Magister Educationis - MEd (Mathematics and Science Education)

Permanent URI for this collectionhttps://hdl.handle.net/10566/15446

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Exploring school algebra constructs with a computer Algebra system
(Univesity of the Western Cape, 2025) Petersen, Magdalene Veronica Ann; Julie, Cyril Martin
This was a replication study. Replication studies are defined as “the repetition of a previous piece of research, generally under a different situation, i.e., different data or time period, to determine if the key findings from the original study can indeed be applied to such other situations” (McMillian, 2017, p. 1). This study was a partial replication of a 1999 study by Stevens and Konvalina in which they gauge the perspectives of pre-college students on the use CAS. The findings of Stevens and Konvalina (1999) shown the experimental group having positive perspectives on the use CAS. This study investigated the perspectives of Grade 10 mathematics learners on the use of CAS for productive practice of algebraic procedures. The research question is: “What perspectives do Grade 10 learners hold on the use of the Computer Algebra System (CAS) for practising algebraic procedures?” This was a quantitative study, like the study that are being replicated, it followed the same survey design as the original study. The instrument is a questionnaire that was developed with the assistance of my supervisor. The ICT activities for the purpose of “productive practice” are GeoGebra applets that are developed by GeoGebra users and are freely available online or via the GeoGebra classroom. The findings of this study shown that learners had positive perspectives on the use of the GeoGebra’ CAS feature. And they felt more confident about simplifying algebraic expressions due to the individual repeated practice sessions that supported their procedural knowledge. These individual sessions allowed learners to navigate their own learning process and to engage with mathematics on a deeper level. This deeper engagement with mathematics allowed learners to cement their procedural fluency in simplifying algebraic expressions. The findings also revealed that the use of Information and Communication Technology (ICT) tools are beneficial to both the teacher and the learner.
Exploring the embedding of UWC graduate attributes in BSc programmes: a case study in the Faculty of Natural Sciences.
(Univesity of the Western Cape, 2025) Parker, Khadija; Marshall, Delia
The Faculty of Natural Sciences at the University of Western Cape (UWC) currently only offers named degree programmes, where most of the electives are offered at first year level. The Faculty has embarked on a project to integrate the Physical Sciences stream of degrees (Chemical Sciences, Physical Science, Mathematics & Statistical Sciences, Computer Science) into one general Bachelor of Science (BSc) degree with majors in the various areas of study. The new degree must be compliant with DHET standards, and must align with the Graduate Attribute (GA) Charter of the University thus fitting into the Institutional Operating Plan (IOP) of the University. The purpose of this study is to examine the extent to which the modules in these programmes ascribe to the UWC Graduate Attributes Charter. The study is framed using Bigg’s concept of constructive alignment as a conceptual framework (Biggs, 1999; Biggs, 2003), as well as Barrie’s framework of perceptions of graduate attribute development (Barrie, 2004; 2007). The study explores the perceived relevance of graduate attributes, and how graduate attributes are developed in modules, through alignment with teaching and learning activities and assessment. This study employs a mixed methods design, incorporating both qualitative and quantitative data. Firstly, module descriptors in the yearbook were scrutinised to check whether the modules do indeed fulfil what the UWC Graduate Attribute Charter espouses to do.
