I denna studie undersöker vi matematiklärares olika synsätt utifrån deras erfarenheter kring visuella representationer och aspekter av bråk. Lärare intervjuades om en uppsättning problem som deras elever fick lösa, vilka var de metoderna som användes. Vi undersöker därav även om det finns några samband mellan klassresultaten och matematiklärarnas olika synsätt. Tre olika teorier användes som ramverk vid både metodövervägandet och analysen; Fyra utvecklingsstadier vid inlärning med representationer, fem aspekter av bråk och pedagogical content knowledge. Resultaten visar att det finns likheter och skillnader i deltagande lärares synsätt. Några likheter var att alla deltagande lärare finner visuella representationer att vara ett användbart hjälpmedel vid utveckling av elevers förståelse samt att de alla uttrycker en större bekantskap med aspekten del-helhet jämfört med de övriga aspekterna. Några skillnader mellan lärarnas synsätt var att de har olika inställningar till huruvida bråk uppfattas som svårt för eleverna men också i vilken utsträckning visuella representationer ska användas i årskurs 9. Vid analys av klassresultaten finner vi några samband där eleverna har svårare för uppgifter med areamodeller som inte är indelade i lika stora delar, uppgifter med tallinjer och uppgifter där eleverna ska uttrycka sig med ord. Lärarnas kommentarer till detta är att eleverna saknar förståelse för den matematik som ska användas och att eleverna på så sätt använder en felaktig metod. Lärarna kommenterar även att vissa uppgifter i studien kräver en större begreppsförståelse vilket påverkar elevernas resultat på uppgifterna. / In this study we examine mathematics teacher’s verity in their reasoning and their different perspectives about visual representations and sub-constructs of fractions. Teachers were interviewed about task that their students solved, which were the methods used in this study. Therefor we also examine if there are any correlations between the teacher’s different perspectives of visual representations and the classes results on fraction tasks. The study was based on three theoretical frameworks; Four stages through the use of representations, five sub-constructs of fraction and pedagogical content knowledge. The methodological consideration and analysis were based on these theories. The result showed that there were some similarities and differences in the teacher’s reasoning. A similarity was that all participated teachers found visual representations as a helpful tool when teaching fractions for student’s conceptual understanding. Another similarity was that all teachers were more familiar with the part-whole sub-construct than the other subconstructs of fraction. Some differences in the teacher’s reasoning were if they found fraction difficult or not for the students and in which extent visual representation should be used in the ninth grade. When the class results were analyzed some connections were found. Student have difficulties with tasks where area models are not equaled partitioned, with tasks related to number lines and to tasks were the students must change from a visual to a textual representation. According to the teachers comments these difficulties accurse due to that the students lacks a curtain mathematical understanding that is required when solving these tasks. This leads to the use of an inaccurate method. The teacher’s also comments that some tasks in this study requires a higher conceptual understanding which affects the classes’ results.

Teaching students with Intellectual Disability (ID) is a relatively new endeavor. Beginning in 2001 with the passage of the No Child Left Behind Act, the general education curriculum integrated algebra across the K-12 curriculum (Kendall, 2011; National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010), and expansion of the curriculum included five intertwined skills (productive disposition, procedural fluency, strategic competence, adaptive reasoning, and conceptual understanding) (Kilpatrick, Swafford, & Findell, 2001). Researchers are just beginning to explore the potential of students with ID with algebra (Browder, Spooner, Ahlgrim-Delzell, Harris & Wakeman, 2008; Creech-Galloway, Collins, Knight, & Bausch, 2013; Courtade, Spooner, Browder, & Jimenez, 2012; Göransson, Hellblom-Thibblin, & Axdorph, 2016). Most of the research examines the development of procedural fluency (Göransson et al., 2016) and few researchers have explored high school level skills. Using a single-case multiple-baseline across participants design, the study proposes to teach two algebra skills to six high school students with ID, creating an equation (y=mx+b) from a graph of a line and creating a graph from an equation. The six high school students with ID will be recruited from a school district in central Virginia. The intervention package modeled after Jimenez, Browder, and Courtade (2008), included modeling, templates, time delay prompting, and a task analysis. Results showed that all six individuals improved performance during intervention for the target skills over baseline; results also indicated that in three out of the six cases some generalization to the inverse skill occurred without supplemental intervention. The ability of individuals with ID to generalize the learning without intervention provides some evidence that individuals with ID are developing conceptual understanding while learning procedural fluency.

