The aim of this study was firstly to evaluate the impact of two interactive-engagement models of instruction, namely Whole Class Discussions (WCD) and Computer Simulations (CS) on first year physics student-teachers’ conceptual understanding of Newtonian mechanics, and on their epistemological beliefs about physics. The force concept inventory was used to evaluate the impact on conceptual understanding while the Epistemological Beliefs About Physical Science questionnaire was used to evaluate the impact on their epistemological beliefs. The findings suggest that interactive engagement models had a positive impact on students’ conceptual understanding of Newtonian mechanics, and on their epistemological beliefs about physics. The study also contributed WCD and CS activities that can be used or adapted with an aim of enhancing conceptual understanding in physics. The study did not show any direct relationship between students’ conceptual understanding of Newtonian mechanics and their epistemological beliefs about physics. Key words: Interactive-engagement, Whole Class Discussion, Computer Simulations, epistemological beliefs about physics. / Physics / M.Sc. (Physics Education)

Magister Educationis - MEd / Two grade 11 classes of two high schools in Cape Town were taught some concepts of sound by means of indigenous musical instruments. The purpose was to find out the relative effects (or none) of two instructional strategies. Toulmin (1958)’s Argumentation Pattern, Ogunniyi (1997)’s Contiguity Argumentation Theory and Reiner et al. (2000)’s Substance Schema formed the theoretical framework. A pre-post-test quasi-experimental design was employed and data collated using questionnaires, a sound conceptual test, argumentation worksheets, and classroom observation schedules. One teacher taught the experimental group using dialogical argumentation while another teacher taught the comparative group using lecturedemonstration method, coupled with the use of ICTs for duration of four weeks. Data were analysed using a mixed (quantitative and qualitative) methods approach. The findings revealed that many the learners held some scientifically valid conceptions of sound prior to formal instruction. However, the learners also held many scientifically invalid conceptions in relation to the speed of sound in air, sound propagation, and sound produced by stringed instruments. The alternative conceptions of learners in the C group remained largely unchanged after instruction, while those of the E group changed appreciably, but not completely. The E group learners changed the alternative conceptions that were worked into structured argumentation activities better than those which were not. Also, the learners in both groups seemed to hold indigenous beliefs in relation to sound which did not seem to change after instruction. Most learners had a positive attitude towards the use of indigenous knowledge in the science class. No significant difference was found between male and female learners with respect to conceptual understanding of sound, indigenous beliefs, and interest in the integration of science and indigenous knowledge.

