In our presentation we will show how technology enables us to improve the teaching and learning of linear functions at the middle school level. Through various classroom activities that involve technology such as dynamic geometry software, graphing calculators and Excel, students explore functions and discover basic facts about them on their own. Students then work with real life data and on real life problems to draw graphs and form linear models that correspond to given situations as well as draw inferences based on their models. Participants will receive complete classroom materials for the unit on linear functions.

In our presentation we will show how technology enables us to improve the teaching and learning of linear functions at the middle school level. Through various classroom activities that involve technology such as dynamic geometry software, graphing calculators and Excel, students explore functions and discover basic facts about them on their own. Students then work with real life data and on real life problems to draw graphs and form linear models that correspond to given situations as well as draw inferences based on their models. Participants will receive complete classroom materials for the unit on linear functions.

O presente trabalho tem como objetivo estudar se atividades de investigação matemática podem ajudar a desenvolver a capacidade de argumentação e demonstração matemática nos alunos do Ensino Médio, abordando o tema do crescimento e decrescimento de funções afins. Para isso, propõe uma reflexão sobre o papel da argumentação e da demonstração na formação integral do aluno do ensino médio. Enfoca, em particular, a transição entre o ensino básico e o superior, estudando algumas de suas dificuldades. O trabalho explora a diferença entre esses níveis de ensino, considerando que, enquanto a escola básica trata a matemática baseada em procedimentos aritméticos e algébricos, do ponto de vista prático tais como contas, medições, equações, análise de dados , o ensino superior exige mais abstração por parte do aluno onde a argumentação, o raciocínio lógico (dedutivo e indutivo) e as demonstrações são condições necessárias para a produção do conhecimento. Ao final, faremos uma proposta de atividade matemática através de uma abordagem investigativa, refletindo sobre como a demonstração, abordada dessa forma, pode contribuir para a formação integral do aluno e criar aproximações entre a forma como a matemática é tratada na escola básica e no ensino superior. / This present work aims to study if mathematical research activities can help to develop mathematical argumentation and demonstration capacity in high school students, addressing the theme of growth and decrease of linear functions. For this, it proposes a reflection about the role of argumentation and demonstration in the integral formation of the high school student. It focuses on the transition between basic and higher education, studying some of its difficulties. The study explores the difference between these levels of education, whereas, while the basic school treats mathematics based on arithmetic and algebraic procedures from a practical point of view - such as arithmetic, measurements, equations, data analysis -, higher education demands from the student a greater abstraction level, where argumentation, logical reasoning (deductive and inductive) and demonstrations are necessary conditions to knowledge construction. In the end, we will propose a mathematical activity through an investigative approach, reflecting on how the demonstration, addressed in this way, can contribute to the integral formation of the student and create approximations between the way mathematics is treated in elementary school and in higher education.

The aim of the diploma theses is to find out if the study in a form of an e-learning course is more beneficial to the student than the isolated study text. To find the benefits, the research among nineth grade students from two different schools was done. A study text called Functions for the nineth grade was prepared together with a collection of solved and unsolved problems. Based on this text an e-learning course called Functions for the nineth grade was prepared. The e-learning course was placed on the website linked http://funkcepro9r.maweb.eu/. For the communication of the students and the tutor of the course, for the completion of the compulsory tasks and for the carrying out the study agenda, the modul iTřída on the web link itrida.dumy.cz was selected. Both forms of the study materials were tested by the nineth grade students on two selected schools. The efectivity of both of the forms was tested by the written test. The evaluation of the both forms was done by the electronic questionnaire. According to the results of the written test, the students which used the isolated study text were more succesful than the students studying the e-learning course. The results of the questionnaire verified that the study text was more acceptable for the students. The e-learning course did not suit the...

