FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / nÃo hà / AvaliaÃÃes recentes atravÃs de testes como o SAEB (Sistema Nacional de AvaliaÃÃo da EducaÃÃo BÃsica) mostra que os alunos nÃo tÃm um desempenho em matemÃtica que seria esperado para a sÃrie que estÃo concluindo. Geralmente isso acontece devido ao ensino que à desprovido de significado e prioriza a manipulaÃÃo simbÃlica. Pesquisas em InformÃtica Educativa e EducaÃÃo MatemÃtica tÃm mostrado que ambientes computacionais ajudam no desenvolvimento de conceitos matemÃticos. Tais ambientes permitem aos alunos realizarem simulaÃÃes e construir novas formas de representaÃÃo mental. Alguns desses ambientes mais recentes sÃo chamados de objetos de aprendizagem. O presente estudo tem o objetivo de investigar como esses objetos podem contribuir no desenvolvimento de pensamento algÃbrico em alunos de sÃries iniciais, um aspecto novo, jà que geralmente os alunos comeÃam a estudar Ãlgebra somente no sÃtimo ano do ensino fundamental. A pesquisa foi realizada em uma escola pÃblica de Fortaleza com alunos do terceiro e quinto ano do Ensino Fundamental. Os alunos foram investigados atravÃs de uma entrevista clÃnica que analisa o desenvolvimento do raciocÃnio algÃbrico dos alunos durante as atividades em materiais elaborados para tais fins. O estudo concluiu que os alunos superam suas dificuldades iniciais sobre as atividades propostas e elaboram estratÃgias de resoluÃÃo que facilitam a compreensÃo de conceitos algÃbricos exigidos em sÃries mais avanÃadas. Os resultados do estudo devem contribuir na elaboraÃÃo de atividades para auxiliar no desenvolvimento do pensamento algÃbrico em alunos das sÃries iniciais. / Recent results from standardized tests such as SAEB (National Evaluation System of K-12 Education) shown that students are not developing mathematical concepts accordingly to the grade they are concluding. Generally, the reason is a teaching approach meaningless which emphasize only symbolic manipulation. Results from research in Mathematics Education and Instructional Technology shown that computer environments can help in the development of mathematical concepts. These environments allow for the students to conduct simulations and new ways of mental representation. Some of these current environments are called learning objects. The present study aims to investigate how these objects can contribute in the development of the algebraical thought in the early school years, a recent trend, since usually students are introduced to algebra in the 7th grade. The research was conducted in a public school in the city of Fortaleza, CE, Brazil with 3rd and 5th grade students. Students were interviewed while solving algebraical problems using a variety of manipulative, written problems and learning objects, specifically design for the study. The results shown that students can overcome conceptual obstacles about algebra during the proposed situations. They developed a series of solving problems strategies which can later facilitate the comprehension of more advanced algebraical concepts. The results should contribute to the design of activities for helping the development of algebraical concepts in the early school years.

Research in early algebra is critical because a smooth transition from arithmetic to algebra will influence future algebra learning that is central to school mathematics. This study investigated learners’ interpretation of letters in different levels of generalised arithmetic activities. Thirty grade nine learners from one inner city school participated in this study. All learners engaged with seventeen paper and pencil tasks encompassing six different interpretations of letters and six learners were then interviewed. Analysis of the data showed that the overall performance of learners was very poor and most learners have not been successful in making the transition from arithmetic to algebra. Learner responses suggested a strong arithmetical influence and a poor understanding of algebraic letter and basic manipulative skills. Throughout the data a number of misconceptions surfaced which suggested that most learners in this sample were lacking ‘symbol sense’.

