This is a report of an investigation of children’s beliefs about the nature of mathematics, the nature of learning and helping factors for learning mathematics. The study aimed to investigate whether beliefs held by eight learners of eight to nine years of age could be articulated and portrayed. It aimed also to develop procedures to facilitate this process, to portray children’s beliefs from their responses to the research procedures, to provide insights into possible complexities and subtleties of young learners’ beliefs, to reflect upon the significance for the mathematics classroom of the insights gained, and to reflect upon the value of the procedures developed for the study. The research took the form of individual case studies of four girls and four boys of eight to nine years of age from two schools in suburban Melbourne. Four children were teacher-perceived low achievers in mathematics and four were teacher-perceived high achievers in mathematics. The children were each interviewed on ten occasions over a five-month period using thirty semi-structured, creative interviewing procedures that were developed or adapted for the study that included drawing, writing, discussing scenarios presented through photographs, video snippets and other children’s drawings, ordering of descriptors, and responding to questionnaires presented verbally. The interview data consisted of transcripts and artefacts. Some class administered tasks, lesson observations and interviews with the mathematics teachers provided background information. Analysis of interview responses was undertaken through a criss-cross examination in which themes were drawn from each child’s data. Responses were not judged for correctness or for a match to any predetermined categories and the researcher sought to take a stance of neutrality to the phenomena under study. The research suggests that teachers and others involved in the education of young learners of mathematics should know that: it is possible to gain insights into children’s beliefs about maths (the term used most commonly by the children), learning, and helping factors for learning maths; to gain insights into young children’s beliefs, it is important to have dialogue with the children to avoid making assumptions about their interpretations or meanings; the creative interviewing procedures developed for the present research are helpful as they can stimulate reflection and prompt conversation; young children’s beliefs can be complex, subtle, broad and deep; young children’s beliefs are individually constructed and differ from child to child; children may not see mathematics concepts in the same ways as their teachers and other adults; beliefs are sufficiently diverse and significant to affect the way children see the mathematics learning situation; although the beliefs of children of eight to nine years of age may, on the surface, appear simplistic and naïve, they are not necessarily so. Young learners are able to reflect on their own and others’ experiences and often construct complex beliefs. There is a lot happening in the minds of these children. The research suggests also that it is important that educators do not to make assumptions about: what children see as maths (or mathematics); what children see as learning; and what children see as helping factors for learning maths. A key factor facilitating children’s reflection and expression was the range of visual, verbal, and text-based creative interviewing strategies developed for the present study. The individual procedures provided suitable prompts to allow young children to articulate or represent their beliefs. The semi-structured procedures, through which ideas were explored on multiple occasions, followed by theme-based, criss-cross analysis of interview transcripts and artefacts, resulted in rich and trustworthy portrayals of beliefs, increasing the validity of the findings. The research provides the education community with insights into young children’s beliefs that are unlikely to emerge within the day to day activity of the classroom and, through the availability of the research procedures, facilitates further gaining of insights into beliefs either by classroom teachers or other researchers.

Syftet med arbetet var att åskådliggöra innebörden av att vara i matematiksvårigheter och elevers syn på hur de hanterar sin vardag. Undersökningen utgår från ett elevperspektiv och ambitionen var att elevens egna tankar och känslor skulle komma fram. Jag har använt mig av en kvalitativ ansats med observationer och semistandardiserade intervjuer. För alla eleverna i studien påverkas vardagen och skolan av att de har svårigheter i matematik. Det är allt från att planera sina fritidsaktiviteter till att veta när ett arbete i skolan ska vara inlämnat. Detta gör att föräldrar måste hjälpa till i denna planering i stor utsträckning även nu när barnen är tonåringar och borde klara sig själva lite mer. Eleverna själva är ofta inte medvetna om sina strategier för att klara de svårigheter som uppstår på grund av matematiksvårigheterna. Föräldrarna kan dock berätta mer om hur dessa strategier tar sig ut. Känslorna inför matematiklektionerna är ofta en uppgivenhet, ska de misslyckas idag igen och de känner sig stressade inför att kanske inte uppnå målen. Lärarens bemötande spelar här en stor roll och ett icke positivt bemötande gör att eleverna upplever att de är till besvär för läraren. Det leder till att eleverna får ett svagt självförtroende och olustkänslor inför matematiken. I jämförelse med andra studier så upplever eleverna i min studie inte att de blir stämplade till följd av att de får särskilt stöd, utan tvärtom så är det en lättnad att gå till stödgruppen. / This study focuses on pupil’s view how they handle mathematics learning disabilities in their daily life. The study is from pupil’s point of view and the ambition was to capture the pupil’s own thoughts and feelings. Observations and interviews were carried out with three pupils, in the age of 13, and their parents. The results show that all three of the pupils have difficulty in their daily life due to the mathematics learning disabilities. They have different problems from to plan their free time to plan when schoolwork should be submitted. The result of this is that the parents have to plan for them, even now when they are teenager’s and should be able to handle things like this by themselves. The pupils are not aware of the strategies they use to cope with the difficulties in mathematics. However, their parents are able to explain the strategies their children have developed. The emotions the pupils feel when they think about math class are that they feel dejected and they are under stress with thoughts as “will I fail again today”. It is very important how the teacher do talk to the pupils and this will affect the pupil’s self-confidence and they develop discomfort to mathematics. This study, unlike some other studies, does not show that pupils feel uncomfortable because they have special support. Instead they feel relief in visiting the support group.