Practical work is regarded as an essential part of learning; hence most tertiary institutions have included a practical component in their physics courses. There is a concern about the effectiveness of the practical work in most universities. The present study is a case study that assessed the contributions of simulations on Foundation Physics students’ practical work. In assessing the contribution of simulations, two tests, Determining and Interpreting Resistive Electric Circuits Concepts Test (DIRECT) and the Test of Integrated Science Process Skills (TISP) were used. A class test, observations and worksheets from students’ practical work were analyzed and interviews with a selected group were conducted. There were 20 Foundation Physics students participating from the University of Limpopo. Results indicated that the simulations contributed positively on students’ understanding of electric circuits. However the study revealed that the students who did simulations do not differ from those who did not do the simulations with regards to the development of process skills. / Institute of Science and Technology Education / M.Sc. (Physics Education)

The study constituted an investigation for concept images and mathematical reasoning of Grade 11 learners on the concepts of reflection, translation and stretch of functions. The aim was to gain awareness of any conceptions that learners have about these transformations. The researcher’s experience in high school and university mathematics teaching had laid a basis to establish the research problem. The subjects of the study were 96 Grade 11 mathematics learners from three conveniently sampled South African high schools. The non-return of consent forms by some learners and absenteeism during the days of writing by other learners, resulted in the subsequent reduction of the amount of respondents below the anticipated 100. The preliminary investigation, which had 30 learners, was successful in validating instruments and projecting how the main results would be like. A mixed method exploratory design was employed for the study, for it was to give in-depth results after combining two data collection methods; a written diagnostic test and recorded follow-up interviews. All the 96 participants wrote the test and 14 of them were interviewed. It was found that learners’ reasoning was more based on their concept images than on formal definitions. The most interesting were verbal concept images, some of which were very accurate, others incomplete and yet others exhibited misconceptions. There were a lot of inconsistencies in the students’ constructed definitions and incompetency in using graphical and symbolical representations of reflection, translation and stretch of functions. For example, some learners were misled by negative sign on a horizontal translation to the right to think that it was a horizontal translation to the left. Others mistook stretch for enlargement both verbally and contextually. The research recommends that teachers should use more than one method when teaching transformations of functions, e.g., practically-oriented and process-oriented instructions, with practical examples, to improve the images of the concepts that learners develop. Within their methodologies, teachers should make concerted effort to be aware of the diversity of ways in which their learners think of the actions and processes of reflecting, translating and stretching, the terms they use to describe them, and how they compare the original objects to images after transformations. They should build upon incomplete definitions, misconceptions and other inconsistencies to facilitate development of accurate conceptions more schematically connected to the empirical world. There is also a need for accurate assessments of successes and shortcomings that learners display in the quest to define and master mathematical concepts but taking cognisance of their limitations of language proficiency in English, which is not their first language. Teachers need to draw a clear line between the properties of stretch and enlargement, and emphasize the need to include the invariant line in the definition of stretch. To remove confusion around the effect of “–” sign, more practice and spiral testing of this knowledge could be done to constantly remind learners of that property. Lastly, teachers should find out how to use smartphones, i-phones, i-pods, tablets and other technological devices for teaching and learning, and utilize them fully to their own and the learners’ advantage in learning these and other concepts and skills / Mathematics Education / D.Phil. (Mathematics, Science and Technology Education)

The study investigated the influence of Process Oriented Guided Inquiry Learning Approach (POGIL) on Science Foundation students’ achievements in stoichiometry versus traditional lecture centered pedagogy. Two intact science foundation class groups at the University of Namibia were used as a case study. A quasi-experimental non-randomized pre and posttests control group design was used to investigate the achievement in stoichiometry. Data on student achievements were collected and analyzed using descriptive statistics and Analysis of Covariance (ANCOVA). The ANCOVA results showed that there was a significant statistical difference in achievements when comparing the adjusted mean score (54.5%) obtained by the control group and the adjusted mean score (60.5%) obtained by students in the POGIL group; (F (1,75) = 17.990, p < 0.05). The POGIL group also showed the highest average improvement (65%) on questions related to reaction stoichiometry and limiting reagents, whereas the control group recorded improvements of about 53% in the same section. The results from the analysis of student’s test solutions revealed that the POGIL group students were able to give concrete reasons for their answers that they had obtained through numerical calculations or multiple choices and demonstrated enhanced understanding of linking various stoichiometry concepts. / Science and Technology Education / M. Sc. (Chemistry Education)