Efterfrågan på matematiskt kunnande ökar i samhället. Algebraiskt kunnande gynnar lärandet inom andra delar av matematiken. Algebra innebär att upptäcka matematiska strukturer och kunna generalisera dessa. Mönsterundervisning ses som en säker väg till algebra. Samtidigt visar forskning kopplat till upprepade mönster att de flesta elever inte kan generalisera upprepade mönster innan de börjar skolan, och menar att det finns ett behov av en strukturerad undervisning kopplad till upprepade mönster. Stora delar av matematikundervisningen i Sverige utgår från matematikböcker. Därför syftar denna studie till att beskriva de upprepade mönster elever kan möta genom ett läromedel och därigenom dra slutsatser om vad elever får för möjlighet att lära om algebra genom dessa. Följande frågor besvaras: Vilken variation och omfattning av mönster och mönsteruppgifter får eleverna möta i matematikböckerna kopplade till upprepade mönster? och Vilken ytterligare vägledning ger lärarhandledningen för att elever ska utveckla förståelse av upprepade mönster?  Den teoretiska ansats studien utgår från är tidigare forskning om undervisning i växande mönster. Studien är avgränsad till matematikböcker för år 1–3. Urvalet av material innefattar 6 matematikböcker i serien Koll på matematik samt tillhörande lärarhandledningsböcker. Samtliga böcker har analyserats för att finna upprepade mönster. Dessa mönster sorterades efter kategorierna riktning, uttrycksformer, mönsterstruktur och uppgifts typ. Därefter genomsöktes lärarhandledningen efter förslag på undervisning kopplade till de upprepade mönstren som fanns i elevböckerna. Resultatet visar att det främst finns en variation på uttrycksformer och mönsterstrukturer i matematikböckerna och en mindre variation på riktning. Uppgiftstyperna varierar i viss mån. Vidare visar resultatet att lärarhandledningen tillförde flera kommunikativa aspekter kopplade till mönster och även fler metoder vilket inte framgick i elevböckerna. / The request for mathematical knowledge is increasing in today's society. Algebraic knowledge favour further learning in other areas of mathematics. Algebra involves finding mathematical structures and being able to generalize these. Teaching about patterns are described as a safe path to learn algebra. At the same time, research on repeating patterns proves that most children cannot make generalization about repeating patterns before they enter school, therefore structured teaching connected to repeated patterns is needed. Teaching in Sweden is mostly based on mathematics schoolbooks. Therefore, the aim for this study is to describe the repeated patterns students can encounter through teaching material and thereby draw conclusions about which opportunities students have to learn about algebra through these. The questions answered in this study are: Which variation and amount of patterns and pattern tasks do the pupils get exposed to in the mathematics books connected to repeating patterns? and What does the additional teacher guides provide with for pupils understanding of repeating patterns? The study is limited to mathematics schoolbooks for schoolyear 1-3.  The theoretical approach in this study is based on previous research on teaching methods connected to repeated patterns. The selection of mathematics books includes 6 mathematics books from the teaching materials Koll på matematik. All pages have been scanned to find repeated patterns. The repeating patterns were categorized by the pattern's direction, different expression of pattern, structure, and task type. Afterwards, the teacher's manual was searched for teaching suggestions linked to the repeating patterns that were found in the student books. The result indicates that the variation is primally connected to different expression of pattern and pattern’s structure in the mathematics books and that there are minimal variation of the patterns direction. The task types vary to a certain extent. Furthermore, the result indicate that the teacher's guide contributes with several communicative aspects connected to repeating patterns and more methods that were not apparent in the student books.