The purpose of this study is to investigate the children’s mathematics learning through play. The questions to be answered are: how can teachers facilitate children’s mathematics learning through play? How can play affect children’s math learning? Are the children aware of the role of play in their mathematics learning? My method is based on interviews and observations. My informants are three teachers who work in two different preschools. I have interviewed the teachers and have been allowed to observe their work with the children. I have chosen to document the interviews and the observations with the help of a Dictaphone and digital camera. In the interviews, the teachers tell me that they try to capture and use mathematics in different contexts in everyday activities. They add that their duty is to actively work to draw the children’s attention to mathematical problems in an enjoyable and fun way. In my study, I clearly see that plays can facilitate the learning of mathematics.

<p>Syftet med arbetet var att åskådliggöra innebörden av att vara i matematiksvårigheter och elevers syn på hur de hanterar sin vardag. Undersökningen utgår från ett elevperspektiv och ambitionen var att elevens egna tankar och känslor skulle komma fram. Jag har använt mig av en kvalitativ ansats med observationer och semistandardiserade intervjuer. För alla eleverna i studien påverkas vardagen och skolan av att de har svårigheter i matematik. Det är allt från att planera sina fritidsaktiviteter till att veta när ett arbete i skolan ska vara inlämnat. Detta gör att föräldrar måste hjälpa till i denna planering i stor utsträckning även nu när barnen är tonåringar och borde klara sig själva lite mer. Eleverna själva är ofta inte medvetna om sina strategier för att klara de svårigheter som uppstår på grund av matematiksvårigheterna. Föräldrarna kan dock berätta mer om hur dessa strategier tar sig ut. Känslorna inför matematiklektionerna är ofta en uppgivenhet, ska de misslyckas idag igen och de känner sig stressade inför att kanske inte uppnå målen. Lärarens bemötande spelar här en stor roll och ett icke positivt bemötande gör att eleverna upplever att de är till besvär för läraren. Det leder till att eleverna får ett svagt självförtroende och olustkänslor inför matematiken. I jämförelse med andra studier så upplever eleverna i min studie inte att de blir stämplade till följd av att de får särskilt stöd, utan tvärtom så är det en lättnad att gå till stödgruppen.</p> / <p>This study focuses on pupil’s view how they handle mathematics learning disabilities in their daily life. The study is from pupil’s point of view and the ambition was to capture the pupil’s own thoughts and feelings. Observations and interviews were carried out with three pupils, in the age of 13, and their parents. The results show that all three of the pupils have difficulty in their daily life due to the mathematics learning disabilities. They have different problems from to plan their free time to plan when schoolwork should be submitted. The result of this is that the parents have to plan for them, even now when they are teenager’s and should be able to handle things like this by themselves. The pupils are not aware of the strategies they use to cope with the difficulties in mathematics. However, their parents are able to explain the strategies their children have developed. The emotions the pupils feel when they think about math class are that they feel dejected and they are under stress with thoughts as “will I fail again today”. It is very important how the teacher do talk to the pupils and this will affect the pupil’s self-confidence and they develop discomfort to mathematics. This study, unlike some other studies, does not show that pupils feel uncomfortable because they have special support. Instead they feel relief in visiting the support group.</p>

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.