The effect of an argumentation-based instruction on Grade 11 learners' understanding of chemical reactions used in extracting gold
(University of the Western Cape, 2011) Van der linde, Phillip Charles; Ogunniyi, M.B; Dinie, Shafiek
This purpose of this research study was first, to identify some of the many conceptual problems that science learners have. An area that students generally find difficult according to a plethora of studies is chemical reactions. In the light of the new curriculum stressing the need for learners to know about the application of chemistry in industries, the extraction of gold being one of them, I decided to select this topic as my study focus for two reasons: (1) knowledge about the methods used in the extraction of gold is particularly useful for learners to learn in that gold is the leading mineral exported by South Africa.; and (2) learners find the chemical processes involved in the extraction of gold particularly difficult. It is perhaps for the same reasons why this topic is in the curriculum. Specifically, this study focused on determining: (1) Grade 11 learners' conceptions of chemical reactions and in particular, the extraction of gold; (2) whether subjects exposed to an argumentation-based instruction would hold more valid conceptions of chemical extraction of gold than those not so exposed; and (3) whether or not the subjects' conceptions of chemical extraction of gold are related to their age, gender and socio-economic backgrounds. This study was underpinned by Toulmin's (1958) Argumentation Pattern (TAP) and Ogunniyi's (2007a) Contiguity Argumentation Theory (CAT).These two theoretical frameworks were used to elicit and address the learners' scientific and indigenous knowledge conceptions of chemical reactions with respect to chemical reactions used in the extraction of gold. The study was based on a quasi-experimental pre-post-test control design depicted as follows: The pre-test was administered simultaneously to both the experimental and control groups. The experimental group was then exposed to the argumentation-based instruction underpinned by TAP and CAT alluded to earlier, while the control group was exposed to traditional expository instruction. It was after this that the post-test of Chemical Reactions Questionnaire (CRQ) was simultaneously administered in both the experimental and control groups. The data collected through the questionnaire and interviews were analyzed and described both quantitatively and qualitatively. The study confirmed that many learners had misconceptions. These include: failing to recognise everyday reactions as in fact chemical reactions; misconceptions in reaction rates; exo- and endothermic reactions; balancing of equations; precipitation reactions; oxidation number changes and hence oxidation -reduction misconceptions. Many learners' also displayed a lack of terminology and the correct usage of science terms. The study showed that the argumentation-based instruction underpinned by TAP and CAT used in the experimental group was more effective in enhancing Grade 11 learners' understanding of chemical reactions involved in gold extraction than was the case in the control group where learners were exposed to traditional expository instruction. The finding has thus confirmed earlier studies using similar argumentation based instruction (e.g. Erduran, et ai, 2004; Simon, et ai, 2006). The study also seemed to have increased the learners' knowledge and awareness about the industrial and indigenous methods of extracting gold than was previously the case. However, the study did not reveal much conceptual differences among the learners in both groups on account of differences in their gender, age or socio-economic backgrounds. Despite the generally poor socio-economic background of many learners in the experimental group they became particularly fascinated with the argumentation-based instruction and indicated that the approach had enhanced their interest in studying science and in pursuing science-related careers. Overall, the study seemed to have confirmed earlier findings that: (1) an argumentation-based instruction was not only feasible in the classroom but also effective for enhancing learners' understanding the concept of chemical reactions; (2) teachers could be trained as I was, to implement an inclusive science-indigenous knowledge curriculum, thus making school science relevant to learners' socio-cultural environment (e.g. Ogunniyi, 2007a &b).
Exploring the use of productive practice for the development of procedural fluency with algebraic fractions in grade 10.
(University of the Western Cape, 2025) Hassiem , Courtney
Many learners in South Africa and globally struggle with poor performance in algebraic fractions. This widespread underachievement has prompted extensive research by scholars in mathematics education. Recently, there has been a notable shift in educational discourse: researchers and educators are moving away from the traditional classroom setting and teacher-learner approach to including more emphasis on learner-centred approaches. In response to the persistent challenges of learners underperforming in algebraic fractions, this study aimed to design and implement instructional activities as part of an intervention. Specifically, this study investigated whether the use of “productive practice” could enhance Grade 10 learners’ procedural and conceptual knowledge in the topic of algebraic fractions. The central research question guiding this study was: How should productive practice activities be designed and implemented to engender learner engagement for the consolidation of conceptual and procedural knowledge of algebraic fractions? This study was a qualitative approach and utilised a design approach – also known as, educational design research or developmental research – as its methodology. Data was collected by observational notes, audio recordings of the learners’ discussions and written responses of the activities. The framework that guided this study was productive practice. The results of this study indicated that many learners tend to forget previously taught constructs, and face challenges in understanding algebraic fractions and relevant terminology. Regular use of productive practice can reinforce prior learning, helping learners to retain information more effectively. The benefits of this study include the improvement of learners’ ability to simplify algebraic fractions and to enhance their mathematical thinking, which can contribute to learners performing better in the National Senior Certificate (NSC) examinations. Moreover, this study may provide teachers with valuable strategies to support learners’ success in this subject.