Syftet med denna licentiatuppsats är att undersöka aspekter i undervisningen som skapar förutsättningar för att elever i yngre åldrar (årskurs 2 och 3) ska engageras i ett algebraiskt arbete. Learning study har använts som metod (ansats) för att producera data. Ett forskarlag bestående av två grundskollärare i matematik och en lärarforskare har i  learning study-processen arbetat i en kolloborativ och intervenerande process.   I designen och analysen har Davydovs lärandeverksamhets teori, variationsteorin och Radfords definition av algebraiska mönstergeneraliseringar använts som teoretiska utgångspunkter. Det empiriska  datamaterialet består av (1) videoinspelade intervjuer med åtta elever samt transkriptioner av dessa; (2) videoinspelningar av tre forskningslektioner; (3) lektionsplaneringar; (4) synopsis av videoinspelade forskningslektioner; (5) transkriptioner av delar av forskningslektioner.   Resultatet består av tre identifierade kritiska aspekter som elever kan behöva urskilja för att kunna uttrycka och argumentera för en mönstergeneralisering algebraiskt: (a) att urskilja relationen mellan ett elements position och antalet komponenter; (b) att urskilja hur man kan använda relationen mellan ett elements position och antalet komponenter för att förutsäga ett godtyckligt element i mönstret; (c) att urskilja konstanten (den komponent som inte ändras utan är densamma i samtliga element) i mönstret.  Resultatet ger även exempel på vilka funktioner de lärandeverksamhetsteoretiska principerna (Davydov), problemsituation, lärandemodell och motsättningar, kan ha för att ett algebraiskt arbete ska etableras och upprätthållas. Vidare kan resultatet bidra till att fördjupa förståelsen gällande vad det innebär att kunna uttrycka och argumentera för mönstergeneraliseringar algebraiskt i yngre åldrar. Resultatet kan även bidra till kunskap som kan användas av lärare för att iscensätta och realisera en undervisning inom ramen för early algebra. / The purpose of this licentiate thesis is to study the aspects of the teaching that enable students of younger ages to be engaged in algebraic work. Learning study has been used as the method to produce data. A research team consisting of two primary school teachers in mathematics and a teacher researcher worked collaboratively, designing interventions iteratively during the learning study process.  In the design as well as analysis, Davydov's learning activity theory, Variation theory and Radford's definition of algebraic pattern generalizations have been used as theoretical starting points. The empirical data consists of (1) video-recorded interviews with eight students as well as transcriptions thereof; (2) video recordings of three research lessons; (3) lesson plans; (4) synopsis of video recordings of three research lessons; (5)  transcriptions of parts of video recorded research lessons. Results consists of three identified critical aspects that students may need to discern in order to express and justify for a pattern generalization algebraically: (a) to discern the relationship between the position of an element and the number of components; (b) to discern how to use the relationship between the position of an element and the number of components to predict an arbitrary element in the pattern; (c) to discern the constant (the component that does not change but is the same in all elements) in the pattern. Results give examples of what functions the theoretical principles of Davydov´s learning activity, problem situation, learning model and contradictions, may have for algebraic work to be established and maintained. Furthermore, the results may contribute to a deepened understanding of what it means to be able to express and justify for pattern generalizations algebraically at younger ages. The results may also contribute to knowledge that can be used by teachers to stage and carry out a teaching within the frame of early algebra.

In mathematics curricula teachers often find the more or less implicit request to link the taught subjects to the previous knowledge of the students, for example using word problems from everyday life. But in today’s multicultural and multisocial society teachers can no longer assume that the children they teach have a more or less equal background and thus everyday live can have a very different meaning for different children. Furthermore there is evidence that good previous knowledge in arithmetic can hinder the approach to other mathematical subjects, like algebra. In this paper I want to provide a brief overview on how previous knowledge in arithmetic can affect student\'s access to algebra and therefore present an early algebra teaching project which introduces elementary school children to algebraic notation by measurement in an action-oriented way. Thereby the chosen approach to algebra explicitly does not come back to the student\'s previous arithmetical knowledge but additionally may support non-proficient students in obtaining more insight in the structure of calculations and hence may help them to have more success in solving calculations and word problems.

