­­­­­­­­­­I flera decennier har likhetstecknets betydande roll i matematikämnet studerats av en rad olika forskare. Där det tydligt framgår att den operativa uppfattningen av likhetstecknet dominerar i klassrummen, därav syftar vår studie till att undersöka hur elevers uppfattning och förståelse av likhetstecknets relationella funktion förändras om olika delar av undervisningsinnehållet varieras. Syftet kommer besvaras genom frågeställningarna; Hur visar sig elevernas förståelse för likhetstecknets funktion i förtestet, hur påverkar variationen av undervisningsinnehållet elevernas förståelse av likhetstecknet och vilken uppfattning av likhetstecknet visas av eleverna i eftertestet efter en variation av undervisningsinnehållet. Studien har sin utgångspunkt i variationsteorin som fokuserar på vad som ska läras och hur lärandet kan möjliggöras genom teorins centrala begrepp, kritiska aspekter och variationsmönster. För att uppfylla studiens syfte och besvara frågeställningarna kommer designbaserad forskning användas som forskningsmetod, i form av en lektionsserie på tre lektioner som baserats på ett förtest.  Resultatet visar att variation av undervisningsinnehållet kan med hjälp av variationsmönster; kontrastering, generalisering och fusion, synliggöra och urskilja de kritiska aspekterna för att kunna utveckla elevernas förståelse av likhetstecknet. / For several decades, the significant role of the equals sign in the subject of mathematics has been studied by several different researchers, where the operative conception of the equal sign dominates in the classroom. Therefore, our study aims to investigate how students' conception and understanding of the equal sign relational function changes if different parts of the teaching content are varied. In accordance with this aim, the following research questions are addressed in the present study; What is the students' understanding of the function of the equal sign in the pre-test, how does the variation of the teaching content affect the students' understanding of the equal sign and what perception of the equal sign is shown by the students in the post-test after a variation of the teaching content. The study has its starting point in variation theory, which focuses on what is to be learned and how learning can be made possible through the theory's central concepts, critical aspects, and patterns of variation. To fulfil the purpose of the study and answer the questions, design-based research will be used as a research method, in the form of a series of lessons (3) based on a pre-test.  The results show variation of the teaching content can be by means of patterns of variation; contrasting, generalizing, and fusion, making visible and distinguishing the critical aspects to develop the students' understanding of the equal sign.

Från och med höstterminen 2018 är programmering en del av kursplanen i matematik, och har då placerats under det centrala innehållet för algebra. Trots att det inte är obligatoriskt förrän höstterminen har lärare redan börjat undervisa i programmering, och appar är då en populär lärresurs. Hur väl stämmer apparna för programmering överens med kursplanen i matematik, och hur väl lämpar de sig för undervisning på lågstadiet? Med utgångspunkt i dessa frågor gjordes en analys av 14 programmeringsappar som riktar sig till lågstadieåldrarna, 6-10 år. Studien är en kvalitativ innehållsanalys med en analysmodell som baserats på kursplanen i matematik, teorier om pre-algebra samt kriterier som lärare anser viktiga för en bra app. Studien visar att apparnas lämplighet för matematikundervisning varierar, men att det är de appar som är utformade som spel som stämmer bäst överens med kursplanen i matematik. Det är också dessa appar som har en pedagogisk utformning som lämpar sig bäst för undervisning på lågstadiet. Resultatet av studien ifrågasätter dock valet att placera programmering under algebra i läroplanen, då kopplingen däremellan inte är helt tydlig. / Starting in the fall of 2018, programming is a part of the Swedish curriculum for mathematics and has been placed under the core content for algebra. Even though it is not obligatory until the fall, some teachers have already started teaching about programming using apps as a digital learning resource. How well do the programming apps fit with the curriculum for mathematics, and how suitable are they to be used in primary school education? With these questions in mind an analysis was made of 14 programming apps for children 6-10 years old. The study is a qualitative content analysis with an analysis model based on the curriculum for mathematics, theories about pre-algebra and criteria that teachers believe to be important in the choice of a good app. The study shows that the apps’ suitability for mathematics education vary, but it is the apps that are designed as games that fit best with the curriculum for mathematics. These apps also have the pedagogical design that suits best for primary school education. The study’s results question the choice of placing programming under algebra in the curriculum, since the connection between the areas is not quite clear.

