Responding constructively ‘in-the moment’ to student offers is described as a critical, and yet difficult, aspect of skilful and responsive teaching. South African evidence points to limited evaluation of student offers in schools serving poor communities. In this paper, we present and discuss an ‘elaboration’ framework emerging from a grounded analysis of data drawn from video recordings of 18 mathematics lessons prepared and conducted by four in-service primary school teachers in South Africa. This analysis led to a categorization of the situations in which teacher responses to student offers occurred, and the nature and range of these responses. Three response situations are identified within the framework: breakdown, sophistication, and individuation/collectivization, with a range of response (and non-response) categories in each situation. Literature on responsive feedback is drawn in to explore hierarchies and relationships between the emergent categories within situations of elaboration. The elaboration framework provides a tool for lesson observation, and a model for thinking about developments in responsive teaching.

The cardinal and ordinal aspects of number have been widely written about as key constructs that need to be brought together in children’s understanding in order for them to appreciate the idea of numerosity. In this paper, we discuss similarities and differences in the ways in which understandings not only of ordinality, cardinality but also additive and multiplicative relations have been theorized. We examine how the connections between these can be considered through a focus on number line representations and children positioning and comparing numbers. The responses of a cohort of South African Grade 1 learners’ (6- and 7-year-olds) to a numerical magnitude estimation task and to a numerical comparison task are analysed and the findings compared to those in the international literature, some of which argue that children’s early, informal, understandings of cardinality and ordinality are underpinned by an intuitive logarithmic model relating number order and size. A main finding presented here is that the responses from learners in this study exhibited a better fit with an exponential model of the relationship between cardinality and ordinality. These findings raise questions about whether some of the findings in previous research are as universal as sometimes claimed.

From a sociocultural perspective that a teacher’s use of mediational means is central to student learning, this paper presents an analysis of six teachers and their mediating, across a two-three year time gap. Drawing on the Mediating Primary Mathematics framework – developed to examine the type and quality of mediational means – we propose two composite assessments of quality of mediation – extent and depth – that indicate the extent to which teaching addresses mathematical structure and generality. The findings reveal a range of differences in these two assessments for each of the six teachers, but that all six teachers were more coherent in their use of mediational means in the later lesson than in the earlier one. These findings have implications for other schooling systems and researchers seeking to improve the quality of mathematics instruction.

Background: Given the context of low attainment in primary mathematics in South Africa, improving learners’ understanding of multiplicative reasoning is important as it underpins much of later mathematics.

Aim: Within a broader research programme aiming to improve Foundation Phase (Grades 1–3, 7–9-year-olds) learners’ mathematical performance, the aim of the particular research reported on here was to improve learners’ understanding of and attainment in multiplicative reasoning when solving context-based problems.

Setting: The research was conducted in a suburban school serving a predominantly historically disadvantaged learner population, and involved teachers and learners from three classes in each of Grades 1–3.

Methods: A 4-week intervention piloted the use of context-based problems and array images to encourage learners to model (through pictures and diagrams) the problem situations, with the models produced used both to support problem solving and to support understanding of the multiplicative structures of the contexts.

Results: Cleaning the data to include those learners participating at all three data points – pre-, post-and delayed post-test – provided findings based on 233 matched learners. These findings show that, on average, Grade 1 learners had a mean score average increase of 22 percentage points between the pre-test and the delayed post-test, with Grades 2 and 3 having mean increases of 10 and 9 percentage points, respectively.

Conclusion: The findings of this study demonstrate that young learners can be helped to better understand and improve their attainment in multiplicative reasoning, and suggest the usefulness of trialling the intervention model more broadly across schools. 

The paper presents the synthesis of a series of studies on support measures for pupils with additional languages in Sweden, with a particular focus on the consequences of language support measures for teaching and learning school mathematics. The analysis identifies several critical aspects in mathematics teaching when the pupils transition from one school system to another, and particularly when they do not yet have a shared language to communicate among each other or with their teachers. Three aspects with relevance to language support will be discussed and illustrated with examples from the studies: the notion of mathematics as a formal language; the notion of levels and absolute progression in mathematics; the notion that learning mathematics means understanding "concepts".