FREIRE, Raquel Santiago. Objetos de aprendizagem para o desenvolvimento do pensamento algébrico no ensino fundamental. 2007. 132f. Dissertação (Mestrado em Educação) – Universidade Federal do Ceará, Faculdade de Educação, Programa de Pós-Graduação em Educação Brasileira, Fortaleza-CE, 2007. / Submitted by Maria Josineide Góis (josineide@ufc.br) on 2012-07-12T16:27:26Z No. of bitstreams: 1 2007_Dis_RSFREIRE.pdf: 941608 bytes, checksum: 57527231c1b3c4b4dc23b27630cfc744 (MD5) / Approved for entry into archive by Maria Josineide Góis(josineide@ufc.br) on 2012-07-13T10:51:51Z (GMT) No. of bitstreams: 1 2007_Dis_RSFREIRE.pdf: 941608 bytes, checksum: 57527231c1b3c4b4dc23b27630cfc744 (MD5) / Made available in DSpace on 2012-07-13T10:51:51Z (GMT). No. of bitstreams: 1 2007_Dis_RSFREIRE.pdf: 941608 bytes, checksum: 57527231c1b3c4b4dc23b27630cfc744 (MD5) Previous issue date: 2007 / Recent results from standardized tests such as SAEB (National Evaluation System of K-12 Education) shown that students are not developing mathematical concepts accordingly to the grade they are concluding. Generally, the reason is a teaching approach meaningless which emphasize only symbolic manipulation. Results from research in Mathematics Education and Instructional Technology shown that computer environments can help in the development of mathematical concepts. These environments allow for the students to conduct simulations and new ways of mental representation. Some of these current environments are called learning objects. The present study aims to investigate how these objects can contribute in the development of the algebraical thought in the early school years, a recent trend, since usually students are introduced to algebra in the 7th grade. The research was conducted in a public school in the city of Fortaleza, CE, Brazil with 3rd and 5th grade students. Students were interviewed while solving algebraical problems using a variety of manipulative, written problems and learning objects, specifically design for the study. The results shown that students can overcome conceptual obstacles about algebra during the proposed situations. They developed a series of solving problems strategies which can later facilitate the comprehension of more advanced algebraical concepts. The results should contribute to the design of activities for helping the development of algebraical concepts in the early school years. / Avaliações recentes através de testes como o SAEB (Sistema Nacional de Avaliação da Educação Básica) mostra que os alunos não têm um desempenho em matemática que seria esperado para a série que estão concluindo. Geralmente isso acontece devido ao ensino que é desprovido de significado e prioriza a manipulação simbólica. Pesquisas em Informática Educativa e Educação Matemática têm mostrado que ambientes computacionais ajudam no desenvolvimento de conceitos matemáticos. Tais ambientes permitem aos alunos realizarem simulações e construir novas formas de representação mental. Alguns desses ambientes mais recentes são chamados de objetos de aprendizagem. O presente estudo tem o objetivo de investigar como esses objetos podem contribuir no desenvolvimento de pensamento algébrico em alunos de séries iniciais, um aspecto novo, já que geralmente os alunos começam a estudar Álgebra somente no sétimo ano do ensino fundamental. A pesquisa foi realizada em uma escola pública de Fortaleza com alunos do terceiro e quinto ano do Ensino Fundamental. Os alunos foram investigados através de uma entrevista clínica que analisa o desenvolvimento do raciocínio algébrico dos alunos durante as atividades em materiais elaborados para tais fins. O estudo concluiu que os alunos superam suas dificuldades iniciais sobre as atividades propostas e elaboram estratégias de resolução que facilitam a compreensão de conceitos algébricos exigidos em séries mais avançadas. Os resultados do estudo devem contribuir na elaboração de atividades para auxiliar no desenvolvimento do pensamento algébrico em alunos das séries iniciais.

In mathematics curricula teachers often find the more or less implicit request to link the taught subjects to the previous knowledge of the students, for example using word problems from everyday life. But in today’s multicultural and multisocial society teachers can no longer assume that the children they teach have a more or less equal background and thus everyday live can have a very different meaning for different children. Furthermore there is evidence that good previous knowledge in arithmetic can hinder the approach to other mathematical subjects, like algebra. In this paper I want to provide a brief overview on how previous knowledge in arithmetic can affect student\''s access to algebra and therefore present an early algebra teaching project which introduces elementary school children to algebraic notation by measurement in an action-oriented way. Thereby the chosen approach to algebra explicitly does not come back to the student\''s previous arithmetical knowledge but additionally may support non-proficient students in obtaining more insight in the structure of calculations and hence may help them to have more success in solving calculations and word problems.