This study collected evidence to address the assumptions underlying the use of the Utah Core Assessment to Pre-Algebra (UCAP) to (a) measure student achievement in pre-algebra, and (b) assist teachers in making adjustments to instruction. An evaluative argument was defined to guide the collection of evidence. Each of the assumptions in the evaluative argument was addressed using data from a suburban northern Utah school district. To collect the evidence, test content was examined including item match to course objectives, reliability, and subtest intercorrelations. Analyses of correlations of the UCAP with convergent and discriminant measures were completed using student test data (N = 1,461), including an examination of both the pattern of correlations and tests of statistical significance. Pre-algebra teachers (N = 12) were interviewed to ascertain the degree to which UCAP results were used to make necessary adjustments to instruction. It was found that the UCAP was technically sound, but measured only 65% of course objectives. Correlation coefficients were analyzed using pattern comparisons and tests of statistical significance. It was found that the pattern of correlation coefficients and the distinction of convergent and discriminant measures supported the UCAP as a measure of mathematics. Teacher interview data revealed that teachers did not make substantive adjustments to the instruction of pre-algebra based on test scores. Based on these results it was concluded that the underlying assumptions concerning the use of the UCAP were not fully supported. The lack of complete coverage of the pre-algebra course objectives calls into question the ability of the UCAP scores to be used as measures of student achievement, in spite of the technical quality of the test. There was support for the assumption that the UCAP measures mathematics. There was little evidence that teachers use the UCAP score reports to make meaningful and appropriate adjustments to instruction. More evidence is needed to understand the factors that may have led to this lack of use. The evaluative argument framework defined in this study provides guidance for future research to collect evidence of the validity of decisions based on UCAP scores.

Detta är en empirisk studie som har syftat till att synliggöra vilka strategier elever i årskurs två har använt när de löst öppna utsagor samt vilka missuppfattningar som förekommit i samband med detta. Studien har även syftat till att synliggöra hur eleverna har kunnat relatera öppna utsagor till vardagliga sammanhang. Vid datainsamlingen, som utfördes på två olika skolor, genomfördes 15 kvalitativa intervjuer vilka kombinerades med observationer. Eftersom det i uppsatsens syfte finns en underförstådd relation till fenomenografin och hermeneutiken i form av såväl teorier som metodologier har dessa forskningsansatser varit en stor inspirationskälla i vår undersökning. Vidare har vi genom vår studie visat att de strategier som elever använder när de löser slutna utsagor till stora delar även används när eleverna löser uppgifter i form av öppna utsagor. Resultatet har visat på en bred variation i elevernas lösningsstrategier och det framgår att eleverna, vid deras lösningar av öppna subtraktionsutsagor, knappt använder de tre vanligaste subtraktionsstrategierna. Istället använder de bland annat en strategi som ger upphov till flera missuppfattningar. Det visade sig även att samtliga elever hade svårt att relatera öppna utsagor till vardagliga sammanhang.

Det här är en läromedelsgranskning som har syftet att granska hur tre olika matematikläromedel i årskurs 1 behandlar likhetstecknets innebörd och dess koppling till pre-algebra. Tidigare forskning visar att många elever har svårigheter inom just detta område. En djup och relationell förståelse för likhetstecknet är av stor vikt för att kunna utveckla ett algebraiskt tänkande och för att klara av högre studier inom matematik. För att nå denna typ av förståelse lyfter forskning flera olika undervisningsstrategier.  För att besvara studiens frågeställningar kring hur läromedel behandlar dessa undervisningsstrategier och pre-algebraiska uppgifter används ett analysverktyg, framtaget av oss studenter, vid granskningen. Varje läromedel undersöktes i relation till utvalda frågor som byggde på studiens teoretiska ramverk kring hur förståelsen för likhetstecknet påverkas av undervisning. Resultatet visade att alla tre läromedel innehöll uppgifter som stödjer en djup och relationell förståelse för likhetstecknet, men i olika utsträckningar och på olika sätt. Samtliga läromedel visade både likheter och skillnader till tidigare forsknings rekommenderade undervisningstrategier. Med denna insikt kan studien vara en hjälpande hand för lärare vid val av läromedel och vid undervisning av likhetstecknet.

Syftet med vår undersökning var att utvärdera om utomhusmatematik påverkade elevers förståelse för algebra samt hur denna inlärningsmiljö kunde påverka gruppdynamiken. Vi genomförde två undervisningsförsök, ett utomhus och ett inomhus, med elever i skolår 4 och observationer av dessa. Dessutom använde vi oss av gruppsamtal före och kvalitativa intervjuer efter undervisningsförsöken. Resultatet visade att eleverna fick en ökad förståelse för algebraiska likheter vid undervisning utomhus. Resultatet visade även att samarbetet ökade mellan eleverna, det blev ett lugnare klimat i gruppen och eleverna blev mer engagerade under utomhusundervisningsförsöket.