Cooperative learning has emerged to be a preferred teaching-learning model in South Africa since the inception of Curriculum 2005 (C2005) emphasising Outcomes-based education (OBE). However, the documented success rate of cooperative learning in mathematics was experienced in small group sizes (emanating h m small class size) of about five learners. This study, therefore, aims at affording mathematics teachers and learners of crowded classes an opportunity to effectively use cooperative learning, namely the Hour-glass model in mathematics lessons. The prevalence of crowded classes in the majority of South African schools seems to inhibit the effectiveness of cooperative learning in mathematics. The big cooperative group size of about eight learners in South African context results in very complex lines of communication between learners. The teacher spends more time trying to manage off-task behaviour of learners instead of engaging them in active participation in the learning of mathematics. The combined quantitative and qualitative research methods were used. For the former, the study orientation in mathematics (SOM) questionnaire and the mathematics academic achievement test were used to collect data with regard to the influence of the Hour-glass model on the learners' learning skills in mathematics, and on the mathematics academic achievement respectively. A specific true experimental design, namely, the Solomon Four-group design, was used because of a large sample size (n > 500), and its credited ability to control the sources of threats to internal validity. For the latter the lesson observation and interviews were conducted to collect information about the influence of the Hour-glass model on learners' social skills during cooperative learning in mathematics. The groups that received the treatment (i.e. Hour-glass model) achieved higher scores of practical significance in mathematics academic achievement test than the groups that did not receive the treatment The Hour-glass model also yielded positive social skills among learners during mathematics learning. The teachers who applied the Hour-glass model revealed that they coped easier with crowded mathematics classes when using cooperative small groups. However, the Hour-glass model did not significantly influence learners' learning skills in mathematics. Certain logistical and administrative limitations emerged with regard to the implementation of the Hour-glass model in the usual school setting. / Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2004.

Worldwide the teaching and learning of mathematics pose a great challenge to mathematics teachers as learners' performance in the subject leaves much to be desired. This is particularly the case in South Africa where there was a great disparity in the development of teachers in the past. Extensive research has shown that many teachers in South Africa are under-qualified, especially in the teaching of mathematics at secondary schools. Those who are regarded as well qualified for teaching mathematics at secondary schools still experience problems in teaching certain sections of the syllabus, for example geometry, which is not offered at tertiary institutions. It is for this reason that the researcher, together with colleagues at an experimental school, joined forces to share the teaching of mathematics in what they referred to as "collaborative teaching". This work therefore involves a case study, which resulted after three teachers successfully achieved good matric results on employing this approach between 1993 and 1996. The study is based on an experimental design where both quantitative and qualitative methods were used. The aim of the study was to measure the extent to which collaboration between teachers affects the learning of mathematics in Grades 12. Two schools, the experimental school and a control school were involved. Learners from the experimental school were taught according to a collaborative approach whereas learners at the control school were taught conventionally (one teacher teaching all sections alone). This happened over a period of six months in 2001. Learners who were taught collaboratively outperformed those who were taught conventionally especially in the most problematic areas of the syllabus, namely geometry and trigonometry. The teachers who were involved in this approach, that is, collaborators, loved it to the extent that one of them applied it in another school where it improved their Grade 12 results tremendously. Learners who were taught according to this approach greatly appreciated it and wished they had been taught the same way in other subjects. This approach did not, however, significantly influence learners in their problem solving and information processing skills. In addition, one of the most serious limitations of this approach is to find a substitute for a teacher who leaves the team. / Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2006.

Cooperative learning has emerged to be a preferred teaching-learning model in South Africa since the inception of Curriculum 2005 (C2005) emphasising Outcomes-based education (OBE). However, the documented success rate of cooperative learning in mathematics was experienced in small group sizes (emanating h m small class size) of about five learners. This study, therefore, aims at affording mathematics teachers and learners of crowded classes an opportunity to effectively use cooperative learning, namely the Hour-glass model in mathematics lessons. The prevalence of crowded classes in the majority of South African schools seems to inhibit the effectiveness of cooperative learning in mathematics. The big cooperative group size of about eight learners in South African context results in very complex lines of communication between learners. The teacher spends more time trying to manage off-task behaviour of learners instead of engaging them in active participation in the learning of mathematics. The combined quantitative and qualitative research methods were used. For the former, the study orientation in mathematics (SOM) questionnaire and the mathematics academic achievement test were used to collect data with regard to the influence of the Hour-glass model on the learners' learning skills in mathematics, and on the mathematics academic achievement respectively. A specific true experimental design, namely, the Solomon Four-group design, was used because of a large sample size (n > 500), and its credited ability to control the sources of threats to internal validity. For the latter the lesson observation and interviews were conducted to collect information about the influence of the Hour-glass model on learners' social skills during cooperative learning in mathematics. The groups that received the treatment (i.e. Hour-glass model) achieved higher scores of practical significance in mathematics academic achievement test than the groups that did not receive the treatment The Hour-glass model also yielded positive social skills among learners during mathematics learning. The teachers who applied the Hour-glass model revealed that they coped easier with crowded mathematics classes when using cooperative small groups. However, the Hour-glass model did not significantly influence learners' learning skills in mathematics. Certain logistical and administrative limitations emerged with regard to the implementation of the Hour-glass model in the usual school setting. / Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2004.

