In the early 1990s Graham Nuthall and Adrianne Alton-Lee developed a model of the learning and remembering process which has profound implications for teaching and learning at all levels. Using their model they were able to predict what selected primary school students would and would not learn and remember from the teaching of a series of Science and Social Studies units and to do so with an accuracy of between 80 to 85 per cent. The Nuthall model states that for a student to learn and remember a new fact or concept he or she needs three to four learning opportunities with the complete set of information needed to learn the new fact or concept, and a gap of no more than two days between any pair of those two learning opportunities. It had always been Graham Nuthall's intention to test the model he developed with Adrienne Alton-Lee in a series of experiments. Tragically, Professor Graham Nuthall died before this was possible. The ten experiments in this thesis put the Nuthall model to the test.

South Africa has recently introduced a new curriculum – curriculum 2005 and its revised version, the Revised National Curriculum Statement to replace the pre-apartheid education curriculum. C-2005 had come with some major changes in the teaching and learning of most subjects in the school curriculum. Among its major changes, the new curriculum focused on Environmental Education (EE) as a theme to be offered across the entire basic education or General Education and Training (GET) level curriculum. The one implication of that new focus on Environmental Education was that all the children at the basic education level were to be introduced to EE concepts and content. While the new version of C-2005, in the form of the RNCS places arguably less focus on Environmental Education as a theme, it retains the focus on environmental learning through (for example) the first principle listed for the RNCS: “Social justice, a healthy environment, human rights and inclusively.” Furthermore, a number of learning area outcomes with an environmental focus have also been retained. The whole focus on environmental learning, whether as a principle or a theme, is an unprecedented change in the curriculum of South Africa. Not many teachers, policymakers, learners, and other education stakeholders however, have the necessary experience to make such a major focus on Environmental learning workable within the current provincial systems of the country. It is against this background that a research was initiated to find out how the various provinces have responded to this new curriculum focus on EE. The study sought to explore the distribution of Environmental Education Opportunities to learn in the Mpumalanga province. Data for the study were collected mainly through qualitative interviews and document analysis with Environmental Education coordinator, subject advisers and the subject teachers within the province. An important finding in the study among others is that the Mpumalanga Education Department (MDE) on its own appears to have limited intellectual and material resources with which to build the schools’ instructional capacity for the implementation of the new Environmental Education policies and programs. I concluded the study by positing possible approach for providing Opportunities to Learn Environmental Education through an interaction between governmental and non-governmental resources and programmes in Mpumalanga. / Dissertation (MEd (Curriculum and Instructional Design and Development))--University of Pretoria, 2008. / Curriculum Studies / unrestricted

Ett arbete om hur textbaserade problemlösningsuppgifter inom delmomentet area framställs i två matematikböcker. Syftet med arbetet är att skapa en djupare förståelse om läromedlen möjliggör ett lärande för eleverna. Metoden för arbetet är en kvalitativ innehållsanalys som präglas av Taflins (2007) beskrivningar av uppgifter och hennes sju kriterier för att ett problem ska anses vara ett rikt problem. Vidare används även opportunities to learn (OTL) för att undersöka lärandet med frågor. Teorin som formar arbetet är variationsteorin där lärandeobjektet, kritiska aspekter och variationsmönster är övergripande teman. Genom att använda en kvalitativ innehållsanalys tillsammans med variationsteorin och OTL ges möjligheter att enklare urskilja teman, mönster och begrepp i matematikböckerna. Fynden som gjorts i matematikböckerna visar att det finns flera textuppgifter som uppfyller samtliga av Taflins (2007) sju kriterier för rika problem och blir sålunda textbaserade problemlösningsuppgifter. Dessa framställs på olika sätt då de har olika lärandeobjekt, kritiska aspekter och variationsmönster. Olika teman påverkar även elevernas möjligheter till lärande då olika uppgifter fokuserar på olika saker. Slutligen påverkas elevernas lärande av mängden uppgifter och instruktioner i läromedlen. Instruktionerna ger eleverna information om vad de ska arbeta med men det är eleven själv som behöver sin problemlösningsförmåga, förståelse och sina kunskaper samt erfarenheter för att kunna lösa uppgifterna och skapa sig ett individuellt lärande.

