Este trabalho de pesquisa investigou o processo de aprendizagem de geometria em uma turma do sexto ano do Ensino Fundamental de uma escola da rede municipal de Porto Alegre. Durante dois meses do ano de 2016, foram desenvolvidos os conceitos de quadriláteros, triângulos e de Transformações Geométricas (translação, rotação e reflexão) sob a perspectiva da Investigação Matemática em sala de aula, metodologia de ensino que possui potencial para desencadear o processo de construção do conhecimento. Durante este período, os estudantes realizaram a investigação de quadriláteros e de triângulos, utilizando o software GeoGebra como recurso das Tecnologias da Informação e Comunicação (TIC). Os alunos construíram estes polígonos no GeoGebra, através de orientações passo-a-passo que foram disponibilizadas através de formulários online. Ao longo destas construções, os estudantes responderam a questionamentos também contidos nestes formulários online, de forma a identificar as propriedades contidas em cada construção, referentes a cada figura geométrica. Posteriormente, registraram as propriedades de cada polígono em uma tabela de características, de forma a organizar as propriedades de cada quadrilátero e de cada triângulo estudado. Para a investigação das Transformações Geométricas, desenvolveu-se um trabalho fazendo-se uso de tesselações no plano (coberturas para o plano). Para esta etapa da investigação, utilizou-se o applet “Design a Tessellation”, que é um recurso online e gratuito no qual o usuário pode criar diferentes coberturas para o plano através de uma unidade de tesselação quadrada. Os alunos fizeram uso de formulários online para responder aos questionamentos sobre as Transformações Geométricas estudadas, assim como folhas com atividades e malhas impressas para a criação de tesselações. Para a análise do processo de aprendizagem dos estudantes, foi utilizada a perspectiva dos níveis de Van Hiele, que classifica os níveis de pensamento geométrico, utilizando também uma abordagem que admite a existência de níveis intermediários. Além disso, este trabalho também formulou uma complementação para os níveis de Van Hiele quanto às Transformações Geométricas, de forma a analisar os dados obtidos com a pesquisa de uma maneira mais detalhada. Com a pesquisa finalizada, conclui-se que houve progresso dos níveis de Van Hiele para os estudantes analisados. / This research investigated the learning process of geometry in a class of the sixth grade of Elementary School, of a municipal school in Porto Alegre. During two months of 2016, the concepts of quadrilaterals, triangles and Geometric Transformations (translation, rotation and reflection) were developed from the perspective of Math Investigation in the classroom, teaching methodology which has the potential to develop the process of knowledge construction. During this period, students performed the investigation of quadrilaterals and triangles, using GeoGebra software as a resource of Information and Communication Technologies (TIC). The students constructed these polygons in GeoGebra, through step-by-step guidelines, which were made available through online forms. Throughout these constructions, the students answered the questions also contained in these online forms, in order to identify the properties contained in each construction, referring to each geometric figure. Later, they registered the properties of each polygon in a table of characteristics, in order to organize the properties of each quadrilateral and of each triangle studied. For the investigation of the Geometric Transformations, a work was developed making use of tessellations in the plane (covers for the plane). For this stage of the research, the "Design a Tessellation" applet was used, which is an online and free resource, where the user can create different covers for the plane, through a square tessellation unit. Students used online forms to answer questions about Geometric Transformations studied, as well as sheets with activities, and printed meshes for the creation of tessellations. For the analysis of the students' learning process, the Van Hiele levels perspective was also used, which classifies the levels of geometric thinking, using an approach that admits the existence of intermediate levels. In addition, this work also formulated a complementation for the Van Hiele levels, regarding Geometric Transformations, in order to analyze the data obtained with the research in a more detailed way. With the research completed, it is concluded that there was progress of the levels of Van Hiele for the analyzed students.