Investigating the benefits of integrating GeoGebra as a teaching aid in circle geometry in Grade 11
(University of the Western Cape, 2025) Van Niekerk, Emile John
This research investigated learners’ comprehension of the various grade 11 circle theorems and addressed possible common misconceptions that learner may have experienced in the experimental group. Common misconceptions related to tangents, chords and cyclic quadrilaterals were identified through pre-test analysis. This was done with the aid of GeoGebra, a dynamic mathematics software program that was deployed in combination with self-directed learning. The study further examined how learners at different performance levels (low, medium, and High) responded to the intervention. The integration of technologies such as GeoGebra into the teaching and learning of circle geometry has the potential to act as a constructive stimulus to learning. In this study, learners exposed to GeoGebra-based instruction demonstrated higher post-test performance than those taught through conventional methods. This study showed how high school learners' conceptual understanding of circle geometry was affected when GeoGebra was used as a technological teaching tool in conjunction with self-directed learning. This study employed a constructivism theoretical framework, namely Piaget's cognitive constructivism and Vygotsky's social constructivism. These have been selected because both theories are based on the idea that learners are ultimately masters of their own learning experience.
Exploring the lived experiences of selected grade 9 teachers’ integration of indigenous knowledge in their science lessons in the Western Cape province.
(University of the Western Cape, 2025) Magade, Athandile
This study investigated the lived experiences of Grade 9 Natural Sciences teachers with the infusing of Indigenous Knowledge Systems (IKS) in their teaching. Anchored in a phenomenological research design and drawing from Giorgi’s analytic approach to data analysis, the study sought to gain insight on how teachers conceptualise, make sense of and work with the inclusion of IKS in a knowledge world that is predominantly constructed and constituted by Western scientific knowledge. Unstructured interviews were conducted with six teachers who were purposively selected, and their stories were analysed in detail to find the central themes and connections. The findings indicate that teachers recognise the cultural significance and the educational relevance of IKS but experience some challenges to integrate Indigenous Knowledge System effectively. The challenges include limited time (due to curriculum pace), lack of clear teaching approaches, lack of support by the Department of Education and the non-existence of a professional development focusing on Indigenous Knowledge System. Because of these constraints, some teachers integrate Indigenous Knowledge System informally and at times conduct oral conversations with traditional knowledge holders on Indigenous Knowledge System for personal experiences. The findings of this can study contribute to proactive integration of KIS in the science curriculum as it argues for more cultural and contextually relevant pedagogical approaches. The study recommends focused curriculum revision, ongoing professional development of teachers, and shared partnerships with Indigenous Knowledge System holders for effective integration of IKS in Natural Sciences education. These are the efforts necessary to establish equitable and cultural-responsive science education in South Africa. At the end, the research contributes to transformation of science education, and demands attention to more holistic, contextualised teaching practices.
Investigating novice mathematics educators’ reflection through videostimulated recall by observing their own and an experienced teacher’s lessons
(University of the Western Cape, 2024) Ismail, Radiyah
Reflection is of utmost importance in improving one’s practice as an educator. An educator who does not reflect will struggle to improve on their previously taught lessons. Reflection can benefit the educator first and, ultimately, their learners. This study investigated how novice mathematics educators’ professional development could be enhanced by observing their own lessons as well as those of experienced educators, using video-stimulated recall (VSR). The study entailed three cycles in which three lessons of each novice and experienced educator were video-recorded. The researcher watched the video-recordings, trying to establish if anything specific or noteworthy happened that could constitute a possible development area for the educator. Subsequently, the researcher created playback episodes in which certain parts of the novice educator’s and experienced educator’s lessons were presented. Thereafter, the novice educator and the researcher watched the recorded episodes. This viewing was followed up with a semi-structured interview. In this interview, the researcher prompted the novice educator to reflect on what they noticed in the playback episodes. During the observation phase of the playback episodes, the novice educators commented on the video recordings. This was to ensure immediate reflection as they continued watching the recordings. All interviews were audio-recorded, to ensure the accurate capturing of the data. The first cycle of observations entailed the novice educator reviewing their own lessons and then observing those of the experienced educator. Thus, novice educators do not just reflect on their own lessons, but also observe how experienced colleagues teach the same topic – at the same time and in the same school. The second cycle proceeded with the researcher replacing the novice educator as reviewer of the videos. This was to ascertain whether that sequence would better equip the novice educators to adapt and implement the various useful aspects they observed. The novice educators observed the video-recorded lessons of the experienced educators before their own. After watching the video-recorded lessons of their experienced colleagues, the novice educators reflected on their own lessons. This was to determine if there were new or diverse ways of teaching the same topics. The third cycle was similar to the first cycle of the video-stimulated semistructured interviews. The novice educators reviewed their own lessons first, before observing those of the experienced educators. Each playback episode in every cycle was followed by a semi-structured interview. Here, the novice educator was prompted by the researcher to reflect on what they saw and how they could improve the lesson in future, after reflection.