Students only learn what they get the opportunity to learn. This means, for example, that students do not develop their reasoning- and problem solving competence unless teaching especially focuses on developing these competencies. Despite the fact that it has for the last 20 years been pointed out the need for a reform-oriented mathematics education, research still shows that in Sweden, as well as internationally, an over-emphasis are placed on rote learning and procedures, at the cost of promoting conceptual understanding. Mathematical understanding can be separated into procedural and conceptual understanding, where conceptual understanding can be connected to a reform oriented mathematics education. By developing a reasoning competence conceptual understanding can also be developed. This thesis, which deals with students’ opportunities to learn to reason mathematically, includes three studies (with data from Swedish upper secondary school, year ten and mathematics textbooks from twelve countries). These opportunities have been studied based on a textbook analysis and by studying students' work with textbook tasks during normal classroom work. Students’ opportunities to learn to reason mathematically have also been studied by examining the relationship between students' reasoning and their beliefs. An analytical framework (Lithner, 2008) has been used to categorise and analyse reasoning used in solving tasks and required to solve tasks. Results support previous research in that teaching and mathematics textbooks are not necessarily in harmony with reform-oriented mathematics teaching. And that students indicated beliefs of insecurity, personal- and subject expectations as well as intrinsic- and extrinsic motivation connects to not using mathematical reasoning when solving non-routine tasks. Most commonly students used other strategies than mathematical reasoning when solving textbook tasks. One common way to solve tasks was to be guided, in particular by another student. The results also showed that the students primarily worked with the simpler tasks in the textbook. These simpler tasks required mathematical reasoning more rarely than the more difficult tasks. The results also showed a negative relationship between a belief of insecurity and the use of mathematical reasoning. Furthermore, the results show that the distributions of tasks that require mathematical reasoning are relatively similar in the examined textbooks across five continents. Based on the results it is argued for a teaching based on sociomathematical norms that leads to an inquiry based teaching and textbooks that are more in harmony with a reform-oriented mathematics education. / Elever kan bara lära sig de det de får möjlighet att lära sig. Detta innebär till exempel att elever inte utvecklar sin resonemangs- och problemlösningsförmåga i någon större utsträckning om inte deras undervisning fokuserar på just dessa förmågor. Forskning, nationellt och internationellt visar att det finns en överbetoning på utantillinlärning och på procedurer. Detta verkar ske på bekostnad av en konceptuell förståelse, trots att det under 20 års tid pekats på behovet av en reforminriktad matematikundervisning. Matematisk förståelse kan delas in i procedurell- och konceptuell förståelse där en konceptuell förståelse kan kopplas till en reforminriktad matematikundervisning. Genom att utveckla förmågan att resonera matematiskt utvecklas också den konceptuella förståelsen. Denna avhandling, som inbegriper tre studier (med empiri från gymnasiet år ett och matematikläroböcker från tolv länder) behandlar elevers möjlighet att lära sig att resonera matematiskt. Dessa möjligheter har studerats utifrån att undersöka vilka möjligheter läroboken ger att lära sig matematiska resonemang, dels via en läroboksanalys och dels genom att studera elevers arbete med läroboksuppgifter i klassrumsmiljö. Elevers möjligheter att lära sig att resonera matematiskt har också studerats genom att undersöka relationen mellan elevers matematiska resonemang och deras uppfattningar om matematik. Ett analytiskt ramverk (Lithner, 2008) har används för att kategorisera och analysera resonemang som använts för att lösa uppgifter och som behövs för att lösa en uppgift. Resultaten från studierna har givit stöd åt tidigare forskning vad gäller att undervisning och läroböckerna inte nödvändigtvis harmonierar med en reforminriktad matematikundervisning. Och att elever har uppfattningar om matematik som bygger på osäkerhet, förväntan på ämnet och sin egen förmåga samt motivation och att dessa uppfattningar delvis kan kopplas till att eleverna inte använder matematiska resonemang för att försöka lösa icke-rutinuppgifter. Det vanligaste sättet att lösa läroboksuppgifter var att välja andra strategier än att använda sig av matematiska resonemang. Ett vanligt sätt att lösa uppgifter var att låta sig guidas, av främst en annan elev. Eleverna arbetade framförallt med de enklare uppgifterna i läroböckerna. Bland dessa enklare uppgifter var det mer sällsynt med uppgifter som krävde matematiska resonemang för att lösas relativt de svårare uppgifterna. Resultaten visade även att det fanns en negativ relation mellan en uppfattning av osäkerhet hos elever och ett användande av matematiska resonemang. Resultaten visade vidare att fördelningen av uppgifter som krävde matematiska resonemang var relativt lika i alla undersökta läroböcker från fem världsdelar. Utifrån resultaten argumenteras för en förändrad undervisning mot en undersökande undervisning och läroböcker som är mer i harmoni med en reforminriktad matematikundervisning.