Det finns ett stort utbud av digitala läromedel idag vilket kräver ökad kompetens att kunna värdera och välja lämpligt digitalt läromedel till undervisningen. Flera digitala läromedel marknadsförs som program innehållandes färdighetsträning. Färdighetsträning har handlat om att utveckla procedur- och begreppsförmågan frånskilt från de matematiska förmågorna: problemlösnings-, resonemangs- och kommunikationsförmågan.  Enligt forskning främjas utvecklingen av procedur- och begreppsförmågan om de tränas parallellt med de ovan nämnda matematiska förmågorna. Forskare menar att procedur- och begreppsförmågan utvecklas främst genom rika uppgifter där procedurer och begrepp diskuteras. Forskare menar dock att färdighetsträning i digitala läromedel ofta är designade som frågesportslekar vilket tenderar till att enbart främjar ytligt lärande. Forskare säger även att designen i digitala läromedel kan innehålla överflöd av visualiseringar som påverkar koncentrationen negativt. De säger också att återkopplingen oftast kommenterar svarets korrekthet istället för att stödja elevens utveckling av sin matematiska förståelse. Syftet med studien är att ta reda på om det digitala läromedlet Bingel erbjuder färdighetsträning som involverar dessa matematiska förmågor vid träning av procedur- och begreppsförmågan. Vidare undersöks hur designvalet kan påverka utvecklingen av procedur- och begreppsförmågan. För att besvara studiens syfte använder studien två olika ramverk som dels ser till designen av det digitala läromedlet, dels hur det digitala läromedlet involverar matematiska förmågor. Resultatet visar att designen av uppgiftstypen inte involverar fler matematiska förmågor förutom procedur- och begreppsförmågan och träningen av procedurer och begrepp främjar enbart ytligt lärande. Vidare visar designen gällande återkopplingen att den inte uppmuntrar till att utveckla djupare förståelse. / There is a big variety of digital teaching materials today which requires higher digital competence to choose the appropriate digital teaching material for the education. Several digital teaching materials are marketed as containing skills training. Skills training has been about developing procedural and conceptual knowledge apart from the other mathematical competences: communication, reasoning and problem-solving competency. Studies have shown that the development of procedural and conceptual knowledge fosters when practiced parallel with the other mathematical competencies mentioned above. Procedural and conceptual knowledge foster through rich tasks where procedures and concepts are discussed, according to researchers. Furthermore, researchers claim that skills training in digital teaching materials are often designed as pop-quizzes which only foster surface learning. Researchers also claim that the design often contains an abundance of visual effects which can impair concentration. They also say that the feedback given is often in the form of comments on the correctness of the answers rather than the improvement of the student’s own development of mathematical understanding. The purpose of this study is to examine if the digital teaching material Bingel offers skills training that involves these competencies in the practicing of procedural and conceptual knowledge. Furthermore, this study examines if the design can affect the development of the procedural and conceptual knowledge. This study uses two different theoretical frameworks to be able to look into the design of the digital teaching material and also how the mathematical competencies are integrated. Results show that the design of the task does not involve more mathematical competencies other than the procedural and conceptual knowledge which are trained to foster surface learning. Furthermore, the design of the feedback does not foster deeper understanding.

This multiple-case study examines the explicit and implicit assumptions of six veteran calculus instructors from three types of educational institutions, comparing and contrasting their views on the iteration of conceptual understanding and procedural fluency of pre-calculus topics. There were three components to the research data recording process. The first component was a written survey, the second component was a "think-aloud" activity of the instructors analyzing the results of a function diagnostic instrument administered to a calculus class, and for the third component, the instructors responded to two quotations. As a result of this activity, themes were found between and among instructors at the three types of educational institutions related to their expectations of their incoming students’ prior knowledge of pre-calculus topics related to functions. Differences between instructors of the three types of educational institutions included two identifiable areas: (1) the teachers’ expectations of their incoming students and (2) the methods for planning instruction. In spite of these differences, the veteran instructors were in agreement with other studies’ findings that an iterative approach to conceptual understanding and procedural fluency are necessary for student understanding of pre-calculus concepts.

