More than half a century ago, Piaget concluded from an investigation of children’s representational thinking about the nature of space that the development of children’s representational thought is topological before it is Euclidean. This conclusion, commonly referred to as the “topological primacy thesis,” has essentially been rejected. By giving emphasis to the ideas that develop rather than the order in which they develop, this work set out to develop a new form of non-metric geometry from young children’s early and intuitive topological, or at least non-metric, ideas. I conducted an eighteen-week teaching experiment with two children, ages six and seven. I developed a new dynamic geometry environment called Configure that I used in tandem with clinical interviews in each of the episodes of the experiment to elicit these children’s non-metric conceptions and subsequently support their development. I found that these children developed significant and authentic forms of geometric reasoning. It is these findings, which I refer to as qualitative geometry, that have implications for the teaching of geometry and for research into students’ mathematical reasoning. / text

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Primarily via constructivist teaching experiment methodology, units coordination (Steffe, 1992) has emerged as a useful construct for modeling students' psychological constructions pertaining to several mathematical domains, including counting sequences, whole number multiplicative conceptions, and fractions schemes. I describe how consideration of units coordination as a Piagetian (1970b) structure is useful for modeling units coordination across contexts. In this study, I extend teaching experiment methodology (Steffe and Thompson, 2000) to model the dynamics of students' units coordinating activity across contexts within a teaching experiment, using the construct of propensity to coordinate units. Two video-recorded teaching experiments involving pairs of sixth-grade students were analyzed to form a model of the dynamics of students' units coordinating activity. The modeling involved separation of transcriptions into chunks that were coded dichotomously for the units coordinating activity of a single student in each dyad. The two teaching experiments were used to form 5 conjectures about the output of the model that were then tested with a third teaching experiment. The results suggest that modeling units coordination activity via the construct of propensity to coordinate units was useful for describing patterns in the students' perturbations during the teaching sessions. The model was moderately useful for identifying sequences of interactions that support growth in units coordination. Extensions, modifications, and implications of the modeling approach are discussed. / Ph. D.

This Independent work is about testing a method to create meaningful learning and a deeper comprehension in matter of basic hairdressing techniques. I got the idea for this method by reading literature related to my education to become a teacher for hairdressers, the new schoolreform and from my own experience of a teaching method that I saw an opportunity to develope. My teaching method was based on first teaching the pupils the basic techniques, and then give them an assignment were they were to find out their own final result by using a picture for inspiration. They were given an assignment description with certain frames but with freedom of choice how to get to their own final result. By making a job description and a headdrawing to show how the work should be done, and during the process try different ways to achieve the best result, my idea was that a deeper understanding and a feeling of meaningfulness would appear. The result shows that the pupils by freedom within frames experienced meaningful learning and thereby deeper understanding when it comes to basic hairdressing techniques. They also developed skills to write, evaluate their own work and to try new paths for further learning. The conclusion is that this teaching method can be used to achieve the goals according to the be changed and added on to develop the method further, but it is a good foundation as a starting point. / Utvecklingsarbetet i form av ett undervisningsförsök handlade om att försöka skapa meningsfulla lärsituationer och ge djupare förståelse för frisörens grundläggande tekniker. Idén till undervisningsförsöket fick jag genom den valda litteraturen från lärarutbildningen, styrdokumenten samt egna erfarenheten av ett undervisningssätt som jag såg möjligheter att utveckla. Undervisningsförsöket gick ut på att ge eleverna grundteknikerna och utifrån dem ge dem i uppgift att själva hitta ett mål, med en inspirationsbild till hjälp. De fick en uppgiftsbeskrivning, men med frihet att själv hitta vägar att nå sitt mål. Genom att göra en arbetsbeskrivning och en huvudskiss på hur arbetet skulle gå till väga, och under processen prova sig fram till det bästa sättet att nå sitt mål, var tanken att djupare förståelse och en känsla av meningsfullhet skulle uppnås. Resultatet visar att eleverna upplevde att genom frihet inom ramar gav ett meningsfullt lärande och därmed större förståelse för frisörens grundläggande tekniker. De utvecklade även förmågan att skriva, utvärdera sitt eget arbete och pröva nya vägar för fortsatt lärande. Slutsatsen är att detta undervisningssätt kan användas för att uppnå målen i styrdokumenten för Hantverk och Hantverk-Frisör 1 (GY 11 Skolverket). Det finns saker som kan ändras och läggas till för att utveckla undervisningsförsöket ytterligare, men är en god grund att utgå ifrån.

A classroom teaching experiment was conducted to document prospective teachers' development of whole number concepts and operations. The purpose of this mixed-methods study was to examine the collective understanding of prospective teachers in an elementary mathematics content course. Design research methodology, specifically a classroom teaching experiment was the methodology selected for this study since it allows learning to be documented in a classroom environment and is iterative in nature. A revised hypothetical learning trajectory and instructional tasks from a previous classroom teaching experiment were used in this study (Andreasen, 2006). Research about children's development of whole number concepts and operations was used in developing instructional learning goals. In addition, research regarding prospective teachers' development supported the instructional modification that all tasks would be presented and expected to be reasoned about in base-8. Both qualitative data and quantitative data were collected. Qualitative data included whole class dialogue that was videotaped and transcribed, as well as student work samples. Quantitative data included items from the Content Knowledge for Teaching Mathematics database that were administered prior to and subsequent to the instructional sequence in base-8 (Hill, Schilling, & Ball, 2005). It should be noted that the items selected from the database were in base-10. The emergent perspective served as the interpretive framework of the collected qualitative data. This perspective reflexively coordinates the social or group perspective simultaneously with psychological or individual perspective. As stated, this study sought to describe the communal mathematics understanding of prospective teachers in an elementary mathematics content course. Toulmin's (1969) model of argumentation and Rasmussen and Stephan's three-phase methodology served to document normative ways of group reasoning called classroom mathematical practices. The following classroom mathematical practices were identified as taken-as-shared by prospective teachers: (a) developing small number relationships using Double 10-Frames, (b) developing two-digit thinking strategies using the open number line, (c) flexibly representing equivalent quantities using pictures or Inventory Forms, and (d) developing addition and subtraction strategies using pictures or an Inventory Form. Quantitative results indicated that prospective teachers were able to apply mathematical understandings grounded in base-8 to whole number concepts in base-10. In the end, counting and calculating in base-8 provides a meaningful context for prospective teachers to reconstruct their knowledge of whole number concepts and operations.

