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)

Text in English with abstracts in English, Afrikaans and isiZulu / The formalisation of first-order logic and axiomatic set theory in the first half of the 20th century—along with the advent of the digital computer—paved the way for the development of automated theorem proving. In the 1950s, the automation of proof developed from proving elementary geometric problems and finding direct proofs for problems in Principia Mathematica by means of simple, human-oriented rules of inference. A major advance in the field of automated theorem proving occurred in 1965, with the formulation of the resolution inference mechanism. Today, powerful Satisfiability Modulo Theories (SMT) provers combine SAT solvers with sophisticated knowledge from various problem domains to prove increasingly complex theorems. The combinatorial explosion of the search space is viewed as one of the major challenges to progress in the field of automated theorem proving. Pioneers from the 1950s and 1960s have already identified the need for heuristics to guide the proof search effort. Despite theoretical advances in automated reasoning and technological advances in computing, the size of the search space remains problematic when increasingly complex proofs are attempted. Today, heuristics are still useful and necessary to discharge complex proof obligations. In 2000, a number of heuristics was developed to aid the resolution-based prover OTTER in finding proofs for set-theoretic problems. The applicability of these heuristics to next-generation theorem provers were evaluated in 2009. The provers Vampire and Gandalf required respectively 90% and 80% of the applicable OTTER heuristics. This dissertation investigates the applicability of the OTTER heuristics to theorem proving in the hybrid theorem proving environment Rodin—a system modelling tool suite for the Event-B formal method. We show that only 2 of the 10 applicable OTTER heuristics were useful when discharging proof obligations in Rodin. Even though we argue that the OTTER heuristics were largely ineffective when applied to Rodin proofs, heuristics were still needed when proof obligations could not be discharged automatically. Therefore, we propose a number of our own heuristics targeted at theorem proving in the Rodin tool suite. / Die formalisering van eerste-orde-logika en aksiomatiese versamelingsteorie in die eerste helfte van die 20ste eeu, tesame met die koms van die digitale rekenaar, het die weg vir die ontwikkeling van geoutomatiseerde bewysvoering gebaan. Die outomatisering van bewysvoering het in die 1950’s ontwikkel vanuit die bewys van elementêre meetkundige probleme en die opspoor van direkte bewyse vir probleme in Principia Mathematica deur middel van eenvoudige, mensgerigte inferensiereëls. Vooruitgang is in 1965 op die gebied van geoutomatiseerde bewysvoering gemaak toe die resolusie-inferensie-meganisme geformuleer is. Deesdae kombineer kragtige Satisfiability Modulo Theories (SMT) bewysvoerders SAT-oplossers met gesofistikeerde kennis vanuit verskeie probleemdomeine om steeds meer komplekse stellings te bewys. Die kombinatoriese ontploffing van die soekruimte kan beskou word as een van die grootste uitdagings vir verdere vooruitgang in die veld van geoutomatiseerde bewysvoering. Baanbrekers uit die 1950’s en 1960’s het reeds bepaal dat daar ’n behoefte is aan heuristieke om die soektog na bewyse te rig. Ten spyte van die teoretiese vooruitgang in outomatiese bewysvoering en die tegnologiese vooruitgang in die rekenaarbedryf, is die grootte van die soekruimte steeds problematies wanneer toenemend komplekse bewyse aangepak word. Teenswoordig is heuristieke steeds nuttig en noodsaaklik om komplekse bewysverpligtinge uit te voer. In 2000 is ’n aantal heuristieke ontwikkel om die resolusie-gebaseerde bewysvoerder OTTER te help om bewyse vir versamelingsteoretiese probleme te vind. Die toepaslikheid van hierdie heuristieke vir die volgende generasie bewysvoerders is in 2009 geëvalueer. Die bewysvoerders Vampire en Gandalf het onderskeidelik 90% en 80% van die toepaslike OTTER-heuristieke nodig gehad. Hierdie verhandeling ondersoek die toepaslikheid van die OTTER-heuristieke op bewysvoering in die hibriede bewysvoeringsomgewing Rodin—’n stelselmodelleringsuite vir die formele Event-B-metode. Ons toon dat slegs 2 van die 10 toepaslike OTTER-heuristieke van nut was vir die uitvoering van bewysverpligtinge in Rodin. Ons voer aan dat die