The main purpose of the research was to investigate the influence of scientific calculators on Grade 8 South African learner's understanding of fractions in learning mathematics. Quasi-experimental quantitative research methods were used. A sampling frame was selected using non probability sampling technique. A total of 15 learners in each group were randomly selected for an experimental and control group for the study. Both groups were taught fraction concepts by different teachers for the same duration and at the same time. The experimental group used a calculator as a learning aid while the control group used the traditional paper pencil method. Two tasks (post-test and assignment) were administered to both groups and a questionnaire to the experimental group. The results indicated that the scientific calculator has a positive influence in learner's conceptual understanding of fractions in mathematics as reflected in their performance. / Mathematics Education / M. Ed. (Mathematics Education)

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Recently, there has been an increasing interest in high school mathematics education, especially in the teaching and learning of calculus. For example, studies conducted by Bressoud (2010); Judson and Nishimori (2005); Koh and Divaharan (2011); and St. Jarre (2008) all looked at how to improve the understanding of calculus students and what roles the educator must take to ensure that their students are successful. The purpose of this study was to determine if there was a significant difference between instruction using computer algebra system (CAS) compared to instruction using the graphing calculator in high school calculus on students’ conceptual and procedural understanding. This study explored and compared two different types of instruction based on the use of two different types of technology, CAS and graphing calculator. The total population for this study consisted of 333 students. There were 187 students classified as using the graphing calculator and 146 students classified as using CAS. The data for this study were collected from four Advanced Placement (AP) calculus AB courses from high schools in Florida. The study used observations and two sets of calculus tasks in order to gather data. The research questions for this study looked at comparing the grades of students categorized based on the type of instruction received during the learning of calculus. The statistical procedure that was used was a simple oneway analysis of variance (ANOVA). The results indicated that there was no significant difference between the two types of instruction on the students’ procedural knowledge, iii however, there was statistical significance on the students’ conceptual understanding in favor of the CAS students. The study introduces a framework on how to obtain information about the effects of different types of instruction on students’ understanding of calculus. The results of this study contribute in assisting teachers and future researchers on how to analyze student work in order to obtain information about the students’ conceptual and procedural understanding of first semester calculus.

This study aims to investigate the effects of cooperative learning on pupils’ knowledge and development with a particular focus on concepts and conceptual understanding in mathematics. By employing both qualitative and quantitative research methods, the study compared the effectiveness of cooperative learning to traditional teaching approaches in grade 4. The following question was formulated to research the aim of this study: <li aria-level="1">How does pupils’' understanding of concepts in mathematics differ after traditional teaching compared to teaching with cooperative learning in grade 4? Svanelid’s (2014) three aspects were used as an analysis model in this intervention study. With the analysis model, teachers can assess whether pupils have understood various mathematical concepts. The results of the aim showed that cooperative learning had a significant positive impact on pupils’ learning and their understanding of geometrical concepts. Through collaborative activities and interactive discussions, pupils actively participated in the learning process, leading to improved comprehension and retention of geometric concepts. However due to introduction of methodological weaknesses and potential biases the results can not be considered as statistically sound. The study indicated development in pupils’ learning and understanding of mathematics. In an attempt to pursue the main aim of this study, traditional teaching and cooperative learning were used as teaching techniques for pupils to study the development of their learning. After each teaching method, tests were used to identify the results of the analysis. The results indicated progression in the pupils’ learning by using cooperative teaching. Teachers are encouraged to further investigate cooperative learning techniques in their lessons as it potentially can increase pupils’ learning. More research is needed to establish a correlation between the use of cooperative learning in the teaching of mathematical concepts.

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