The effectiveness of revision, based on prior knowledge and productive practice for the achievement of grade 8 mathematics learners in school- based examinations
(University of the Western Cape, 2024) Links, Melissa Catherine
This research evaluated whether the use of a teaching strategy, namely productive practice, will enhance the performance of Grade 8 Mathematics learners residing in a low socio-economic area. Two classes were selected to participate in this specific study. The learners selected, formed part of a convenience sample as the researcher taught both grade 8 classes Mathematics. The content area for the study was exponents. The data used for the statistical analysis in the study was only data from learners who wrote both pre-test and post-test. The research utilized a quasi-experimental quantitative approach. Research participants was exposed to a pre-test post-test design. Learner test scores represented the statistical data. The data analysis was done with the statistics software package IBM SPSS version 28.0. Initially, the data explored with descriptive statistics to determine if there were no severe violations of the statistical requirements. A paired-samples t-test used to determine if there is a significant difference between the means of pre- and post-test scores. The results show that for both classes there were statistically significant differences between the pre- test and post-test scores. Class 1 showed a small effect size whereas class 2 showed a medium effect size. The results indicate that productive practice is more effective with higher Mathematics performing classes than with lower Mathematics performing classes
Exploring the use of “productive practice” in learning congruency and similarity in grade 9 Mathematics
(University of the Western Cape, 2024) Sakupapa, Catherine Longwe; Julie, Cyril
Poor performance in geometry has been experienced by many learners within South Africa, and internationally. This wide underperformance has led to a number of research by various scholars. There is a shift in the discourse regarding teaching and learning amongst researchers and educators. Many researchers are now underscoring and supporting student learning, rather than teacher teaching. This shift has resulted in promoting a student learning approach. In this regard, this study focused on designing some instructional activities as part of the intervention to a complex problem of learner underperformance in geometry. The purpose of this study was to investigate whether the use of “productive practice” can improve learner performance in learning congruence and similarity in grade 9 Mathematics. The research question asked was: How does implementing “productive practice” as a pedagogical strategy influence Grade 9 learner’s understanding, retention, and performance in congruence and similarity in geometry? This study was qualitative in nature, and employed design research, also known as educational design research/developmental research as an approach. Data was collected from participants using observational notes, audio recordings of learners’ discussion when they engaged with activities, and learners’ written work based on the designed activities. The framework that informed this study is “productive practice” and the Van Hiele theory of geometric thinking.
Exploring shifts in teachers’ perspectives of reflection in a context of CPD using video-stimulated recall
(University of the Western Cape, 2023) Slamdien, Yasser; Nel, Benita
Reflective practice is understood as being a vital part of professional development and is seen as a necessary skill in mathematics teaching. The poor mathematics results in South Africa can be addressed by supporting mathematics teachers to reflect more effectively. This study focused on how exposure to Video Stimulated Recall (VSR) affected mathematics teachers' perspectives of reflection and their practice, and to what extent teachers developed, after exposure to VSR. The aim was to gain a better understanding of how teachers reflect and develop through reflection. The research question which guided the study was: How, and to what extent, does the exposure to VSR affect the mathematics teachers’ perspective of reflection? The sub-questions which accompanied this question were: 1. How, and to what extent, did the exposure of mathematics teachers to VSR, shift their perspective of their own practice? 2. How, and to what extent, did the mathematics teachers develop through their exposure to VSR as a reflective tool? The study was conducted in two public primary schools in the Western Cape and focused on two mathematics teachers from each school.
Problem-solving in geometry in collaborative small group Settings: how learner’s appropriate mathematical tools while working in small groups.
(University of the Western Cape, 2011) Cooper, P; Mbekwa, M.; Julie, C.
Problem-solving in Mathematics is an important skill. The poor perforrnance of South African learners in international tests such as the Trends in Intemational Mathematics and Science Study (TIMSS) and in schools in general indicates that emphasis should be placed on problem-solving in the teaching and leaming of Mathematics. The new national senior certificate curriculum in South Africa encourages Soup work amongst learners. The thesis proposes that learning is enhanced in a small-goup setting, since learners actively engage with the problems. Furthermore, Euclidean Geometry is perceived by learners to be a 'diffrcult' section of Mathematics. However, Geometry is important since the skills acquired while doing Geometry can be applied to various fields of study.