OTTER-heuristieke grotendeels ondoeltreffend was toe dit op Rodin-bewyse toegepas is. Desnieteenstaande is heuristieke steeds nodig as bewysverpligtinge nie outomaties uitgevoer kon word nie. Daarom stel ons ’n aantal van ons eie heuristieke voor wat in die Rodin-suite aangewend kan word. / Ukwenziwa semthethweni kwe-first-order logic kanye ne-axiomatic set theory ngesigamu sokuqala sekhulunyaka lama-20—kanye nokufika kwekhompyutha esebenza ngobuxhakaxhaka bedijithali—kwavula indlela ebheke ekuthuthukisweni kwenqubo-kusebenza yokufakazela amathiyoremu ngekhomyutha. Ngeminyaka yawo-1950, ukuqinisekiswa kobufakazi kwasuselwa ekufakazelweni kwezinkinga zejiyomethri eziyisisekelo kanye nasekutholakaleni kobufakazi-ngqo bezinkinga eziphathelene ne-Principia Mathematica ngokuthi kusetshenziswe imithetho yokuqagula-sakucabangela elula, egxile kubantu. Impumelelo enkulu emkhakheni wokufakazela amathiyoremu ngekhompyutha yenzeka ngowe-1965, ngokwenziwa semthethweni kwe-resolution inference mechanism. Namuhla, abafakazeli abanohlonze bamathiyori abizwa nge-Satisfiability Modulo Theories (SMT) bahlanganisa ama-SAT solvers nolwazi lobungcweti oluvela kwizizinda zezinkinga ezihlukahlukene ukuze bakwazi ukufakazela amathiyoremu okungelula neze ukuwafakazela. Ukukhula ngesivinini kobunzima nobunkimbinkimbi benkinga esizindeni esithile kubonwa njengenye yezinselelo ezinkulu okudingeka ukuthi zixazululwe ukuze kube nenqubekela phambili ekufakazelweni kwamathiyoremu ngekhompyutha. Amavulandlela eminyaka yawo-1950 nawo-1960 asesihlonzile kakade isidingo sokuthi amahuristikhi (heuristics) kube yiwona ahola umzamo wokuthola ubufakazi. Nakuba ikhona impumelelo esiyenziwe kumathiyori ezokucabangela okujulile kusetshenziswa amakhompyutha kanye nempumelelo yobuchwepheshe bamakhompyutha, usayizi wesizinda usalokhu uyinkinga uma kwenziwa imizamo yokuthola ubufakazi obuyinkimbinkimbi futhi obunobunzima obukhudlwana. Namuhla imbala, amahuristikhi asewuziso futhi ayadingeka ekufezekiseni izibopho zobufakazi obuyinkimbinkimbi. Ngowezi-2000, kwathuthukiswa amahuristikhi amaningana impela ukuze kulekelelwe uhlelo-kusebenza olungumfakazeli osekelwe phezu kwesixazululo, olubizwa nge-OTTER, ekutholeni ubufakazi bama-set-theoretic problems. Ukusebenziseka kwalawa mahuristikhi kwizinhlelo-kusebenza ezingabafakazeli bamathiyoremu besimanjemanje kwahlolwa ngowezi-2009. Uhlelo-kusebenza olungumfakazeli, olubizwa nge-Vampire kanye nalolo olubizwa nge-Gandalf zadinga ama-90% kanye nama-80%, ngokulandelana kwazo, maqondana nama-OTTER heuristics afanelekile. Lolu cwaningo luphenya futhi lucubungule ukusebenziseka kwama-OTTER heuristics ekufakazelweni kwamathiyoremu esimweni esiyinhlanganisela sokufakazela amathiyoremu esibizwa nge-Rodin—okuyi-system modelling tool suite eqondene ne-Event-B formal method. Kulolu cwaningo siyabonisa ukuthi mabili kuphela kwayi-10 ama-OTTER heuristics aba wusizo ngenkathi kufezekiswa isibopho sobufakazi ku-Rodin. Nakuba sibeka umbono wokuthi esikhathini esiningi ama-OTTER heuristics awazange abe wusizo uma esetshenziswa kuma-Rodin proofs, amahuristikhi asadingeka ezimweni lapho izibopho zobufakazi zingazenzekelanga ngokwazo ngokulawulwa yizinhlelo-kusebenza zekhompyutha. Ngakho-ke, siphakamisa amahuristikhi ethu amaningana angasetshenziswa ekufakazeleni amathiyoremu ku-Rodin tool suite. / School of Computing / M. Sc. (Computer Science)

CAPES / Apresentamos neste trabalho alguns paradoxos lógico-matemáticos, como o paradoxo de Galileu, e também alguns paradoxos geométricos, como os paradoxos de Curry, de Hooper e de Banach-Tarski. Empregamos os paradoxos de Curry e de Hooper para motivar o estudo de conceitos de Geometria e de Teoria dos Números, tais como área, semelhança de triângulos, o Teorema de Pitágoras, razões trigonométricas no triângulo retângulo, o coeficiente angular da reta e a sequência de Fibonacci, e organizamos atividades lúdicas para a sala de aula no Ensino Fundamental e no Ensino Médio. / We present in this work some logical-mathematical paradoxes, as Galileo's paradox, and also some geometric paradoxes, such as Curry's paradox, Hooper's paradox and the Banach-Tarski paradox. We employ the Curry and Hooper paradoxes to motivate the study of concepts of Geometry and Number Theory, such as area, triangle similarity, Pythagorean Theorem, trigonometric ratios in the right triangle, angular coefficient of the line, and Fibonacci sequence, and we organize recreation activities for the classroom in Elementary and High School.