The tensions between tendencies to support and reject that can be observed in reception measures for newly-arrived students in Sweden are a consequence both of the general climate of hostility towards immigration in Europe, and features particular to Sweden. The paper summarises the historical background to the present situation, from the 1960s onwards, and points to some of the impacts for teachers and for civil society. The analysis will above all focus on three diverging but parallell trends and which overlap: dynamics of integration, dynamics of competition and social segregation; dynamics of 'securitisation'. These dynamics evolve within different conceptualisations of education, but also relate to social developments and conflicting visions for society as a whole.

In this paper we direct attention to 5–6-year-olds’ learning of arithmetic skills through a thorough analysis of changes in the children’s ways of encountering and experiencing numbers. The foundation for our approach is phenomenographic, in that our object of analysis is differences in children’s ways of completing an arithmetic task, which are considered to be expressions of their ways of experiencing numbers and what is possible to do with numbers. A qualitative analysis of 103 children’s ways of encountering the task gives an outcome space of varying ways of experiencing numbers. This is further analyzed through the lens of variation theory of learning, explaining why differences occur and how observed changes over a prolonged period of time can shed light on how children learn the meaning of numbers, allowing them to solve arithmetic problems. The results show how observed changes are liberating new and powerful problem-solving strategies. Emanating from empirical research, the results of our study contribute to the theoretical understanding of young children’s learning of arithmetic skills, taking the starting point in the child’s lived experiences rather than cognitive processes. This approach to interpreting learning, we suggest, has pedagogical implications concerning what is fundamental to teach children for their further development in mathematics.

In this paper we describe the empirical and theoretical meaning behind how finger patterns are taught to facilitate the development of preschool children’s perception of the first ten natural numbers. An intervention programme, informed by Variation theory of learning, included 65 five-year-olds and teachers at seven preschool departments in Sweden. The programme aimed at developing teaching activities and artefacts to promote children discerning necessary aspects of the first ten numbers. The design of the programme is significant to describe and evaluate as basis for forthcoming analyses of the learning outcomes, as a pedagogical approach that stands in contrast to common preschool teaching practice in Sweden is adopted.

We report here on an intervention implementing a structural approach to arithmetic problem-solving in relation to learning outcomes among preschoolers. Using the fundamental principles of the variation theory of learning for developing the intervention and as an analytical framework, we discuss teaching and learning in commensurable terms. The research question is how teaching grounded on a structural approach and designed based on principles of variation theory is reflected in children’s learning of numbers. To answer this, three analyses were conducted, addressing: i) how the children’s ways of experiencing numbers changed after participating in the intervention, ii) how the theoretical ideas were afforded in the intervention program, and iii) synthesizing how the affordance was associated with the children’s arithmetic learning. One group of eight children participating in the intervention program was chosen for thorough analysis. Progression was observed in how the children changed their ways of experiencing numbers during the intervention that allowed them to enact more advanced arithmetic strategies, which was associated with the structural approach in teaching. The results also show how analysis focusing on aspects discerned in learning and aspects afforded in teaching provides a way of describing arithmetic learning with significant implications for teaching practices.

The idea of using fingers as a key component in arithmetic development has received a great deal of support, much of which is based on neuroscientific evidence. However, this body of work pays limited attention to how fingers are used and possible different outcomes in arithmetic problem solving. The aim of our paper, based on an analysis of 126 observations of 4–5-year-olds solving a simple subtraction task, is to discuss different ways of using fingers, with some of the ways appearing more, and others less, powerful. The analysis suggests there is much more complexity to children’s finger-related strategies than prior research has indicated. Empirical findings in our study point to the decisive effects of different ways of using fingers, and in particular for either keeping track of counted units or for presenting a structured awareness of number. Three ways of using fingers emerge in the analysis, which are discussed in relation to their rate of success in solving the subtraction task and with attention to why the differences matter for the success rate. Through this discussion we suggest that the complexity of how fingers are used must be considered. 

The aim of this paper is to develop greater understanding of how the way students experience a task is related to which, and how, number relations are discerned. We study how 42 Grade 1 students solved a word problem in a number range that was new to them: 32–25 = __. The variation theory of learning has informed our analysis, opening for thorough analyses of what constitutes differences in the students’ acts in solving the arithmetic word problem and how they experience the task. Observations of their strategies and ways of reasoning revealed that how the students discern the semantic structure and number relations relates to their ways of encountering the task and consequently their success in solving it. The study offers a complementary approach to understanding arithmetic skills that contribute knowledge as to why some students develop powerful ways of solving arithmetic tasks while others get stuck in cumbersome strategies. 