Resumo: O objetivo do presente trabalho foi estimar conjuntamente os parâmetros das curvas de crescimento de animais da raça Nelore, seus componentes de (co)variâncias e os efeitos genéticos e ambientais que atuaram sobre eles. As funções de Brody, Von Bertalanffy, Gompertz e Logística foram empregadas no primeiro estágio de um modelo hierárquico Bayesiano. Os efeitos genéticos e ambientais foram considerados em um modelo animal no segundo estágio de hierarquia. Diferentes abordagens para a variância do erro de ajuste foram avaliadas: constância ao longo da trajetória, aumento linear até os três anos de idade e aumento exponencial. Amostras aleatórias das distribuições marginais foram obtidas aplicando-se os algoritmos de Metropolis-Hastings e amostragem de Gibbs. A presença de animais que não atingiram a maturidade no conjunto de dados não prejudicou a predição dos pesos adultos. Grande parte da variância fenotípica observada neste peso foi devida a efeitos genéticos aditivos. O parâmetro a das curvas de Brody, Von Bertalanffy e Gompertz poderia ser utilizado como critério de seleção para controlar o aumento de peso adulto. O ambiente materno influenciou não somente o crescimento inicial dos animais mas também os pesos maduros e deve ser considerado na avaliação de todas as etapas do crescimento. Os modelos linear e exponencial empregados para a variância do erro de ajuste não representaram de forma adequada este parâmetro no início da curva. A seleção para alterar a pendente da curva de crescimento mantendo o peso adulto constante seria ineficiente, uma vez que, é alta e positiva a correlação genética entre o peso assintótico e a taxa de maturação. / Abstract: The objective of this work was to estimate the joint posterior distribution of Nelore growth curve parameters, their (co)variance components and the environmental and additive genetic components affecting them. The Brody, Von Bertalanffy, Gompertz and Logistic functions were applied in the first stage of a hierarchical Bayesian model. The environmental and genetic effects were described by an animal model in the second stage. Different approaches for describing the adjustment error variance along the growth curve were evaluated: constancy throughout the trajectory, linear increasing until three years of age and exponential increasing. Random samples of the marginal distributions were drawn using Metropolis-Hastings and Gibbs sampling algorithms. Even thought the curve parameters were estimated for animals with records just from the beginning of the growth process, the adult weights were accurately predicted. A high additive genetic variance for mature weight was observed. The parameter a of Brody, Von Bertalanffy and Gompertz models could be used as a selection criterion to control adult weight increases. The effect of maternal environment on growth was carried through to maturity and it should be considered while evaluating all weights. The adjustment error variances at the beginning of growth curve were not adequately described by the linear and exponential models. Selection to change the growth curve slope without modifying adult weight would be inefficient, since their genetic correlation is high. / Orientadora: Lúcia Galvão de Albuquerque / Coorientador: Henrique Nunes de Oliveira / Banca: Joanir Pereira Eler / Banca: Paulo Sávio Lopes / Banca: Humberto Tonhati / Banca: Maurício Mello de Alencar / Doutor

O presente trabalho tem como objetivo estudar se atividades de investigação matemática podem ajudar a desenvolver a capacidade de argumentação e demonstração matemática nos alunos do Ensino Médio, abordando o tema do crescimento e decrescimento de funções afins. Para isso, propõe uma reflexão sobre o papel da argumentação e da demonstração na formação integral do aluno do ensino médio. Enfoca, em particular, a transição entre o ensino básico e o superior, estudando algumas de suas dificuldades. O trabalho explora a diferença entre esses níveis de ensino, considerando que, enquanto a escola básica trata a matemática baseada em procedimentos aritméticos e algébricos, do ponto de vista prático tais como contas, medições, equações, análise de dados , o ensino superior exige mais abstração por parte do aluno onde a argumentação, o raciocínio lógico (dedutivo e indutivo) e as demonstrações são condições necessárias para a produção do conhecimento. Ao final, faremos uma proposta de atividade matemática através de uma abordagem investigativa, refletindo sobre como a demonstração, abordada dessa forma, pode contribuir para a formação integral do aluno e criar aproximações entre a forma como a matemática é tratada na escola básica e no ensino superior. / This present work aims to study if mathematical research activities can help to develop mathematical argumentation and demonstration capacity in high school students, addressing the theme of growth and decrease of linear functions. For this, it proposes a reflection about the role of argumentation and demonstration in the integral formation of the high school student. It focuses on the transition between basic and higher education, studying some of its difficulties. The study explores the difference between these levels of education, whereas, while the basic school treats mathematics based on arithmetic and algebraic procedures from a practical point of view - such as arithmetic, measurements, equations, data analysis -, higher education demands from the student a greater abstraction level, where argumentation, logical reasoning (deductive and inductive) and demonstrations are necessary conditions to knowledge construction. In the end, we will propose a mathematical activity through an investigative approach, reflecting on how the demonstration, addressed in this way, can contribute to the integral formation of the student and create approximations between the way mathematics is treated in elementary school and in higher education.