Este trabalho de pesquisa investigou o processo de aprendizagem de geometria em uma turma do sexto ano do Ensino Fundamental de uma escola da rede municipal de Porto Alegre. Durante dois meses do ano de 2016, foram desenvolvidos os conceitos de quadriláteros, triângulos e de Transformações Geométricas (translação, rotação e reflexão) sob a perspectiva da Investigação Matemática em sala de aula, metodologia de ensino que possui potencial para desencadear o processo de construção do conhecimento. Durante este período, os estudantes realizaram a investigação de quadriláteros e de triângulos, utilizando o software GeoGebra como recurso das Tecnologias da Informação e Comunicação (TIC). Os alunos construíram estes polígonos no GeoGebra, através de orientações passo-a-passo que foram disponibilizadas através de formulários online. Ao longo destas construções, os estudantes responderam a questionamentos também contidos nestes formulários online, de forma a identificar as propriedades contidas em cada construção, referentes a cada figura geométrica. Posteriormente, registraram as propriedades de cada polígono em uma tabela de características, de forma a organizar as propriedades de cada quadrilátero e de cada triângulo estudado. Para a investigação das Transformações Geométricas, desenvolveu-se um trabalho fazendo-se uso de tesselações no plano (coberturas para o plano). Para esta etapa da investigação, utilizou-se o applet “Design a Tessellation”, que é um recurso online e gratuito no qual o usuário pode criar diferentes coberturas para o plano através de uma unidade de tesselação quadrada. Os alunos fizeram uso de formulários online para responder aos questionamentos sobre as Transformações Geométricas estudadas, assim como folhas com atividades e malhas impressas para a criação de tesselações. Para a análise do processo de aprendizagem dos estudantes, foi utilizada a perspectiva dos níveis de Van Hiele, que classifica os níveis de pensamento geométrico, utilizando também uma abordagem que admite a existência de níveis intermediários. Além disso, este trabalho também formulou uma complementação para os níveis de Van Hiele quanto às Transformações Geométricas, de forma a analisar os dados obtidos com a pesquisa de uma maneira mais detalhada. Com a pesquisa finalizada, conclui-se que houve progresso dos níveis de Van Hiele para os estudantes analisados. / This research investigated the learning process of geometry in a class of the sixth grade of Elementary School, of a municipal school in Porto Alegre. During two months of 2016, the concepts of quadrilaterals, triangles and Geometric Transformations (translation, rotation and reflection) were developed from the perspective of Math Investigation in the classroom, teaching methodology which has the potential to develop the process of knowledge construction. During this period, students performed the investigation of quadrilaterals and triangles, using GeoGebra software as a resource of Information and Communication Technologies (TIC). The students constructed these polygons in GeoGebra, through step-by-step guidelines, which were made available through online forms. Throughout these constructions, the students answered the questions also contained in these online forms, in order to identify the properties contained in each construction, referring to each geometric figure. Later, they registered the properties of each polygon in a table of characteristics, in order to organize the properties of each quadrilateral and of each triangle studied. For the investigation of the Geometric Transformations, a work was developed making use of tessellations in the plane (covers for the plane). For this stage of the research, the "Design a Tessellation" applet was used, which is an online and free resource, where the user can create different covers for the plane, through a square tessellation unit. Students used online forms to answer questions about Geometric Transformations studied, as well as sheets with activities, and printed meshes for the creation of tessellations. For the analysis of the students' learning process, the Van Hiele levels perspective was also used, which classifies the levels of geometric thinking, using an approach that admits the existence of intermediate levels. In addition, this work also formulated a complementation for the Van Hiele levels, regarding Geometric Transformations, in order to analyze the data obtained with the research in a more detailed way. With the research completed, it is concluded that there was progress of the levels of Van Hiele for the analyzed students.

Over 50% of secondary students failed the geometry end-of-course test in a Florida school district, indicating a need to improve academic performance. Secondary school students’ learning characteristics and the effectiveness of teachers’ instructional strategies are imperative to educational success. In this qualitative case study, geometry teachers’ instructional strategies, as defined by the Marzano Causal Teacher Evaluation Model, were explored once teachers were informed of students’ multiple intelligences and trained in multiple-intelligence-based lessons. Participants were 2 geometry teachers and 15 secondary geometry students in a traditional public school. Using Howard Gardner’s multiple intelligences theory and the van Hiele model of learning geometry, the researcher analyzed interviews, observations, and teachers’ lesson plans to shed light on teachers’ use of multiple intelligence data and training. Significant conclusions emerged from the findings of the case study. First, teachers’ dominant intelligences shape the use of instructional strategies. Second, multiple intelligences were used to personalize instruction, create a student-centered classroom environment, and nurture student engagement among secondary geometry learners. Lastly, when instructors taught based on students’ van Hiele levels, 5 of 8 intelligences are excluded. Teachers used strategies steeped in spatial, logical, and linguistic intelligences to teach students how to draw, think, and write. Strategies for students with interpersonal, intrapersonal, musical, naturalist, and kinesthetic intelligences were excluded. Based on the conclusions of the study, educators have new information on ways to make geometry instruction more inclusive for their diverse learning population. Education stakeholders are also enlightened with what may be missing in geometry classrooms and impeding student success.