A comparison between the contexts grades 10 learners prefer for mathematical literacy and those reflected in the PISA survey
(University of the Western Cape, 2008) Hartzenberg, Louis Reginald; Julie, Cyril; Mbekwa, Monde
The study focuses on the contexts grade 10 learners prefer to deal with in Mathematical Literacy. These preferred contexts of the leamers were then compared with the contextual situations found in the Organization for Economic Cooperation and Development's (0ECD) Programme for International Student Assessment (PISA). The most important findings of the study are that the grade l0 leamers from low socioeconomic environments regard the intra-mathematical clusters as the most favoured items. In the extra-mathematical cluster learners favoured technology and health. PISA designers favoured the mathematical cluster followed by physical science and the finance cluster. The context preferred by learners may differ from what designers of tests may perceive to be relevant or interesting and vice versa.
Effects of a mathematics teaching strategy based on distributed interleaved practice on procedural and conceptual knowledge
(University of the Western Cape, 2022) Carelse, Cindy; May, Bruce
This study investigated the effects a teaching strategy based on interleaved distributed practice had on learners in three grade 6 classes’ procedural and conceptual knowledge within a, previously disadvantaged, Cape Town Primary School. An inference one can draw from the pace setters of the South African Curriculum (CAPS) is that mass practice is the suggested practice type. Mass practice is a teaching strategy where learners would practice problems requiring one or more specific skill(s) immediately after the presented lesson on that skill(s). The aforementioned practice allowed learners to predict the type of problems they would encounter in these activities. The interleaved distributed practice presents an alternative teaching strategy to massed practice.
Investigating the effects of continuous professional development on ICT integration in the classroom of Foundation Phase Mathematics
(University of the Western Cape, 2022) Senosi, Chloé; Simons, Marius
The use of technology is becoming the norm in the 21st century classroom to enhance teaching and learning. This rapid move to incorporate technology in the classroom brought the need for teachers to be upskilled on the use of technology as a pedagogical tool. Through this study the researcher sought to investigate the effects that Continuous Professional Development (CPD) has on Information and Communications Technology (ICT) integration in the classroom, with a specific focus on the teaching and learning of mathematics in the Foundation Phase. The mixed method research design will be underpinned by the conceptual approach of the Technological Pedagogical Content Knowledge (TPACK) and Substitution, Augmentation, Modification and Redefinition (SAMR) frameworks and supported by a constructivist learning theory approach to the teaching and learning of mathematics.
Pre-service science teachers’ conceptual and procedural difficulties in solving mathematical problems in physical science
(University of the Western Cape, 2014) Iwuanyanwu, Paul Nnanyereugo; Ogunniyi, Meshach
Students frequently leave first-year physical science classes with a dual set of physical laws in mind- the equations to be applied to qualitative problems and the entrenched set of concepts, many erroneous, to be applied to qualitative, descriptive, or explanatory problems. It is in this sense that the emphasis of this study is on ‘change’ rather than acquisition. Thus, a blend of theoretical framework was considered according to the aim of the study. Of immediate relevance in this regard within the “constructivist paradigm” are: Posner, Strike, Hewson and Gertzog’s (1982) conceptual change theory and the revised Bloom’s Taxonomy.
An investigation of strategies used by grade 4 learners to solve measurement type word problems
(University of the Western Cape, 2022) Rawoot, Shabbeer; Govender, Rajendran
The aim of this study was to investigate and highlight the problem-solving strategies used by grade 4 learners when solving measurement word problems. Furthermore, it aimed at revealing how learners go about using these strategies to arrive at meaningful solutions. The findings of this study would strengthen the argument that our current curriculum needs to place greater emphasis on problem solving. It would also enhance the claim that learners need to be exposed to a greater variety of problem-solving strategies to allow them to select the most suitable strategies when solving word problems. The study used a case study research design and a mixed methods approach was adopted. The sample of the study was a class of 42 grade 4 learners at a primary school in Cape Town.