Worldwide the teaching and learning of mathematics pose a great challenge to mathematics teachers as learners' performance in the subject leaves much to be desired. This is particularly the case in South Africa where there was a great disparity in the development of teachers in the past. Extensive research has shown that many teachers in South Africa are under-qualified, especially in the teaching of mathematics at secondary schools. Those who are regarded as well qualified for teaching mathematics at secondary schools still experience problems in teaching certain sections of the syllabus, for example geometry, which is not offered at tertiary institutions. It is for this reason that the researcher, together with colleagues at an experimental school, joined forces to share the teaching of mathematics in what they referred to as "collaborative teaching". This work therefore involves a case study, which resulted after three teachers successfully achieved good matric results on employing this approach between 1993 and 1996. The study is based on an experimental design where both quantitative and qualitative methods were used. The aim of the study was to measure the extent to which collaboration between teachers affects the learning of mathematics in Grades 12. Two schools, the experimental school and a control school were involved. Learners from the experimental school were taught according to a collaborative approach whereas learners at the control school were taught conventionally (one teacher teaching all sections alone). This happened over a period of six months in 2001. Learners who were taught collaboratively outperformed those who were taught conventionally especially in the most problematic areas of the syllabus, namely geometry and trigonometry. The teachers who were involved in this approach, that is, collaborators, loved it to the extent that one of them applied it in another school where it improved their Grade 12 results tremendously. Learners who were taught according to this approach greatly appreciated it and wished they had been taught the same way in other subjects. This approach did not, however, significantly influence learners in their problem solving and information processing skills. In addition, one of the most serious limitations of this approach is to find a substitute for a teacher who leaves the team. / Thesis (M.Ed.)--North-West University, Potchefstroom Campus, 2006.

Many authors have expressed concern regarding the extent of underachievement in mathematics. The role of language proficiency as a causal factor in this underachievement has been neglected. Researchers found sufficient evidence to conclude that language proficiency is related to mathematics achievement. In mathematics, symbolic language fills a dual role: It serves as an instrument of communication and as an instrument of thought by making the representation of mathematical concepts, structures and relationships possible (Esty & Teppo, 1996:45). According to Van Hiele (1988:5), language structure is a critical factor in the progression through the Van Hiele levels from the visual, concrete structures to the abstract structures. In this study, the influence of language proficiency on geometric thinking is investigated. 152 grade 8 and 9 learners completed two tests each. One test measured language proficiency in the learners' mother tongue. The second is a geometric test based on a Mayberry-type Van Hiele test for assessing learners' geometric thinking levels. Language proficiency was taken as the independent variable, and geometric thinking as the dependent variable. In the analysis of the results, the top 25 % and bottom 25% performers in the language proficiency test were chosen. Cohen's (1988) d-value was used to determine if there was a practical significant difference in the performance of the more proficient language learners and the less proficient language learners with respect to each of the first three Van Hiele levels. Results showed a practical significant difference between the performance of the more proficient language learners and the less proficient language learners with respect to each of the first three Van Hiele levels, but also with respect to the geometry test as a whole. In particular, two aspects of language proficiency, namely reading comprehension and vocabulary, appeared to be very strong predictors for geometric thinking on the first three Van Hiele levels (d ≥ 0,8). Key terms for indexing: geometry, geometry learning, mathematics learning, geometric thinking, language, language proficiency, geometry and language, mathematics and language. / Thesis (M.Sc. (Education)--North-West University, Potchefstroom Campus, 2004.