Syftet med den här studien är att undersöka hur elever i de tidiga skolåren är medvetna om de lagar, regler och strategier som bygger strukturen i aritmetiska uttryck. Inom matematiken finns det räknelagar, räkneregler och räknestrategier. Därför är det viktigt att elever i skolan ges möjligheten att utveckla kunskaper om dem. Data har samlats in genom 16 semistrukturerade intervjuer med elever från årskurs 1. Eleverna i studien har under fyra deluppgifter fått visa hur de gör när de summerar tre eller fyra tal. Studien har visat att elever väljer tal efter olika principer, placerar tal efter olika principer och ändrar placering av talen vid beräkningar. I den här studien har elever visat att de kan göra motiveringar till varför operationsordningar går att ändra. Studien har visat att eleverna använt associativa lagen trots att de saknar formell undervisning om den. Vid de olika deluppgifterna har räknelagar, räkneregler och räknestrategier används. Studien har visat att flera elever kan göra generaliseringar över kommutativa och associativa lagen. Slutsatsen av studien är att elever redan i årskurs 1 är väl medvetna om de lagar, regler och strategier som bygger strukturen i aritmetiska uttryck. / In mathematics, there are laws, rules and strategies of arithmetic. That is why it is important that young students are given the opportunity to develop knowledge about them. The purpose of this study it to investigate whether students in the early school years are aware of the laws, rules and strategies that build the structure of arithmetic. The data for this study is 16 semi-structured interviews with Swedish 1:st grade students. The students in this study have, during four sub-tasks, shown how they add three or four numbers. The study shows that students choose numbers according to different principles, place numbers according to different principles and change the placement of the numbers in calculations. Students can also give reasons for why the order of number can be changed. Students used the associative law even though they lack formal education about it. For the various sub-tasks, it is obvious that arithmetic laws, rules and strategies have been used. Particularly several students have shown that they can make generalizations of the commutative and associative law. The conclusion of the study shows that students are aware of the laws, rules and strategies that build the structure of arithmetic expressions.

The purpose of this study, based on the third year of a three-year research study, was to examine Grade 6 students’ previously developed abilities to integrate their understanding of geometric growing patterns with graphic representations as a means of further developing their conception of linear relationships. In addition, I included an investigation to determine whether the students’ understanding of linear relationships of positive values could be extended to support their understanding of negative numbers. The theoretical approach to the microgenetic analyses I conducted is based on Noss & Hoyles’ notion of situated abstractions, which can be defined as the development of successive approximation of formal mathematical knowledge in individuals. I also looked to Roschelle’s work on collaborative conceptual change, which allowed me to examine and document successive mathematical abstractions at a whole-class level. I documented in detail the development of ten grade 6 students’ understanding of linear relationships as they engaged in seven experimental lessons. The results show that these learners were all able to grasp the connections among multiple representations of linear relationships. The students were also able to use their grasp of pattern sequences, graphs and tables of value to work out how to operate with negative numbers, both as the multiplier and as the additive constant. As a contribution to research methodology, the use of two analytical frameworks provides a model of how frameworks can be used to make sense of data and in particular to pinpoint the interplay between individual and collective actions and understanding.

This thesis reports on the findings of an international study into the way in which children approach calculations which involve the order of operations. The study involved 203 pupils aged between 12 to 14 years from four different countries: England, The USA (New York State), Japan and The Netherlands. Many pupils in England are taught to use mnemonics such as BODMAS or BIDMAS to remember the correct order of operations, and in the USA pupils are often taught to use PEMDAS. However in Japan and The Netherlands these methods are not used, and the approach to teaching mathematics differs considerably across the countries. In this study pupils from classes in these four countries have been given calculations to perform and their work has been analysed for misconceptions. The analysis of their work has involved use of the Key Recorder software as a data collection tool, in which the pupils’ calculator keystrokes have been recorded and played back to give a unique insight into their thinking. Analysis of the children’s work has resulted in the categorisation of the misconceptions that were observed, and suggests that the nature of the mathematics curriculum and the teaching methods employed may have a significant effect on the way in which children approach calculations of this sort.

A great majority of children in Canada and the United States from Grades 2-6 fail to solve equivalence problems (e.g., 2 + 4 + 5 = 3 + __) despite having the requisite addition and subtraction skills. The goal of the present study was to determine the relative influence of two variables, instructional focus (procedural or conceptual) and use of manipulatives (with or without), in helping children learn to solve equivalence problems and develop an appropriate understanding of the equal sign. Instruction was provided in four conditions consisting of the combination of these two variables. Students in Grade 2 (n = 122) and Grade 4 (n = 151) participated in four sessions designed to assess the effectiveness of four instructional methods for learning and retention. Session 1 included a pretest of equivalence problem solving and three indicators of understanding of the equal sign. In Sessions 2 and 3 instruction was provided in one of the four instructional conditions or a control condition. Students were tested for their skill at solving equivalence problems immediately following instruction and at the beginning of Session 3 to assess what they had retained from Session 2. In Session 4, one month later, children were re-tested on all of the tasks presented in Session 1 to assess whether instruction had a lasting effect. All four instructional groups outperformed the control group in solving equivalence problems, but differences among instructional groups were minimal. Performance on indicators of understanding, however, favoured students who received conceptually focused instruction. Preliminary evidence was found that children’s understanding of problem structure and attentional skill may be associated with the ability to benefit from instruction on equivalence problems. Children clustered into four groups based on their performance across tasks that are consistent with the view that children’s understanding of the equal sign develops gradually, beginning with learning the definition. These findings suggest that a relatively simple intervention can markedly improve student performance in the area of mathematical equivalence, and that these improvements can be maintained over a period of time and show some limited generality to other indicators that children understand equivalence.