Seit mehr als einem Jahr wird der Alltag unzähliger Menschen auf der ganzen Welt maßgeblich durch das Coronavirus Sars-CoV-2 bestimmt. Einzelhandelsgeschäfte, Restaurants, Hotels und kulturelle Einrichtungen unterliegen seit Beginn der Pandemie erheblichen Einschränkungen, die zuletzt zu mehrmonatigen Schließungen der genannten Dienstleistungsbetriebe führten. Es wurden Ausgangssperren und Kontaktbeschränkungen verhängt, die regional noch immer bestehen, sodass das öffentliche und private Leben in Deutschland auf ein Minimum heruntergefahren wurde. Auch die Schulen sind von den Maßnahmen zur Eindämmung der Pandemie betroffen, wodurch sich der Tagesablauf der Kinder grundlegend veränderte. [...] Zum einen erfordert ein gelungener Fernunterricht ein durchdachtes Konzept, welches die Abwesenheit der Lehrperson im Lernprozess zumindest annähernd kompensiert. Denn häufig reicht die Präsentation von Tafelbildern und das Erteilen standardisierter Rechenaufgaben nicht aus, um mathematische Begriffe in ihrer gesamten Vielfalt zu vermitteln. Es bedarf geeigneter Angebote, welche den Lernenden einen angemessenen Zugang zur Mathematik ermöglichen sowie Kompetenzen nachhaltig aufbauen. Andererseits erfordern die Aufbereitung und Umsetzung dieser innovativen Unterrichtsideen ein gewisses Maß an digitalem Know-how. Gleichzeitig wird den Lehrenden durch die notwendige Neuausrichtung ihrer Unterrichtsmaterialien ein zeitlicher Mehraufwand abverlangt, der im normalen Schulalltag kaum zu bewältigen ist. An dieser Stelle setzt die vorliegende wissenschaftliche Arbeit an und untersucht, wie Lehr-Lern-Arrangements in Zeiten ausgeprägter Selbstlernphasen erfolgreich aufbereitet werden können. Dazu wurde in Zusammenarbeit mit der Carl-von-Ossietzky-Schule in Berlin eine Materialreihe zum Thema „Lineare Funktionen“ entwickelt, welche die aktuelle Situation aufgreift und die Bedürfnisse der Schüler*innen berücksichtigt. Der ausgeprägten Heterogenität der Kinder wird durch binnendifferenzierte und klar strukturierte Arbeitsmaterialien Rechnung getragen, die einen individualisierten Aneignungsprozess ermöglichen. Ferner wird das Potential digitaler Medien genutzt, um die Lernenden bei der Auseinandersetzung mit funktionalen Zusammenhängen gezielt zu unterstützen und so pandemiebedingten Lernverlusten bestmöglich vorzubeugen. [Aus der Einleitung.]

Inom den svenska matematikundervisningen på gymnasiet möter eleverna många olika matematiska uppgifter. Uppgifterna skiljer sig i att de testar olika förmågor men även hur uppgifterna är formulerade skiljer sig. Det kan bland annat röra sig om uppgifter som är textbaserade, grafiskt utformade eller som har en algebraisk representationsform. Denna studie avser att undersöka om representationsformen på matematiska uppgifter kan ha någon påverkan på i vilken utsträckning elever kan lösa uppgifterna samt redogöra för vilka de vanligaste misstagen kan vara inom de olika representationsformerna. Studien syftar även till att se om det är någon skillnad på svarsfrekvensen beroende på om uppgifterna testar elevernas förmåga att genomföra beräkningar (procedurell kunskap) eller förmågan att uppfatta begrepp och principer (konceptuell kunskap). Teorin som används vid framtagandet av uppgifter är Hallidayan-modellen om olika sätt att presentera matematik samt principen om procedurell- och konceptuell kunskap. Metoden som används inom studien är insamling av elevlösningar på tre olika prov som tar sin grund i var sin av de olika representationsformerna: textbaserat, grafiskt och algebraiskt samt att alla tre innehåller uppgifter som testar deras procedurella samt konceptuella kunskap. Resultatet visar att representationsformen på uppgifterna har betydelse för i vilken utsträckning eleverna kan lösa dem och att eleverna har speciellt svårt för grafiskt formulerade uppgifter. Resultatet visar även att eleverna är bättre på att genomföra beräkningar än att förstå matematiska principer. I diskussionen presenteras olika tankar och idéer till hur det kan komma sig att resultatet ser ut som det gör samt vad resultatet kan ha för påverkan på matematikundervisningen framöver. / In the Swedish mathematical education on upper secondary school level, the students face many different mathematical tasks. The tasks are being separated by testing different abilities and in how they are designed. They could differ in how they are presented, and they could for example be text based, graphical and algebraic. These are three different ways of form of representation. This study intends to examine if the form of representation could have an impact on to which extent the student can solve the tasks and elucidate the most common mistakes within the different form of representation. The study also aims to determine if there are any difference in the frequency of the response depending on if the task assess student’s ability to perform calculations (procedural knowledge) or the ability to recognize concepts and principles (conceptual knowledge). The theory used in developing the tasks is the Halliday’s model of different ways to present mathematics and the principles of procedural and conceptual knowledge. The method that is used in this study is collection of student’s answer in three different tests, each based on one of the three forms of representation: text based, graphical and algebraic. Additionally, all three tests contain two tasks which will test the students procedural and conceptual knowledge. The results show that the form of representation have an impact on the extent to which students can solve the tasks and that students particularly struggle with graphically formulated tasks. The results also reveal that students are better at performing calculations than understanding mathematical principles. The discussion presents various thoughts and ideas on why the results appear as they do and what impact the results may have on mathematical education in the future.