The purpose of this study was to improve the mean scores on the COMPASS college placement test after guiding students through a PowerPoint presentation referred to as COMPASS orientation. Scores obtained from college placement tests indicated a large number of entering college students were not prepared for college-level classes. During the 2006–2007 academic year, 1,427 COMPASS placement tests were administered to students enrolling in the Big Sandy Community and Technical College (BSCTC). Of that number, over 70% produced scores at or below the level requiring developmental education. Data showed that only 28% of the students taking the COMPASS placement tests required no developmental course work and could enter college-level general education classes and receive college credit. This study proposed that a short orientation immediately preceding the COMPASS placement test would increase COMPASS placement test scores. Findings from the study showed that a 5-minute pre-test orientation did not significantly improve the mean test scores in reading, writing, or pre-algebra on the COMPASS placement tests. The study showed that algebra scores had a significantly improved mean score on the exam after intervention with the COMPASS orientation tutorial PowerPoint presentation. Recommendations from this study suggest that orientations and workshops should be mandatory for all college applicants. Online web sources should contain materials and web links for COMPASS study questions, COMPASS Web site addresses, and college pre-test workshops and orientation dates. Stakeholders such as local advisory committees, community businesses, and high schools should be involved with college application requirements. How underprepared college students impact the available workforce pool and the local economy should be discussed, and these businesses should be enlisted to suggest avenues for improving testing outcomes and college success.

The teaching and learning of algebraic linear graphs in the school have experience problems with regards to understanding this topic. The learner's inability to understand algebraic linear graphs as a result of scaling was evidence in this study. Learners experience difficulties in interpreting, constructing and predicting from linear graphs without the knowledge of scaling. Four tasks were considered necessary for constructing linear graphs. This research focused on scaling task. The objective of the research is to establish the fact that scaling has influence on the understanding of algebraic linear graphs. An empirical method of research was employed to carry out this research. The results proved that scaling has influence on the understanding of algebraic linear graphs at grade 9 (Form B). Learners will find it easy to construct, interpret and make prediction from a graph drawn by scaling. / Mathematical Sciences / M.A. (Mathematics Education)

Made available in DSpace on 2016-04-27T16:58:19Z (GMT). No. of bitstreams: 1 joao.pdf: 7791795 bytes, checksum: 6083730ab3e399494c03a8cffdf14609 (MD5) Previous issue date: 2002-05-22 / This research is mainly concerned with the convenience and viability of introducing nine-year-old students to the study of the integers and of (pre-)Algebra. We begin with the integers and use their additive structure to model and solve additive problems, with special attention to the problems proposed by Vergnaud in 1976. The point here is to show that students can more easily solve problems, if we use the additive structure of the integers and some (very little) amount of algebraic manipulation, than by considering the addition and subtraction of natural numbers. The theoretical background is provided by some ideas of Raymond Duval related to the role that representations play in the understanding of mathematics. The aim is to enable the students to transform intentional treatments upon semiotic representations into quasiinstantaneous treatments. A consistent and sistematic effort is made to achieve this goal. The main emphasis is placed in the sum of integers. We performed a series of activities with thirty-eight students (average age 9) of a private elementary school in the city of São Paulo. We arrived at satisfactory results / O tema central deste trabalho é o estudo da possibilidade e da conveniência de ensinar estudantes de nove anos a trabalhar com números inteiros e com noções de (pré-)Álgebra. Começamos com o estudo dos inteiros e usamos sua estrutura aditiva para modelar e resolver problemas aditivos, dando especial atenção aos que foram propostos por Vergnaud em 1976. Aqui, a principal preocupação é mostrar que os estudantes podem resolver problemas de maneira mais fácil se usarmos os inteiros e uma pequena dose de manipulação algébrica em vez de utilizar a adição e a subtração dos naturais. Do ponto de vista teórico, apoiamo-nos em algumas idéias de Raymond Duval relacionadas ao papel que as representações desempenham na compreensão da Matemática. A meta é tornar os estudantes capazes de transformar tratamentos intencionais de representações semióticas em tratamentos quase-instantâneos. Um esforço sistemático e consistente foi feito nessa direção. A principal ênfase foi posta na adição de inteiros. As atividades foram desenvolvidas com 38 crianças de nove anos de uma escola particular da cidade de São Paulo. Os resultados obtidos foram satisfatórios

The teaching and learning of algebraic linear graphs in the school have experience problems with regards to understanding this topic. The learner's inability to understand algebraic linear graphs as a result of scaling was evidence in this study. Learners experience difficulties in interpreting, constructing and predicting from linear graphs without the knowledge of scaling. Four tasks were considered necessary for constructing linear graphs. This research focused on scaling task. The objective of the research is to establish the fact that scaling has influence on the understanding of algebraic linear graphs. An empirical method of research was employed to carry out this research. The results proved that scaling has influence on the understanding of algebraic linear graphs at grade 9 (Form B). Learners will find it easy to construct, interpret and make prediction from a graph drawn by scaling. / Mathematical Sciences / M.A. (Mathematics Education)