of the SAC, which was addressed in three sub-questions: the impact of the method on the students’ achievement, the impact of the method on their conceptual learning and the impact of the method on their higher order thinking skills. The second question examined whether the heuristic method of teaching the SAC helped students to sustain their learning better than the traditional method (Euclidean method). The third question examined whether the heuristic method of teaching SAC could change students’ readiness level, according to Bloom’s taxonomy. A pre-test and post-test quasi-experimental research design was used in the study that involved a total of 198 Grade 11 students (98 in the experimental group and 100 in the control group) from two schools in Cyprus. The instruments used for data collection were cognitive tests, lesson observations (video-recorded), interviews and questionnaire. Data was analysed using inferential statistics and the Oh (2010) model of the enhanced conflict map. Student achievement within time was the dependent variable and the method of training the independent variable. Therefore, time was the “within” factor and each group was measured three times (pre-test, post-test and delayed). The differences in students’ achievement within each group over time were examined. Results indicated that the average mean score achievement of the students in the experimental group was double that of the students in the control group. The Jun- Young Oh’s model of the enhanced conflict map showed that students in both groups changed from alternative conceptions to scientific conceptions with the experimental group showing greater improvement. It was also observed that from the post-test to delayed test, the Lakatosian method of teaching the SAC has a significant positive effect on students’ achievement at all levels of Bloom’s taxonomy, especially at the higher order thinking (HOT) levels (application and analysis-synthesis levels) as compared to the Euclidean method of teaching. In addition, the Lakatosian method helped the students to sustain their learning over time better than the Euclidean method did and also helped them to change their readiness level, especially at the HOT levels. The Lakatosian method helped students to foster skills that promote active learning. Of great importance was the use of mathematical language, as well as, the enhanced perception in the experimental group in comparison with the control group, through the use of the Lakatosian method. The results of this study are promising. It is recommended that pre-service teachers should be trained on how to effectively implement the Lakatosian heuristic method in their teaching. / Mathematics Education / D. Phil. (Mathematics, Science and Technology Education (Mathematics Education))

The purpose of this study was to examine the effect of using Lakatos’ heuristic method to teach the surface area of the cone (SAC) on students’ learning. The Lakatos (1976) heuristic framework and the Oh (2010) model of “the enhanced-conflict map” were employed as framework for the study. The first research question examined the impact of the Lakatosian heuristic method on students’ learning of the SAC, which was addressed in three sub-questions: the impact of the method on the students’ achievement, the impact of the method on their conceptual learning and the impact of the method on their higher order thinking skills. The second question examined whether the heuristic method of teaching the SAC helped students to sustain their learning better than the traditional method (Euclidean method). The third question examined whether the heuristic method of teaching SAC could change students’ readiness level, according to Bloom’s taxonomy. A pre-test and post-test quasi-experimental research design was used in the study that involved a total of 198 Grade 11 students (98 in the experimental group and 100 in the control group) from two schools in Cyprus. The instruments used for data collection were cognitive tests, lesson observations (video-recorded), interviews and questionnaire. Data was analysed using inferential statistics and the Oh (2010) model of the enhanced conflict map. Student achievement within time was the dependent variable and the method of training the independent variable. Therefore, time was the “within” factor and each group was measured three times (pre-test, post-test and delayed). The differences in students’ achievement within each group over time were examined. Results indicated that the average mean score achievement of the students in the experimental group was double that of the students in the control group. The Jun- Young Oh’s model of the enhanced conflict map showed that students in both groups changed from alternative conceptions to scientific conceptions with the experimental group showing greater improvement. It was also observed that from the post-test to delayed test, the Lakatosian method of teaching the SAC has a significant positive effect on students’ achievement at all levels of Bloom’s taxonomy, especially at the higher order thinking (HOT) levels (application and analysis-synthesis levels) as compared to the Euclidean method of teaching. In addition, the Lakatosian method helped the students to sustain their learning over time better than the Euclidean method did and also helped them to change their readiness level, especially at the HOT levels. The Lakatosian method helped students to foster skills that promote active learning. Of great importance was the use of mathematical language, as well as, the enhanced perception in the experimental group in comparison with the control group, through the use of the Lakatosian method. The results of this study are promising. It is recommended that pre-service teachers should be trained on how to effectively implement the Lakatosian heuristic method in their teaching. / Mathematics Education / D. Phil. (Mathematics, Science and Technology Education (Mathematics Education))

Aim of the theoretical part is to describe a relationship teacher-pedagogical assistant - immigrant child in primary school. I describe the definition of their competences, possible difficulties of their cooperation but also ideal humanities of the assistant and identify the relationship between him and the immigrant student. Continuing with describing special needs of this student, his position in a class and appropriate forms and methods of working with him/her. I slightly touch even the base of culture-free methods. In the practical part, the reader will follow my experiment with three immigrant students using method of conceptual learning and how this culture-free method influences students school achievements. The results of this study are positive not only in a way that the students become more proficient in the troubled school subject but they also gained a lot of self-confidence and autonomy. KEYWORDS Culture-free methods, conceptual learning, pedagogical assistant, prejudice, competences, personality, immigrant student, inclusive environment, autonomy, cooperation, special needs.