Made available in DSpace on 2016-07-27T13:54:01Z (GMT). No. of bitstreams: 1 Adelino Candido Pimenta.pdf: 652136 bytes, checksum: c0e6315dc38b466836b23f3ed67900be (MD5) Previous issue date: 2001-12-07 / This paper presents a new approach in the study of linear functions. In addition, a brief analysis is made of the most relevant aspects of the history of the theme and its articulation with informatics. Initially, a survey was carried out in Goiânia's principal secondary schools in order to identify the textbooks they use. At this stage, a detailed examination of the predominant concepts in these books was undertaken. An attempt was made to give the proposal a theoretical foundation, maintaining, from start to finhish, a dialogue with the ideas of the principal researchers involved with the chosen theme. With regard to the historical and epistemological aspects, these were based on works orientated by Almounloud of the Pontifícia Universidade Católica de São Paulo, while in the context of informatics there was a constant attentive dialogue with the works of Borba and his disciples and other investigators in the field. with regard to the text, the principal objective of the research, this was based on the publications of the orientator Oliveira Filho. Finally, by using the Linear Web Aplett software, a proposal, which dynamically directs the study of linear functions, was drawn up, while at the same time new concepts were added to those already consolidated. / Esta trabalho apresenta uma nova abordagem noestudo de funções lineares. Analisa, também, os aspectos mais relevantes da história do tema e suas articulações com a informática. Inicialmente, procedeu-se a um levantamento nas principais escolas de ensino médio de Goiânia para identificar os livros didáticos. Nessa etapa, foi realizada uma identificação dos conceitos predominantes no livros. Procurou-se fundamentar teoricamente essa proposta mantendo diálogo, do início ao fim, com as idéias dos principais pesquisadores que se preocupam com a temática eleita. No que diz respeito ao aspecto histórico e epistemólogico, este trabalho apoiou-se especialmente nas análises de Almouloud, da Pontifícia Universidade Católica de São Paulo, ao passo que na área de informática a interlocução deu-se, permanente e atentamente, com a produção de Borba e seus discípulos e outros investigadores. O texto, objeto principal desta pesquisa, baseia-se nas publicações de Oliveira Filho. Finalmente, meidante a utilização do software Linear Web Apllet, elaborou-se uma proposta que norteia o estudo de funções lineares de forma dinâmica e iterativa, ao mesmo tempo que agrega novos conceitos

abstract: The focus of the study was to identify secondary school students' difficulties with aspects of linearity and linear functions, and to assess their teachers' understanding of the nature of the difficulties experienced by their students. A cross-sectional study with 1561 Grades 8-10 students enrolled in mathematics courses from Pre-Algebra to Algebra II, and their 26 mathematics teachers was employed. All participants completed the Mini-Diagnostic Test (MDT) on aspects of linearity and linear functions, ranked the MDT problems by perceived difficulty, and commented on the nature of the difficulties. Interviews were conducted with 40 students and 20 teachers. A cluster analysis revealed the existence of two groups of students, Group 0 enrolled in courses below or at their grade level, and Group 1 enrolled in courses above their grade level. A factor analysis confirmed the importance of slope and the Cartesian connection for student understanding of linearity and linear functions. There was little variation in student performance on the MDT across grades. Student performance on the MDT increased with more advanced courses, mainly due to Group 1 student performance. The most difficult problems were those requiring identification of slope from the graph of a line. That difficulty persisted across grades, mathematics courses, and performance groups (Group 0, and 1). A comparison of student ranking of MDT problems by difficulty and their performance on the MDT, showed that students correctly identified the problems with the highest MDT mean scores as being least difficult for them. Only Group 1 students could identify some of the problems with lower MDT mean scores as being difficult. Teachers did not identify MDT problems that posed the greatest difficulty for their students. Student interviews confirmed difficulties with slope and the Cartesian connection. Teachers' descriptions of problem difficulty identified factors such as lack of familiarity with problem content or context, problem format and length. Teachers did not identify student difficulties with slope in a geometric context. / Dissertation/Thesis / Ph.D. Curriculum and Instruction 2011