Exploring Grade 9 Mathematics learners learning of congruency based proofs in geometry via a web-based learning system
(University of the Western Cape, 2022) Chetty, Taariq; Govender, Rajendran
Globally, research and evaluation reports show that students are not learning congruency-based proofs, as part of the Geometry section of mathematics, efficiently. One identifier of student understanding related to geometry is the teacher’s method of instruction. In order to attain success in mathematics the understanding of proofs and writing of proofs are of utmost importance. In this regard, web-based learning could be used in school mathematics to enhance activities involving “proof”. Proofs are the heart of mathematics and digital resources may be used to teach learners effectively, starting from primary school level.
A description of entry level tertiary students' mathematical achievement: Towards an analysis of student texts
(University of the Western Cape, 2006) Jacobs, Mark Solomon; Breiteig, Trygve; Torkildsen, Ole Einar; Julie, Cyril
A DESCRIPTION OF ENTRY LEVEL TERTIARY STUDENTS' MATHEMATICAL ACHIEVEMENT (Towards an analysis of student texts) M.S. Jacobs PhD Thesis, School of Mathematics and Science Education, University of the Western Cape. This research provides insights into the mathematical achievement of a cohort of tertiary mathematics students. The context for the study is an entry level mathematics course, set in an engineering programme at a tertiary institution, the Cape Peninsula University of Technology (CPUT). The participant members are first year, first semester students. The materials for the inquiry are student produced test scores and examination scripts taken from their entry level course. The characteristics of the mathematical achievement of the cohort concern the understanding of procedural and conceptual knowledge and problem solving abilities in mathematics. The facility with mathematics is another central concern of this study as it forms the dominant aspect of mathematical achievement which is accessible to research in the materials employed for the study. This research also develops a mathematical achievement profile for individual members of the cohort. The methodology makes use of content - and textual analytic methods for profiling the students. When viewed across the different kinds of profiling techniques adopted, this study suggests that these techniques complement one another: the profiles developed provide a cohesive and complementary overview of the achievement of the cohort. 111 This study challenges perceptions that responses to constructed response questions offer little information about the mathematical knowledge of students. This study investigates the possibilities of providing a bridge between the assessment of students by means of tests scores and a taxonomy of mathematical objectives, on the one hand, and the critical analysis of student produced texts, on the other. Findings suggest that diagnostic uses of paper and pencil tests can be revealing about the achievement of students. The wide range of responses to test items revealed a distribution of incompleteness in terms of employing algorithmic techniques. This research revealed that even in cases of wrong solutions, participant members' responses were reasonable, meaningful, clear and logical. The participants responded in many ways as predicted by the research literature. Evidence could be found for the use of child methods; poor use of reflective abstraction for coordination; accessing the wrong cognitive frames; not seeing the underlying structure of the mathematics and treating letters as objects. Findings suggest that the use of a textual analytic method, which led to the creation of critical indicators as a way of sign-posting events, enhanced the achievement profile of the students.
An investigation into some causes of cognitive deficiencies among high school pupils in the learning of differential calculus and implications for further study
(University of the Western Cape, 1994) Matsela, Simon; Rhodes, J.S.
Little attention is usually given to the disadvantaged pupils' lack of problem solving skills and thinking skills needed for successful learning of differential calculus. Factors like what makes (disadvantaged) pupils not function (cognitively) very well are usually overlooked. The only factor considered by (some) teachers being only to comment on the impairments that already exists. That is for example labelling students as unintelliqent "foolish" or "dull". In some cases the blame is placed fully on the system (Government and its hidden agenda). While others blame the environment in which the pupils live without finding what impact the situations have on the learners' cognition. There is a need to know where and what causes cognitive impairment in general. This study looks at the domain of differential calculus. This research therefore sets out to find what the causes of cognitive deficiencies are in the learning of differential calculus. To find out some of the causes 67 High School pupils and 40 first year University students completed the questionnaires, 34 pupils wrote the differential calculus test, 10 final year matric teachers ,also contributed some information about some causes of cognitive deficiencies and how they could possibly be remedied. 4 first-year university mathematics students and 7 high school pupils were involved in the thinking - aloud interviews. The interviews were audio-taped and then subjected to a protocol analysis. Special references to this study were made to the works of Feuerstein, Perkins and Sternberg Thus in general the research set out to answer the following questions: 1. Which problems do pupils think they have with regard to learning differential calculus? 2. Which cognitive deficiencies or difficulties do pupils have in the learning of differential calculus? 3. What role do motivation and affective processes and other non-cognitive processes have in the learning of differential calculus? 4. Is there any relationship between the pupils' cognitive deficiencies, motivation and affective processes as well as the problems they think they have with regard to learning differential calculus? 5. What perception do practising teaching have on the above? 6. Is there any change or changes for first year university mathematics students? 7. Which instructional strategies can be used to rectify such deficiencies? 8. What are the implications and recommendations for further study? .... It was found that there are many cases of deficiencies among high' school pupils. These can briefly be summarised as follows: lack of effective institutional processes, poor socio-economic and political climate for blacks, inadequate research (thinking aloud) among the pupils. Finally the list of 28 recommendations and implementation for further study were suggested. These 28 recommendations were further divided into 5 general headings that teachers could experiment with in the classroom and school.
