Denna studie har undersökt överensstämmelsen mellan lärares planering, genomförande och uppfattning av geometriundervisning i årskurs 3. Bakgrunden till undersökningen är att elever i årskurs 4 uppvisar låga resultat i geometri. Bakgrunden är även att lärare som fokuserar på kommunikation, samarbete och utmaningar i geometriundervisningen kan se till att elever får arbeta med annat än enskilt räknande av rutinuppgifter, något som studier efterfrågar. Utifrån detta behövs en studie som undersöker lärares geometriundervisning närmare, gärna ur flera perspektiv. I denna studie samlades data om två lärares geometriundervisning in genom observationer, intervjuer och innehållsanalyser. Dataanalysen har utgått från Van Hieles utvecklingsnivåer. Resultatet visar att lärarna föredrar en balans mellan enskilt arbete med rutinuppgifter och alternativ till detta men har ett omedvetet fokus på olika geometriska aspekter vid varje stadie som tillhör undervisningen (planering, genomförande, uppfattning). Konsekvenserna blir exempelvis att en Van Hiele-nivå som ska bearbetas enligt planering inte alls berörs i genomförandet av lektionen, samt att nivåer hoppas över. Det finns även utmaningar i att skapa balans mellan matematik och aktivitet samt en avsaknad av geometriskt innehåll i en del läroböcker. Elever i årskurs 4 kanske har låga resultat i geometri för att nivåer av geometrilärande hoppas över i undervisningen? / <p>Matematik</p>

The purpose of this study is to observe which method that is used when it comes to teaching geometry. The study aims to take a closer look at four different ways of teaching: illustrations, using practical objects, brainstorming and verbal communication. The method used to get the results is observation. With the result, we can deduce primarily that the educators make use of the four didactic methods to varying degrees. There are similarities between most of the teachers but also differences between them. What we can learn from the study is that communication between teacher and student is aimed at ultimate weight and especially the verbal communication. The pattern found is that all the educators are using verbal communication as their primary tool when teaching math.

This study investigated the use of an advanced technological development (free GeoGebra software) within the secondary educational setting in four relatively under-resourced schools in the Gauteng Province of South Africa. This advancement is viewed as having the potential to promote the teaching and learning of complex ideas in mathematics, even within traditionally deprived communities. The focus in this study was on the teaching and learning of transformation geometry at Grade 9 and attainment was reflected in terms of the van Hieles' levels of geometrical thinking. A mixed methods approach was followed, where data was collected through lesson observations, written tests and semi-structured interviews. Four Grade 9 teachers from four schools were purposively selected, while twenty-four mathematics learners (six from each school) in the Tshwane metropolitan region were randomly selected. The teachers' lesson observations and interview outcomes were coded and categorised into themes, and the learners' test scripts were marked and captured. The analysis of test scores was structured according to the van Hieles' levels of geometric thought development. As far as the use of GeoGebra is concerned, it was found that teachers used the program in preparation for, as well as during lessons; learners who had access to computers or android technology, used GeoGebra to help them with practice and exercises. As far as the effect of the use of GeoGebra is concerned, improved performance in transformation geometry was demonstrated. / Dissertation (MEd)--University of Pretoria, 2016. / Science, Mathematics and Technology Education / MEd / Unrestricted

The purpose of this study has been to contribute knowledge about textbooks in mathematics in the field of geometry. More precisely, our ambition has been to find out if the textbooks offer progression between grades 1-3. To study progression in textbooks, we have used van Hiele’s theory, which is a model for how students learn geometry in five hierarchical levels. The method we chose for our study was a combination of a quantitative and a qualitative content analysis. To meet the aim of the study, we analysed two different textbook series. Six textbooks were analysed in total. The textbooks were selected because of their popularity in Swedish schools, one of the textbook series was printed, Favorit Matematik and the other series was digital, Matematik (NE). The results of our analysis showed that both textbook series contained two dimensional and three-dimensional geometric objects and the tasks in these textbooks could be classified according to van Hiele levels 1-3. The results of our study showed that both textbook series offered progression in degree of difficulty based on van Hiele’s levels of thinking. We found differences between these two-textbook series with the number of tasks and in the degree of difficulty. We also found similarities in the fact that both textbook series in grade 2 had tasks that consisted three-dimensional geometrical objects. Our results also showed that Favorit Matematik had a higher degree of difficulty compared to Matematik (NE) which makes it doubtful when it comes to an equivalent education in Swedish schools, as according to school law, equality and equal conditions for all students is central in each school form. The conclusion that was then made is that Favorit Matematik is more suitable for students with higher learning abilities whilst Matematik (NE) is more suited for students who need tasks with a lower degree of difficulty as well as less tasks. The results of this study can provide increased knowledge about how van Hiele theory is used in the textbooks in primary schools.