Elevers kunskaper inom bevisföring i geometri har länge presenterats som bristande, både nationellt och internationellt. Denna litteraturstudie syftar till att studera vilka svårigheter elever stöter på i arbetet med bevisföring inom geometri för att ge en samlad bild av vanliga missuppfattningar. Artiklarna valdes ut genom sökningar i UniSearch och ERIC där de behövde uppfylla författarnas urvalskriterier. Dessa artiklar sammanfattades för att vidare analyseras utifrån en modifierad version av van Hieles teori om geometriskt lärande bestående av tre olika nivåer. Resultatet visar att elever befinner sig på lägre nivåer än det som förväntas av högstadie- och gymnasieelever enligt forskningen. Vidare visar resultat att elever har svårigheter med att förstå innebörden av ett matematiskt bevis, rita och dra korrekta slutsatser från figurer, använda geometriska definitioner samt att stödja sina påståenden med axiom och satser. Vi tror att lärare kan använda denna information för att utveckla och anpassa sin undervisning för att stärka elevers förmåga att förstå och konstruera geometriska bevis. Forskning om elevers svårigheter med bevisföring inom geometri i Sverige är knapphändig och därför kan nästa steg vara att studera området i svenska gymnasieskolor. / Students understanding with proof in geometry has for a long time been deficient both national and international. The aim of this literature review is to study what difficulties students encounter when working with geometrical proof to give a summary of common misunderstandings. The articles were selected through UniSearch and ERIC and had to fulfill our chosen criterions. These articles where summarized and analyzed with van Hieles modified theory of geometrical learning consisting of three levels. The result shows that students are at a lower level than expected from high- and secondary school students. Furthermore, our result shows that students have difficulties to understand the meaning of a mathematical proof, to draw and make right conclusions from diagrams, to use geometrical definitions and to support claims by using axioms and theorems. We believe that teachers can use this information to develop their teaching to strengthen students’ understanding of proofs and the ability to construct geometrical proofs. Research on students’ difficulties with geometry in Sweden is lacking and therefore the next step could be to study this field in Swedish upper secondary schools.

Detta arbete undersöker svenska gymnasieelevers bevisföringsförmåga inom geometri med van Hieles modifierade modell som ramverk. Studien bygger på vårt tidigare examensarbete Elevers svårigheter med bevisföring inom geometri (Strandler & Mach, 2022), där internationella studier kring området sammanfattades. I föreliggande studie fick 35 gymnasieelever genomföra ett kunskapstest. Kunskapstestet bestod av fem uppgifter på olika svårighetsnivåer valda utifrån van Hieles modell. Syftet med studien var att synliggöra elevers svårigheter utifrån van Hieles modifierade modell. Resultatet visade att svenska elever har svårigheter på den teoretiska nivån. Några vanliga svårigheter var att beviset var otillräckligt och saknade motivering, exempelvis hänvisning till satser eller geometriska definitioner. Ytterligare en brist som visades i flera elevlösningar var att de inte förstod figurers egenskaper på ett korrekt sätt. Resultatet i denna studie överensstämmer med vårt tidigare examensarbete, elever befinner sig på en lägre nivå än det som förväntas. För att stödja elevers förståelse inom området bör undervisningen ge fler tillfällen att utforska geometrin med exempelvis dynamiska hjälpmedel, och inte enbart låta eleverna pröva satsers sanning empiriskt. / This study explores upper secondary school students from Sweden with van Hieles modified model as framework. The study is based on our previous research Students’ Difficulties with Proof in Geometry (Strandler & Mach, 2022) where international studies of this field was summarized. In this study 35 upper secondary students performed a knowledge test. The test contained five questions on different difficulty levels based on van Hieles model. The results showed that a lot of students have difficulties on the theoretical level. A common difficulty was that their proof were insufficient and lacked motivation, for example referring to a mathematical theorem or geometrical definitions. Another difficulty that several students showed was the lack of understanding for different properties of diagrams. The results support our previous findings, students are found to be at a lower level than expected. The education in this area should give more opportunities for students to explore geometry more, for example with dynamical tools, and not only let students experience geometrical proof empirical.