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The purpose of this study, based on the third year of a three-year research study, was to examine Grade 6 students’ previously developed abilities to integrate their understanding of geometric growing patterns with graphic representations as a means of further developing their conception of linear relationships. In addition, I included an investigation to determine whether the students’ understanding of linear relationships of positive values could be extended to support their understanding of negative numbers. The theoretical approach to the microgenetic analyses I conducted is based on Noss & Hoyles’ notion of situated abstractions, which can be defined as the development of successive approximation of formal mathematical knowledge in individuals. I also looked to Roschelle’s work on collaborative conceptual change, which allowed me to examine and document successive mathematical abstractions at a whole-class level. I documented in detail the development of ten grade 6 students’ understanding of linear relationships as they engaged in seven experimental lessons. The results show that these learners were all able to grasp the connections among multiple representations of linear relationships. The students were also able to use their grasp of pattern sequences, graphs and tables of value to work out how to operate with negative numbers, both as the multiplier and as the additive constant. As a contribution to research methodology, the use of two analytical frameworks provides a model of how frameworks can be used to make sense of data and in particular to pinpoint the interplay between individual and collective actions and understanding.

Forskning och internationella analyser har visat att elevers kompetenser inom algebra är bristfälliga. Syftet med denna studie är därför att undersöka vilka kompetenser eleverna på låg och mellanstadiet behöver befästa för att uppnå kunskapskraven och för att kunna ta sig an framtida studier. Syftet kommer att besvaras genom tre framtagna forskningsfrågor: I vilken utsträckning följer läromedlen på lågstadiet forskningen kring early algebra? Hur ser progressionen ut från lågstadiet till mellanstadiet i lärobokserier beträffande matematiska kompetenser inom algebraområdet? och I vilken utsträckning behandlar mellanstadiets läromedel de kompetenser som forskning menar är bristande kunskaper på högstadiet? En läromedelsanalys genomfördes för att få syn på i vilken utsträckning de olika kompetenserna finns representerade i läromedlen. Genom tidigare forskning har studien identifierat vilka kompetenser som eleverna behöver befästa på låg och mellanstadiet för att kunskapskraven ska uppnås.Resultatet från analysen visar på att de kompetenser som efterfrågas finns med till viss del i de analyserade läromedlen men att det finns utvecklings möjligheter. / Research and international analyses have shown that students' skills in algebra are deficient. The purpose of this study is therefore to investigate whichcompetencies students in primary and middle school need to consolidate to achieve the knowledge requirements and to be able to take on future studies.The purpose will be answered through three developed research questions: To what extent do the teaching materials in primary school follow the research about early algebra? What is the progression from primary to middleschool in textbook series regarding mathematical skills in the field of algebra? To what extent do middle school teaching materials deal with the skillsthat research believes are lack of knowledge in high school? A teaching material analysis was implemented to see to what extent the different competencies are represented in the teaching materials. Through previus reasearch, the study has identified which competencies the students needto consolidate at the primary and intermediate leves for the knowledge requirements to be achieved. The results from the analysis show that the competencies that are in demandare included to some extent in the analysed teaching materials, but that thereare development opportunities

This study explores the development of early algebraic reasoning in mathematical communities of inquiry. Under consideration is the different pathways teachers take as they develop their own understanding of early algebra and then enact changes in their classroom to facilitate algebraic reasoning opportunities. Teachers participated in a professional development intervention which focused on understanding of early algebraic concepts, task development, modification, and enactment, and classroom and mathematical practices. Design research was employed to investigate both teaching and learning in the naturalistic setting of the schools and classrooms. The design approach supported the development of a model of professional development and the framework of teacher actions to facilitate algebraic reasoning. Data collection over the school year included participant observations, video recorded observations, documents, teacher interviews, and photo elicitation interviews with students. Retrospective data analysis drew the results together to be presented as cases of two teachers, their classrooms, and students. The findings show that the integration of algebraic reasoning into classroom mathematical activity is a gradual process. It requires teachers to develop their own understanding of algebraic concepts which includes understanding of student reasoning, progression, and potential misconceptions. Task implementation and design, shifts in pedagogical actions, and the facilitation of new classroom and mathematical practices were also key elements of change. The important role which students have in the development of classrooms where algebraic reasoning is a focus was also highlighted. These findings have significant implications for how teachers can be supported to develop their understanding of early algebra and use this understanding in their own classrooms to facilitate early algebraic reasoning.