In line with international developments, the Department of Education in South Africa (SA) recognises environmental education as a key vehicle to respond to the national and global environmental crisis (DoE, 2001, P. 3). For this reason, the post-1994 education provision sought to infuse environmental education into the new curriculum called Curriculum 2005. The White Paper on Education and Training (RSA, 1995) perceived environmental education as a means to a better quality of life for all people and argued that it should be integrated at all levels of the SA Education and Training system. The White Paper further stated that “environmental education, involving an inter-disciplinary, integrated and active approach to learning, must be a vital element of all levels and programmes of the education and training system, in order to create environmentally literate and active citizens and ensure that all South Africans, present and future, enjoy a decent quality of life through the sustainable use of resources” (RSA, 1995, P. 22). How have the schools responded to this challenge by the new integration policy? How is the environmental learning provided for in the primary school curriculum across SA? What resources exist to make environmental policy workable in schools, and how are these resources mobilised and organised to promote learning? These and other questions formed the basis for the present inquiry An Opportunity to Learn (OTL) study was conducted to explore how one primary school in SA provides OTL about the environment. The main aim of the study was to understand the manner in which teachers integrate environmental learning in the school curriculum. A qualitative research approach was used as a mode of inquiry for this study. Interviews, classroom observations and document analysis were used as data collection methods. The findings of the study suggest that Sechaba Primary School has managed to integrate environmental learning in its curriculum through the help of non-governmental organisations (NGOs) and other stakeholders. The research concluded by arguing that OTL about the environment appear to be enhanced where there are strong connections between the schools and NGOs. Specifically, the following set of recommendations was documented: First, local leadership and agency are required to pursue the various opportunities and resources to build the school’s capacity for environmental learning. Schools should be encouraged to designate and support local leaders to take responsibility for driving the integration of environmental learning into their curriculum. Such integration is too important and maybe too demanding to be left to individual teachers independently in their own classrooms. Second, converting the latent capacity and/or physical and intellectual infrastructure for environmental learning into real OTL about the environment for the students will continue to remain a challenge for some of the teachers while others have managed the integration in some exemplary fashion. Opportunities for teachers to observe each other, plan together and work collaboratively on issues of integration should be created at school and district level. Third, it is critical that all teachers undergo in service training regarding the implementation of environmental education, and such training should provide teachers with enough time to learn. One of the major limitations of the Department of Education’s programme of implementation of the new environmental learning policy has been the inability to provide teachers with enough time to learn and implement the new ideas of the revised national curriculum statement. Finally, it is important to reiterate the importance of providing adequate resources for implementation of environmental learning from the Department of Education. Having said that, however, the case of Sechaba has demonstrated how such resources can also be mobilised from elsewhere outside the system. Encouraging beneficial partnerships between schools and NGOs may in itself be a valuable resource to encourage many primary schools in SA for whom government resources in this field continue to remain inadequate. / Dissertation (MEd (Curriculum and Instructional Design and Development))--University of Pretoria, 2008. / Curriculum Studies / unrestricted

Students only learn what they get the opportunity to learn. This means, for example, that students do not develop their reasoning- and problem solving competence unless teaching especially focuses on developing these competencies. Despite the fact that it has for the last 20 years been pointed out the need for a reform-oriented mathematics education, research still shows that in Sweden, as well as internationally, an over-emphasis are placed on rote learning and procedures, at the cost of promoting conceptual understanding. Mathematical understanding can be separated into procedural and conceptual understanding, where conceptual understanding can be connected to a reform oriented mathematics education. By developing a reasoning competence conceptual understanding can also be developed. This thesis, which deals with students’ opportunities to learn to reason mathematically, includes three studies (with data from Swedish upper secondary school, year ten and mathematics textbooks from twelve countries). These opportunities have been studied based on a textbook analysis and by studying students' work with textbook tasks during normal classroom work. Students’ opportunities to learn to reason mathematically have also been studied by examining the relationship between students' reasoning and their beliefs. An analytical framework (Lithner, 2008) has been used to categorise and analyse reasoning used in solving tasks and required to solve tasks.Results support previous