Undervisning och lärande är grundbultar i ett modernt demokratiskt samhälle. Dessa aktiviteter sker i stor utsträckning av lärare vilka bär ansvaret för att utbilda framtidens generationer. Inom utbildningsväsendet existerar en rad olika läromedel såsom böcker och datorer vilka ofta tenderar att vara statiska i sitt förhållande till eleven. Enligt internationella kunskapsmätningar har svenska elevers matematiska kunskap stagnerat de senaste två decennierna, även om en vändning har skett de senaste åren. En möjlig förklaring till detta kan vara att ett för stort fokus hos bl.a. lärare och matematikböcker ägnats åt procedurella färdigheter snarare än begreppslig förståelse. Detta i kombination med lärares upplevda svårigheter kring att tillämpa ett formativt arbetssätt kan innebära hinder för elevers matematiska kunskapsutveckling. Flera digitala verktyg har utvecklats för att stötta lärare och elever i deras lärandeprocess. Ett sådant digitalt verktyg är Mentimeter som utvecklats för att öka interaktionen mellan en presentatör (t.ex. en lärare) och dess publik. Mentimeterär inte avsett specifikt för en lärandekontext och det finns idag ingen etablerad utgivning av läromedel som använder Mentimeter för skolans värld. Det här examensarbetet har med bakgrund av detta haft som mål att designa och konstruera ett matematiskt läromedel med hjälp av Mentimeter, ämnat för att utveckla elevers förståelse för ett matematiskt begrepp (funktioner). Detta har gjorts med en designbaserad forskningsmetodik vilket är en metod som lämpar sig väl för att utveckla tekniska artefakter inom en skolkontext samt undersöka hur den utvecklade artefakten fungerar i en specifik kontext. Syftet med studien har varit att undersöka lärares upplevelse av att använda det utvecklade läromedlet och med den kunskapen, genom en iterativ process utveckla och samtidigt öka förståelse för hur ett digitalt utbildningsverktyg kan möjliggöra ett formativt arbetssätt. Ett grundantagande inom ramen för detta examensarbete är att ett formativt arbetssätt kan bidra till ett ökat lärande och därmed ökad förståelse för funktionsbegreppet, men att det behövs mer kunskap för att förstå sambandet mellan didaktisk teori och hur den kan användas inom en skolkontext. Detta examensarbete utgår både från en konstruktivistisk och en sociokulturell syn på lärande. Motiveringen av denna breda syn på lärande är att det möjliggör en mer induktiv ansats i förhållande till tolkningen av studiens resultat. Studiens resultat är en“Mentimeter-modul” som designats för att möjliggöra ökad förståelse för matematiska begrepp, samt en teoretisk förståelse för hur modulen upplevts av lärare i en skolkontext. Vidare visar studiens resultat på vikten av att lärare använder Mentimeter-modulen som ett komplement till den ordinarie undervisningen istället för att låta verktyget agera som en egen aktör. Slutsatsen är att den utvecklade Mentimeter-modulen upplevs möjliggöra ett formativt arbetssätt om syftet med modulen tydliggörs och lärarna ser på modulen som ett komplement till sin ordinarie undervisning. / Teaching and learning could be considered as cornerstones of a modern democratic society. These activities are largely being practiced by teachers who are in part responsible for educating future generations. In the education system, there are a number of different teaching materials such as books and computers, which often tend to be static with no or little interaction with the student. According to international surveys the mathematical knowledge among Swedish student’s has stagnated over the past two decades, although a turnaround has taken place in recent years. A possible explanation for this may be that too much focus has been devoted to procedural skills rather than conceptual understanding. This, combined with teachers' perceived difficulties in applying a formative approach, can slow down the development of students' mathematical knowledge. Several digital educational tools have been developed to support teachers and students in their learning process. One such tool is Mentimeter, which was developed to increase the interaction between a presenter (e.g. a teacher) and its audience (e.g. a student). Mentimeter is not intended specifically for a learning context and there is currently no mathematical educational materials developed with the software and that is accessible for everyone with a Menitemeter account. This master thesis has aimed to design and construct a mathematical teaching aid with the help of Mentimeter, intended to develop students' understanding of a mathematical concept (functions). This has been done with a design-based research methodology which is suited for exploring learning environments and gaining deeper theoretical understanding by introducing a designed technical artifact in the context that is being observed. The purpose of this Master Thesis has been to investigate teachers' experience of using the developed teaching aid and with that knowledge, through an iterative process develop understanding on how a digital educational tool can enable a formative approach. A basic assumption made by the authors of this study is that a formative approach can contribute to increased learning andthus increased understanding of the concept of a function. But more knowledge is needed to understand the connection between the didactic theory and how it can be applied in a school context. This study is based partly on a constructivist view of learning but also on a socio-cultural perspective. The motivation for this broad view of learning enables a more inductive approach in relation to the interpretation of the study results. The results of this thesis are a “Mentimeter module” designed to enable an increased understanding of mathematical concepts, as well as a theoretical understanding of how the module is experienced by teachers in a school context. Furthermore, the study shows the importance of teachers using the Mentimeter module as a complement to their regular teaching activities instead of letting the digital education tool act on its own. The conclusion is that the developed Mentimeter module is perceived to enable a formative approach if the purpose of the module is clarified and the teachers see the module as a complement to their established experience andprevious knowledge.

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)

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.