Programmering infördes 2018 i Lgr 11 under Algebra i det centrala innehållet för matematik. Sverige är unikt med denna konstruktion och det har medfört svårigheter för matematiklärare då det dels är otydligt vad undervisningen ska innehålla och dels, vad programmering har med algebra att göra. Syftet med denna systematiska litteraturstudie är att tydliggöra vilka kunskaper i algebra som är relevanta för undervisningen av visuell programmering i årskurs 4-6. Detta arbete avgränsar sig till visuell programmering då det dels är den typ av programmeringsmiljö som introduceras i årskurs 4-6 och dels, är den programmeringsmiljö som ska förbereda elever inför olika programmeringsmiljöer i årskurs 7-9.  För att försöka tydliggöra vilka kunskaper i algebra som är relevanta för undervisningen av visuell programmering används så kallade ”Big Ideas” från reflektionsverktyget CoRe. Den systematiska litteraturstudien har granskat 13 artiklar och använt kvalitativ innehållsanalys för att finna kategorier och teman. Resultatet visar att förståelse och användning av variabler kan anses vara en ”Big Idea” i både Algebra och Visuell programmering. Dock är det otydligt huruvida variabler är att anse som en ”algebraisk kunskap” i visuell programmering eller om det handlar om två skilda variabelbegrepp i algebra respektive visuell programmering. Lärare som undervisar i visuell programmering i årskurs 4-6 bör uppmärksamma skillnader mellan variabler i en algebraisk kontext och variabler i en programmeringskontext.

Syftet med licentiatuppsatsen är att bidra med kunskap om vilka aspekter som kan vara nödvändiga för elever att urskilja för att identifiera och analysera en del-helhetsstruktur och i förlängningen bemästra ekvationer. I forskningsprojektet medverkade lärare och forskare i ett kollaborativt och intervenerande arbete. Elever i årskurserna 3, 8 och 9 deltog i semistrukturerade intervjuer och forskningslektioner för att producera data. Forskningslektionerna genomfördes med learning study som forskningsansats, vilken erbjuder iterativa processer. De teoretiska ramverken utgjordes av fenomenografi, variationsteori och lärandeverksamhet. Ansatsen fenomenografi utgjorde ett analysverktyg när intervjuerna analyserades. Variationsteori och lärandeverksamhet utgjorde kompletterande ramverk när forskningslektioner designades och analyserades. Resultatet i Artikel 1 påvisar en kritisk aspekt, vilken formulerades som: två kvantiteter tillsammans (två delar) bildar en tredje kvantitet (en helhet) med samma ”värde” som de två delarna tillsammans. Resultatet i Artikel 2 påvisar fem kritiska aspekter, vilka formulerades som (1) det är en relation mellan alla tal i en ekvation, (2) två delar tillsammans är ekvivalenta med en helhet med samma värde, (3) vad som utgör en helhet respektive delar, (4) samma relation kan formuleras på fyra olika sätt och (5) helheten kan anta ett lägre värde än delarna. En slutsats är att det kan vara en fördel redan för unga elever att delta i undervisning med utgångspunkt i early algebra och lärandeverksamhet där generella matematiska strukturer fokuseras. En ytterligare slutsats är att det kan vara gynnsamt att delta i teoretiska resonemang där ekvationer med negativa tal inkluderade utforskas med stöd av en lärandemodell. / The aim of the licentiate thesis is to contribute with knowledge concerning which aspects may be necessary for students to discern in order to identify and analyze a part-whole structure and in the long term master equations. In the research project, teachers and researchers participated in a collaborative and intervening work. Students in grades 3, 8 and 9 participated in semi-structured interviews and research lessons in order to produce data. The research lessons were conducted with learning study as research approach, which offers iterative processes.The theoretical frameworks consisted of phenomenography, variation theory and learning activity. The phenomenography approach was an analysis tool when the interviews were analyzed. Variation theory and learning activity constituted complementary frameworks when research lessons were designed and analyzed. The finding in Article 1 demonstrates a critical aspect, which was formulated as two quantities together (two parts) build up a third quantity (the whole) with the same “value” as the two parts together. The finding in Article 2 demonstrates five critical aspects, which wereformulated as (1) there is a relationship between all numbers in an equation, (2) two parts together are equivalent to a whole with the same value, (3) what constitutes a whole and parts, (4) the same relationship can be formulated in four different ways and (5) the whole can assume a lower value than the parts. One conclusion is that it may be an advantage already for young students to participate in teaching based on early algebra and learning activity where general mathematical structures are focused. A furtherconclusion is that it may be beneficial to participate in theoretical reasoning where equations with negative numbers included are explored with support by a learning model.