research in that teaching and mathematics textbooks are not necessarily in harmony with reform-oriented mathematics teaching. And that students indicated beliefs of insecurity, personal- and subject expectations as well as intrinsic- and extrinsic motivation connects to not using mathematical reasoning when solving non-routine tasks. Most commonly students used other strategies than mathematical reasoning when solving textbook tasks. One common way to solve tasks was to be guided, in particular by another student. The results also showed that the students primarily worked with the simpler tasks in the textbook. These simpler tasks required mathematical reasoning more rarely than the more difficult tasks. The results also showed a negative relationship between a belief of insecurity and the use of mathematical reasoning. Furthermore, the results show that the distributions of tasks that require mathematical reasoning are relatively similar in the examined textbooks across five continents.Based on the results it is argued for a teaching based on sociomathematical norms that leads to an inquiry based teaching and textbooks that are more in harmony with a reform-oriented mathematics education. Elever kan bara lära sig de det de får möjlighet att lära sig. Detta innebär till exempel att elever inte utvecklar sin resonemangs- och problemlösningsförmåga i någon större utsträckning om inte deras undervisning fokuserar på just dessa förmågor. Forskning, nationellt och internationellt visar att det finns en överbetoning på utantillinlärning och på procedurer. Detta verkar ske på bekostnad av en konceptuell förståelse, trots att det under 20 års tid pekats på behovet av en reforminriktad matematikundervisning. Matematisk förståelse kan delas in i procedurell- och konceptuell förståelse där en konceptuell förståelse kan kopplas till en reforminriktad matematikundervisning. Genom att utveckla förmågan att resonera matematiskt utvecklas också den konceptuella förståelsen. Denna avhandling, som inbegriper tre studier (med empiri från gymnasiet år ett och matematikläroböcker från tolv länder) behandlar elevers möjlighet att lära sig att resonera matematiskt. Dessa möjligheter har studerats utifrån att undersöka vilka möjligheter läroboken ger att lära sig matematiska resonemang, dels via en läroboksanalys och dels genom att studera elevers arbete med läroboksuppgifter i klassrumsmiljö. Elevers möjligheter att lära sig att resonera matematiskt har också studerats genom att undersöka relationen mellan elevers matematiska resonemang och deras uppfattningar om matematik. Ett analytiskt ramverk (Lithner, 2008) har används för att kategorisera och analysera resonemang som använts för att lösa uppgifter och som behövs för att lösa en uppgift.Resultaten från studierna har givit stöd åt tidigare forskning vad gäller att undervisning och läroböckerna inte nödvändigtvis harmonierar med en reforminriktad matematikundervisning. Och att elever har uppfattningar om matematik som bygger på osäkerhet, förväntan på ämnet och sin egen förmåga samt motivation och att dessa uppfattningar delvis kan kopplas till att eleverna inte använder matematiska resonemang för att försöka lösa icke-rutinuppgifter. Det vanligaste sättet att lösa läroboksuppgifter var att välja andra strategier än att använda sig av matematiska resonemang. Ett vanligt sätt att lösa uppgifter var att låta sig guidas, av främst en annan elev. Eleverna arbetade framförallt med de enklare uppgifterna i läroböckerna. Bland dessa enklare uppgifter var det mer sällsynt med uppgifter som krävde matematiska resonemang för att lösas relativt de svårare uppgifterna. Resultaten visade även att det fanns en negativ relation mellan en uppfattning av osäkerhet hos elever och ett användande av matematiska resonemang. Resultaten visade vidare att fördelningen av uppgifter som krävde matematiska resonemang var relativt lika i alla undersökta läroböcker från fem världsdelar.Utifrån resultaten argumenteras för en förändrad undervisning mot en undersökande undervisning och läroböcker som är mer i harmoni med en reforminriktad matematikundervisning. / Elever kan bara lära sig de det de får möjlighet att lära sig. Detta innebär till exempel att elever inte utvecklar sin resonemangs- och problemlösningsförmåga i någon större utsträckning om inte deras undervisning fokuserar på just dessa förmågor. Forskning, nationellt och internationellt visar att det finns en överbetoning på utantillinlärning och på procedurer. Detta verkar ske på bekostnad av en konceptuell förståelse, trots att det under 20 års tid pekats på behovet av en reforminriktad matematikundervisning. Matematisk förståelse kan delas in i procedurell- och konceptuell förståelse där en konceptuell förståelse kan kopplas till en reforminriktad matematikundervisning. Genom att utveckla förmågan att resonera matematiskt utvecklas också den konceptuella förståelsen. Denna avhandling, som inbegriper tre studier (med empiri från gymnasiet år ett och matematikläroböcker från tolv länder) behandlar elevers möjlighet att lära sig att resonera matematiskt. Dessa möjligheter har studerats utifrån att undersöka vilka möjligheter läroboken ger att lära sig matematiska resonemang, dels via en läroboksanalys och dels genom att studera elevers arbete med läroboksuppgifter i klassrumsmiljö. Elevers möjligheter att lära sig att resonera matematiskt har också studerats genom att undersöka relationen mellan elevers matematiska resonemang och deras uppfattningar om matematik. Ett analytiskt ramverk (Lithner, 2008) har används för att kategorisera och analysera resonemang som använts för att lösa uppgifter och som behövs för att lösa en uppgift. Resultaten från studierna har givit stöd åt tidigare forskning vad gäller att undervisning och läroböckerna