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Esta pesquisa aborda uma área da Matemática que trabalha a tomada de decisão. Trata da aplicação e análise de uma sequência de atividades para alunos do Ensino Médio de uma escola estadual da cidade de São Paulo com base em conceitos da Teoria dos Jogos. Tem como objetivos apresentar situações de dilema e jogos e promover a discussão de estratégias ótimas para solucioná-los, bem como apresentar aos alunos elementos desta teoria, tendo em vista a importância de desenvolver a habilidade de tomar decisões em situações de conflito. Design experiment foi o referencial metodológico escolhido. Inicialmente é apresentado um breve resumo da história da Teoria dos Jogos, destacando seus principais contribuidores, bem como uma explanação de seus conceitos e fundamentos, seguida de um resumo do modelo matemático. Na sequência, tem-se uma análise da relação entre o desenvolvimento da habilidade de tomar decisão em situações de conflito na formação escolar e os jogos dessa teoria; são citadas experiências de aplicações em sala de aula realizadas por alguns autores e conclusões obtidas. Em seguida, é feita uma apresentação da metodologia de pesquisa utilizada, justificando-se sua pertinência a este trabalho, bem como as ações de pesquisa de acordo com o embasamento metodológico. Em capítulos posteriores tem-se a descrição das atividades realizadas, comentários e análises. Da pesquisa pode-se concluir que foi possível mostrar aos alunos elementos da Teoria do Jogos, como aplicação da Matemática e que esta é ferramenta para modelar fenômenos que envolvem tomadas de decisão. Pode-se perceber que ocorreu a sensibilização dos alunos para situações de jogos de conflito e como a teoria lida com as mesmas. / This research addresses an area of mathematics that works on decision making. It deals with the application and analysis of a sequence of activities for high school students of a state school in the city of São Paulo based on concepts of Game Theory. It aims to present situations of dilemma and games and to promote the discussion of optimal strategies to solve them, as well as to present to the students elements of this theory, in view of the importance of developing the ability to make decisions in situations of conflict. The design experiment was chosen as methodological reference. Initially, a brief summary of the history of the Game Theory is presented, highlighting its main contributors, as well as an explanation of its concepts and fundamentals, followed by a summary of the mathematical model. In the sequence, there is an analysis of the relationship between the development of the ability to make decisions in situations of conflict in school formation and the games of this theory; experiences of classroom applications made by some authors and conclusions are cited. Then, a presentation of the methodology of research is made, justifying its relevance to this work, as well as the research actions according to the methodological basis. In later chapters there is a description of the activities carried out, comments and analysis. From the research it can be concluded that it was possible to show students the elements of the Game Theory as an application of Mathematics and that this is a tool to model phenomena involving decision making. It can be noticed that students have been sensitized to situations of conflict games and how the theory deals with them.

The aim of this study was to investigate the effects of using a dynamic geometry environment together with open-ended explorations on sixth grade students&rsquo / performance in polygons and congruency and similarity of polygons. Two groups of sixth grade students were selected for this study: (1) An experimental group composed of 66 students whom 34 were boys and 32 were girls / and (2) a control group composed of 68 students whom 35 were boys and 33 were girls. While the students in the control group received instruction via traditional methods, the students in the experimental group studied the same topics by open-ended explorations in a dynamic geometry environment. Geometry Test (GT) and Computer Attitude Scale (CAS) were used as data collection instruments. All students had taken the GT as pre-test, post-test, and delayed post test. However, CAS was administered only to the experimental group at the end of the instruction. Furthermore, some qualitative data were collected through video-taped classroom observations and interviews with selected students. Pre-test scores showed no statistical difference between control and experimental group students in terms of their performances in polygons and congruency and similarity of polygons before the study. On the other hand, the results of the post and delayed-post tests which are analyzed by independent t test showed that experimental group achieved significantly better than the control group students. In addition, a statistically significant correlation between CAS and GT was observed. Those results were also supported by the qualitative data. In conclusion, the results indicated that dynamic geometry environment together with open-ended explorations significantly improved students&rsquo / performances in polygons and congruency and similarity of polygons.

Central to this article is networking in math lessons, whereby concentration is placed on the construction of a student-focused teaching method for the networking of mathematical knowledge in the lower secondary. Firstly, normative standards and descriptive results will be compared. Secondly, several already existing teaching methods for networking in math lessons will be added to the method of „chapter-spanning task variation“. Using this method, attention is be placed on the integration of mathematical content and specific social netowrk-form (e.g. teacher led classes, group-work etc.). This paper will be concluded with the presentation of the testing of the method in the school context).