Studiens syfte är att undersöka hur lärare kan utföra deras undervisning som möjliggör att elever utvecklar deras tänkande i geometri. Den syftar även till att undersöka lärares upplevelser av möjligheter och begränsningar med utomhus– och inomhusundervisning i geometri. Studien baseras på semistrukturerade intervjuer och observationer av två verksamma lärare på lågstadiet. Den insamlade datan har analyserats utifrån en induktiv och deduktiv analys där Van Hieles teori har använts. Teorin består av olika nivåer av tänkande i geometri. Resultatet visar att Van Hieles nivåer behandlas av lärarna både inomhus och utomhus, vilket därmed möjliggör att eleverna utvecklar deras tänkande i geometri både inne och ute. Möjligheten att arbeta praktiskt och förena teori och praktik på en större yta är fördelar som lärarna upplever med utomhusundervisning i geometri, å andra sidan begränsar organisationen lärares möjligheter att genomföra detta. Tryggheten som klassrummet erbjuder är den främsta fördelen som lärarna upplever, dock finns det risk att elever i svårigheter blir utpekade på ett sätt som inte hade visat sig ute. Slutsatser som har dragits är vikten av att arbeta praktiskt utomhus samtidigt som störningsmoment kan ta uppmärksamhet från undervisningen. Det kräver att lärare tillsammans med sina elever arbetar fram förhållningsregler på platsen för att lyckas skapa sitt uteklassrum.

En rad företag som erbjuder läxhjälp har uppstått sedan Rutavdraget för läxhjälp utökades till att gälla för gymnasieelever. Youschool är ett av dessa företag. Det som utmärker Youschool är att de erbjuder läxhjälp där elever och lärare kan kommunicera med varandra både med penna och ljud i realtid via nätet. De tillhandahåller virtuella lektioner med en lärare på 2 till 4 elever åt gången, där en dokumentkamera är det verktyg som utgör grunden i kommunikationstekniken. Eleverna lägger exempelvis sitt skrivblock under sina dokumentkameror och så kan de som är med på lektionen följa varandras resonemang,eftersom de hela tiden kan se vad alla gör med sina pennor. Matematik 2b är en kurs främst för elever som läser på Samhällsvetenskapsprogrammet eller på Ekonomiprogrammet, två högskoleförberedande gymnasieprogram. Statistik för resultaten på de nationella proven i kursen, från totalundersökningar i Sverige, visar att en hög andel elever får underkänt betyg på provet. Denna studie är kvalitativ och utgörs av semistrukturerade telefonintervjuer med elever som läser Matematik 2b och som använder Youschool som läxhjälp samt av observationer från en skärminspelad lektion på Youschool där eleverna jobbar med ett för studien tillrättalagt material som testar deras kunskaper om funktionsbegreppet. Syftet är att undersöka om och i så fall hur läroverktyget Youschool kan utgöra ett stöd i elevers kunskapsutveckling i Matematik 2b. Kursens kunskapskrav kopplade till matematiska förmågor hos eleverna och van Hieles tankenivåer är de analysverktyg som används i diskussionen av resultaten. Lärarens roll som stöttande och utmanande framträder som viktig för att upprätthålla elevernas motivation till att arbeta under lektionerna på Youschool. Vidare kan eventuellt antydas att eleverna tränas i vissa matematiska förmågor mer än andra, och att elever önskar fler uppgifter som stimulerar deras tänkande på de högre av van Hiele-nivåerna. Tekniken som å ena sidan möjliggör undervisningen på Youschool kanbehöva utvecklas eftersom den å andra sidan ofta strular. / There are a growing number of companies in Sweden that provide private tutoring to students in upper secondary school, one of these companies is Youschool. They distinguished themselves from the others by having a tutor communicating with a student via Internet–using pencil and sound in real time. Youschool provide virtual classes with two to four students at each time lead by one teacher using a“document camera”as the main communication equipment. The students put their notebooks under their document cameras and are thereafter able to demonstrate their solutions and follow each other’s. They can literally follow each stroke of each other’s pencils. Matematik 2b is a mathematics unit in the Swedish upper secondary school mainly taken by students in the Business Management and Economics Programme and the Humanities Programme, both theoretical programmes preparing students for university studies. Statistics based upon the Swedish national examinations each year shows that a great number of students fail the tests in this unit. This is a qualitative study based on semi structured telephone interviews of students taking the Matematik 2b unit, and who are using Youschool as private tutoring, as well as observations of a screen filmed class where students practiced solving mathematic problems. The purpose of the study is to research whether Youschool is supportive in studying mathematics or not. In the discussion section of this study, both the curriculum of the unit and the van Hiele-levels are referred to when analysing the findings. The results point out the importance of a supportive and challenging tutor to help students to keep their motivation up during classes in Youschool. Furthermore, some mathematics skills might be better practiced using Youschool than others, therefore students wish to exercise further mathematic problems to stimulate thinking on the higher van Hiele-levels. However, the technology that is supposed to enable learning by Youschool might sometimes be the one thing to hinder a student from learning. Problems with the technology therefore impose Youschool to update their systems to affirm effective learning.