I denna systematiska litteraturstudie är syftet att undersöka olika undervisningsmetoder och klassrumsaktiviteters påverkan på kunskapsutveckling i ämnet geometri. Som bakgrund till detta ligger de svenska elevernas bristfälliga geometrikunskaper. Undervisningsmetoderna och klassrumsaktiviteterna, med dess kunskapsinnehåll, vägs mot Van Hieles utvecklingsnivåer. Vidare vävs det sociokulturella lärandeperspektivet in i synen på kunskapsinhämtande. TIMSS syn på kunskap diskuteras mot betygskriternernas syn på kunskap för årskurs 6 i Lgr 22. 23 artiklar inhämtades, lästes och kategoriserades för att få en överblick inom området. 14 av dessa har djupanalyserats och sammanställts. Efter sammanställningen framgår det att studiens resultat inte pekar på något entydigt svar. Resultatet visar att det går att åstadkomma utveckling inom flertalet olika kunskapsområden genom att arbeta med olika undervisningsmetoder och klassrumsaktiviteter, så som exempelvis GeoGebra eller fysiskt laborativt material. Studiens olika undervisningsmetoder och klassrumsaktiviteter lutar sig främst mot Van Hieles nivå 2, men berör även fler av Van Hieles utvecklingsnivåer. Genom att identifiera vilka kunskaper eleverna besitter kan läraren utforma undervisningsaktiviteter som gynnar kunskapsutvecklingen hos eleverna för att de ska kunna ta sig vidare till nästa Van Hiele nivå. Studiens resultat och dess olika undervisningsmetoder och klassrumsaktiviteter kan ses som ett stöd för läraren i sin planering, och på så vis stödja eleven till kunskapsutveckling. Studien visar även på att motivation kan ses som en bidragande faktor till kunskapsutveckling. / In this systematic literature study, the aim is to examine the influence of different teaching methods and classroom activities on knowledge development in geometry. The background to this is the Swedish students' deficient knowledge of geometry. The teaching methods and classroom activities, with their knowledge content, are compared against Van Hiele's levels of geometric thinking. Furthermore, the view of knowledge acquisition is seen through the socio-cultural learning perspective. The concept of knowledge is discussed between TIMSS' view of knowledge and the grading criteria for grade 6 in the Swedish school curriculum. 23 articles were obtained, read and categorized to get an overview of the field. 14 of these 23 articles were compiled and through the compilation it appears that the results of the study do not point to any straight answer. The result shows that it is possible to achieve development in several different areas of knowledge by working with different teaching methods and classroom activities, such as GeoGebra and physical manipulatives materials. By identifying which knowledge the students possess, the teacher can design teaching activities that favor the knowledge development of the students, so the students can advance from one Van Hiele level to the next. The results of the study and its various teaching methods and classroom activities can be seen as support for the teacher in his planning, and thus support the student's knowledge development. The study also shows that motivation can be seen as a contributing factor to knowledge development.