Mediation and a Problem Solving Approach to Junior Primary Mathematics
(University of the Western Cape, 1996) Dirks, Denise; Rhodes, J.S.
This study argues that not all children in the Junior Primary phase benefit from the Problem Centred Approach in mathematics that was adapted by the Research, Unit for Mathematics at the University of Stellenbosch (RUMEUS). \One of the reasons could be that not all pupils can construct their own knowledge and methods. There are the highly capable pupils who cope well with this approach. These pupils are able to solve mathematical problems with little or no teacher interaction. Then there are the average and weaker pupils who cannot solve a mathematical problem on their own. These pupils need strategies and skills to solve problems and they need the teacher to mediate these strategies and skills to them, which will help these pupils to become autonomous problem solvers. ,Working in groups can, to some extent, supplement mediation or teacher interaction. Peer group teaching can be effective, whereby pupils are placed in groups so that the more capable pupils can teach concepts or make concepts clearer to the average or weaker pupils). There is, however, the possibility that when pupils of mixed abilities are placed in groups of four there might be one pupil who might refuse to work with the group. This pupil will work on her own and will not share ideas with the other members of the group. If this happens, mediation is necessary for those pupils who cannot solve a mathematical problem on their own. The purpose of this study is to investigate how exposure to mediation can improve pupils' problem solving abilities. As directions for my research I've chosen the first six criteria of Feuerstein's Mediated Learning Experiences (MLE). The first three parameters: intentionality and reciprocity, mediation of transcendence and mediation of meaning _are conditions for an interaction to qualify as MLE. Mediation of competence and regulation of behaviour are functions of specific experiences that combine with the first three to make an adult-child interaction one of mediated learning. Mediation of sharing behaviour . can be added. Here the child and the mediator are engaged in a shared quest for structural change in the child. In addition to this, the five mechanisms of mediational teaching, i.e. process questioning; challenging or asking reasons; bridging; teaching about rules; and emphasising order, predictability, system, sequence and strategy are also used in the implementation of mediation as described by Haywood. Two methods of investigation were chosen. The pupils' problem solving abilities were studied by means of eight word sums, of which the first four word sums were done in the pre-test and the other four word sums in the post-test. After the pre-test and before the post-test there was a period of mediational teaching for the experimental group. During this period and during the post-test the control group was denied mediation. After this research, mediation was also available for the control group. Two pupils from the experimental group were then chosen for further in-depth, think-aloud, person-to-person interviews. The aim of the interviews was to determine why these pupils could not solve the problem in the pre-test, but could successfully solve the post-test question. The results of the word sums in the pre-test and the post-test were compared. The role of strategies and thinking skills is concentrated on in the results. Mediation was not equally successful in all of the four different types of problem sums. Questions one and five contained two or more numbers and here pupils tended to either plus or minus these numbers. Questions two and six also contained numbers, but this is a problem situated in a real life situation. Questions three and seven contained no numbers and questions four and eight compelled pupils to first work out a plan. Mediation was most successful in problem sums situated in a real life situation, followed by problem sums which compelled pupils to first work out a plan, and then by problem sums where there were no numbers. Mediation was least; successful in problem sums that contained two or more numbers. Analysis of these results shows that with mediation there is an improvement in the pupils' problem solving abilities; Mediation can be viewed as S-H-O-H-R, in which the human mediator (H) is interposed between the stimulus (S) and the organism (0), and between the organism and the response (R). We can argue that the Problem Centred Approach without mediation can produce individuals who are little, if at all, affected by their encounter and interaction with new situations. Due to the lack of support in the Problem Centred Approach to Mathematics, it is the aim of this mini-thesis to propose mediation as an essential component in the Problem Centred Approach to Mathematics in the Junior Primary phase.

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