inte nödvändigtvis harmonierar med en reforminriktad matematikundervisning. Och att elever har uppfattningar om matematik som bygger på osäkerhet, förväntan på ämnet och sin egen förmåga samt motivation och att dessa uppfattningar delvis kan kopplas till att eleverna inte använder matematiska resonemang för att försöka lösa icke-rutinuppgifter. Det vanligaste sättet att lösa läroboksuppgifter var att välja andra strategier än att använda sig av matematiska resonemang. Ett vanligt sätt att lösa uppgifter var att låta sig guidas, av främst en annan elev. Eleverna arbetade framförallt med de enklare uppgifterna i läroböckerna. Bland dessa enklare uppgifter var det mer sällsynt med uppgifter som krävde matematiska resonemang för att lösas relativt de svårare uppgifterna. Resultaten visade även att det fanns en negativ relation mellan en uppfattning av osäkerhet hos elever och ett användande av matematiska resonemang. Resultaten visade vidare att fördelningen av uppgifter som krävde matematiska resonemang var relativt lika i alla undersökta läroböcker från fem världsdelar. Utifrån resultaten argumenteras för en förändrad undervisning mot en undersökande undervisning och läroböcker som är mer i harmoni med en reforminriktad matematikundervisning.

The purpose of this thesis is to investigate opportunities to learn (OTL) algebra by grade ten learners at three Catholic secondary schools in South Africa. Performance in mathematics is poor and is a great cause for concern. Despite the government’s effort to make education open and available to all, underperformance has continued among the black majority who were previously marginalised in the former regime. This thesis focuses on the OTL which are afforded learners who are given the chance to attend classes. This thesis met its aims through an extensive review of related literature and the implementation of practical research. The latter was carried out through case studies conducted in three schools where lessons were observed and interviews conducted with the respective teachers. Literature on how OTL mathematics are created is lacking in South Africa. Real OTL still needs to be created if the expected level of performance is to be achieved. The research produced a number of key findings: the learners were given the right to attend class but were subjected to different OTL, learning to convert within and between the different registers of representation of algebraic concepts is necessary to provide learners with OTL, it is not enough for learners to master certain facts and procedures, and learning is enhanced if the means to make the conversion necessary for concept building is developed and the OTL provided. The teacher’s approach influences the way OTL are realised and utilised by learners. The main conclusion drawn from this research is that the OTL afforded the grade ten learners were not the same and that different chances to make conversion within and between registers of representation of algebra concepts were given. Giving the teachers guidelines without expounding the meaning of specific terms such as ‘convert’ leaves gaps in their practices and results in some learners receiving adequate OTL and others not. This research argues for a more involved capacity building programme for in-service teachers to acquaint them with the expected learner-centred approaches to lesson delivery as well as familiarise them with the terminology used in defining terms in the syllabus. / Educational Studies / D. Ed. (Curriculum Studies)

The purpose of this thesis is to investigate opportunities to learn (OTL) algebra by grade ten learners at three Catholic secondary schools in South Africa. Performance in mathematics is poor and is a great cause for concern. Despite the government’s effort to make education open and available to all, underperformance has continued among the black majority who were previously marginalised in the former regime. This thesis focuses on the OTL which are afforded learners who are given the chance to attend classes. This thesis met its aims through an extensive review of related literature and the implementation of practical research. The latter was carried out through case studies conducted in three schools where lessons were observed and interviews conducted with the respective teachers. Literature on how OTL mathematics are created is lacking in South Africa. Real OTL still needs to be created if the expected level of performance is to be achieved. The research produced a number of key findings: the learners were given the right to attend class but were subjected to different OTL, learning to convert within and between the different registers of representation of algebraic concepts is necessary to provide learners with OTL, it is not enough for learners to master certain facts and procedures, and learning is enhanced if the means to make the conversion necessary for concept building is developed and the OTL provided. The teacher’s approach influences the way OTL are realised and utilised by learners. The main conclusion drawn from this research is that the OTL afforded the grade ten learners were not the same and that different chances to make conversion within and between registers of representation of algebra concepts were given. Giving the teachers guidelines without expounding the meaning of specific terms such as ‘convert’ leaves gaps in their practices and results in some learners receiving adequate OTL and others not. This research argues for a more involved capacity building programme for in-service teachers to acquaint them with the expected learner-centred approaches to lesson delivery as well as familiarise them with the terminology used in defining terms in the syllabus. / Educational Studies / D. Ed. (Curriculum Studies)