The purpose with this research paper is to examine the students’ shown knowledge in geometry, with a focus on construction and its concepts, and the educational value and teaching the students got in this area. The students’ homes are used as a starting-point. The students shall, from a self-made drawing of their home and a photograph of it, describe what their home looks like. In this paper, the mathematical concepts the students used will be analyzed and compared with the education they received. The analytical framework is based on Van Hieles levels of knowledge and Blooms Taxonomy. The study was done at a Secondary School in Kenya. Four students were selected and interviewed. The lesson observations were made with the purpose to get an understanding for how the education for these students look like and to get examples on how the teaching is conducted for these students. Finally, interviews with the teachers were carried out. The students show a good knowledge in the national exams. However, the study shows that when the students are supposed to use this particular knowledge outside of the classroom, the students experience difficulties. Mostly, the students encounter problems when they are supposed to estimate measurements. Furthermore, they lack the ability to compare scales. The research also shows that the education for these students is monotone and much time during the lessons is spend either with a teacher lecturing in front of the board or students working with examples in the textbook. According to the Variation Theory, the knowledge of the students should deepen if the objects of learning are varying. This variation is not something the students receive in the present situation. / Syftet är att undersöka några gymnasieelevers visade kunskaper i geometri med fokus på konstruktion och begreppsanvändning samt den undervisning som erbjuds eleverna inom området. Elevernas hem används som utgångspunkt. Eleverna ska utifrån en teckning, som de själva ritat, och ett fotografi beskriva hemmet. De matematiska begrepp som eleverna använder analyseras. Analysverktyget bygger på van Hieles kvalitativa kunskapsnivåer och Blooms Taxonomi. Undersökningen genomfördes på en gymnasieskola i Kenya. Fyra utvalda elever intervjuades. Lektionsobservationer genomfördes i syfte att få förståelse för hur elevernas undervisningssituation ser ut och få exempel på hur undervisningen bedrivs. Slutligen intervjuades två av elevernas lärare. Eleverna har goda kunskaper på nationella prov men undersökningen visar att när dessa kunskaper skall överföras till något utanför lektionssalen stöter eleverna på problem. De har svårt att uppskatta längdenheter och svårt att jämföra skala. Det kommer också fram att deras undervisning är ganska monoton. Mycket tid läggs till att läraren undervisar eleverna framme vid tavlan eller att eleverna jobbar med uppgifter i sin övningsbok. Enligt variationsteorin, som beskrivs i arbetet, skulle elevernas kunskaper ges möjlighet att fördjupas om de geometriska objekt som skall förstås varieras. Denna variation erbjuds inte eleverna i nuläget.

The aim of this study has been to analyze how, which, and to what extent two-dimensional geometric figures and tasks are presented in tasks in Swedish textbooks for pupils in elementary school, grades 1–3. The tasks were classified according to the Van Hiele theory, which is a model for how pupils learn geometry by developing knowledge through hierarchical levels 1–5. To reach the aim of the study, a textbook analysis was conducted with 18 textbooks from three different math book series for grades 1–3 in the Swedish elementary school: Matematik Eldorado, Favorit Matematik and Matte Direkt. The results showed that all the textbooks contained two-dimensional geometric figures and tasks that could be classified under one of the Van Hiele levels 1–3. However, the results also revealed big differences in how many, and at what level they were classified. None of the textbook series totally followed Van Hiele’s theory of hierarchical progression. The results from this study can increase the awareness of how Van Hiele’s theory is used in textbooks for Swedish pupils. However, more research is needed to get a clearer picture of how pupils meet Van Hiele’s  geometric levels in the classroom.

In this article, based on Dissection-Motion-Operations, DMO (decomposing a figure into several pieces and composing the resulting pieces into a new figure of equal area), a set of visual representations (models) of mathematical concepts will be introduced. The visual models are producible through manipulation and computer GSP/Cabri software. They are based on the van Hiele’s Levels (van Hiele, 1989) of Thought Development; in particular, Level 2 (Informal Deductive Reasoning) and level 3 (Deductive Reasoning). The basic theme for these models has been visual learning and understanding through manipulatives and computer representations of mathematical concepts vs. rote learning and memorization. The three geometric transformations or motions: Translation, Rotation, Reflection and their possible combinations were used; they are illustrated in several texts. As well, a set of three commonly used dissections or decompositions (Eves, 1972) of objects was utilized.

In this article, based on Dissection-Motion-Operations, DMO (decomposing a figure into several pieces and composing the resulting pieces into a new figure of equal area), a set of visual representations (models) of mathematical concepts will be introduced. The visual models are producible through manipulation and computer GSP/Cabri software. They are based on the van Hiele’s Levels (van Hiele, 1989) of Thought Development; in particular, Level 2 (Informal Deductive Reasoning) and level 3 (Deductive Reasoning). The basic theme for these models has been visual learning and understanding through manipulatives and computer representations of mathematical concepts vs. rote learning and memorization. The three geometric transformations or motions: Translation, Rotation, Reflection and their possible combinations were used; they are illustrated in several texts. As well, a set of three commonly used dissections or decompositions (Eves, 1972) of objects was utilized.

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