The Lesotho Junior Secondary Examination Analysis (2009 and 2010) revealed that students performance in Mathematics in general and Transformation geometry of rotation in particular was generally poor. Only a few number of students that sat for the final Form C Examination passed. This study employed the van Hiele’s levels of learning to investigate and describe the difficulties students have in the learning of rotational transformation geometry. Both a written test and interview were used to solicit information regarding students’ difficulties. This information was collected from 90 students from Qaoling Secondary School in Maseru district in Lesotho. Findings from the study revealed that students had difficulties in identifying and naming transformation of rotation, finding the centre, angle of rotation and locating the exact image of a rotated figure after rotation. Also, they had greater difficulties when using transformation to do proof. The analysis showed that students mostly had difficulties at the level of Abstraction and Deduction. This gave an indication that the vast majority of the students in Form C are reasoning at the lowest two levels of the van Hiele’s model which are Visualization and Description. For these students’ difficulties to be curbed, the analysis demonstrated amongst others that teachers needed to use Manipulative materials and Information Communication Technology (ICT) during the process of teaching and learning. Manipulative materials provide experience in which students can transfer their understanding smoothly from one concept to another. / Mathematics Education / M. Ed. (Mathematics Education)

This study investigated the effect of integration of GeoGebra into the teaching of circle geometry on Grade 11 students’ achievement. The study used a quasiexperimental, non-equivalent control group design to compare achievement, Van Hiele levels, and motivation of students receiving instruction using GeoGebra and those instructed with the traditional ‘talk-and-chalk’ method. Two samples of sizes n = 22 (experimental) and n = 25 (control) drawn from two secondary schools in one circuit of the Vhembe district, Limpopo Province in South Africa were used. A pilot study sample of size n = 15, was carried out at different schools in the same circuit, in order to check the reliability and validity of the research instruments, and statistical viability. The results of the pilot study were shown to be reliable, valid and statistically viable. The study was informed by the action, process, object, schema (APOS) and Van Hiele theories, as the joint theoretical framework, and the literature search concentrated on technology integration, especially GeoGebra, in the teaching and learning of mathematics. The literature was also reviewed on the integration of computer technology (ICT) into mathematics teaching and learning, ICT and mathematical achievement, and ICT and motivation. The study sought to answer three research questions which were hypothetically tested for significance. The findings of this study revealed that there was a significant difference in the achievement of students instructed with GeoGebra compared to those instructed with the traditional teaching method (teacher ‘talk-andchalk’). The average achievement of the experimental group was higher than that of the control group. Significant differences were also established on the Van Hiele levels of students instructed with GeoGebra and those instructed without this software at Levels 1 and 2, while there were no significant differences at Levels 3, 4 and 5. The experimental group achieved a higher group average at the visualisation and analysis Van Hiele levels. It was also statistically inferred from questionnaires through chi-square testing, that students instructed with GeoGebra were more motivated to learn circle geometry than those instructed without the software / Mathematics Education / M. Sc. (Mathematics Education)

The Lesotho Junior Secondary Examination Analysis (2009 and 2010) revealed that students performance in Mathematics in general and Transformation geometry of rotation in particular was generally poor. Only a few number of students that sat for the final Form C Examination passed. This study employed the van Hiele’s levels of learning to investigate and describe the difficulties students have in the learning of rotational transformation geometry. Both a written test and interview were used to solicit information regarding students’ difficulties. This information was collected from 90 students from Qaoling Secondary School in Maseru district in Lesotho. Findings from the study revealed that students had difficulties in identifying and naming transformation of rotation, finding the centre, angle of rotation and locating the exact image of a rotated figure after rotation. Also, they had greater difficulties when using transformation to do proof. The analysis showed that students mostly had difficulties at the level of Abstraction and Deduction. This gave an indication that the vast majority of the students in Form C are reasoning at the lowest two levels of the van Hiele’s model which are Visualization and Description. For these students’ difficulties to be curbed, the analysis demonstrated amongst others that teachers needed to use Manipulative materials and Information Communication Technology (ICT) during the process of teaching and learning. Manipulative materials provide experience in which students can transfer their understanding smoothly from one concept to another. / Mathematics Education / M. Ed. (Mathematics Education)