Internationally, education reform has been directed towards describing educational goals that go beyond topic and content descriptions. The idea of mathematical competencies describes such goals. National tests have been seen as one way of communicating these goals and influence teaching. The present study analyses Swedish national tests in mathematics, and seeks evidence about the extent to which they represent these competencies and may play a role in reforming teaching. The results show that the national tests assess all competencies. However, the study also shows a limited focus on the competencies’ interpretation and evaluation aspects. Thus, the tests do not fully capture the complex nature of the competencies. This may cloud the reform message and restrict the possibilities for the tests to function as levers for reform.

En problembaserad undervisning tillåter matematikundervisningen att vara utmanande och undersökande för eleverna. Till skillnad från den traditionella undervisningen är syftet att eleverna ska ställas inför problem som de får arbeta med för att komma fram till en lösning. Denna undervisningsmetod leder således till andra sätt att lära än den traditionella undervisningen. Syftet med denna litteraturstudie är att belysa hur en problembaserad undervisning kan komplettera den traditionella undervisningen för att utveckla elevers matematiska måluppfyllelse. För att undersöka detta genomfördes en systematisk granskning av forskning inom området. Resultatet är uppdelat efter tre forskningsfrågor som har i uppgift att besvara studiens syfte. Studien har identifierat vad som utmärker en problembaserad undervisning, som att det utgår från elevernas kunskapsnivå samt att det ger eleverna möjlighet att skapa sin egen förståelse. Läraren är en viktig del för lärandet och behöver besitta en hög yrkeskompetens för att maximera elevernas inlärningsmöjligheter. En problembaserad undervisning leder till resonemang och reflektion vilket främjar den djupa förståelsen för matematiken. Men, för att kunna resonera och reflektera krävs att eleverna har en god begreppsförståelse och besitter goda kunskaper inom metodval.

This study has been done within mathematic education. The purpose of the study is to analyse how much space time- and clock assignments are given in relation to other quantities in two popular teaching materials. Analysis and interpretations of assignments from the two teaching materials have been the basis for the study.

 The study has been restricted to containing teaching materials for grade 1-3. This subject was chosen because the curriculum in mathematic can be seen as rather slim when it comes to time and the clock, therefore it can be interesting to see how much space these assignments get in the different teaching materials. The study has its foundation in the sociocultural perspective, which means that humans learn in interplay with one another.  The sociocultural perspective is the most central perspective of the curriculum for the elementary school  (Skolverket, 2018) and is also strongly connected to the five mathematical abilities the study contains. The result also shows that the five mathematical abilities are not clearly shown through the assignments of the teaching materials. The result also shows that the five mathematical abilities are not revealed in a clear way through the assignments in the teaching materials. However, the five mathematical abilities can be made more visible through interpretations by those who work with the teaching material. Teacher manuals proved helpful to some degree when it comes to interpretation of what abilities that can be learned through the assignments.

Children develop an understanding of numbers by, for instance, counting items in smaller or larger sets. When a larger set is placed in a regular arrangement (for example, in rows) subitizing or counting can be used to quantify a subset, and thereby determine the size of the larger set. It becomes more challenging when the items are placed in an irregular arrangement. The aim of this study is to answer the question: How do 6-year-olds estimate and reason about how to determine a quantity of a larger set in an irregular arrangement? In this study, 130 Swedish 6-year-olds were asked: How many cubes do you think there are on the tray? How could you find out? looking at a tray with 47 randomly arranged wooden cubes. In the analysis, students’ answers were summarized. Codes, inductively sprung from the data, were used to describe students’ reasoning. The analysis shows that around half of the students made a reasonable estimation of the number of cubes. In 2/3 of the observations, single-unit counting was in focus in students’ reasoning when determining the size of the set of cubes. Whereas in 1/3 of the observations, decomposing the whole collection into subsets, either of the same size (e.g., groups of five) or different size, was in focus in their reasoning. Hence, the study reveals different ways in which 6-years-olds reason about estimating or determining the size of an uncountable set. Based on this, implications for how to teach quantification and estimation are discussed. 

The data in this paper draws from an eight-month intervention study based on the idea that children need to discern the first ten natural numbers as relations of parts and whole to develop their arithmetic skills. In order to implement educational activities according to this conjecture, a group of Swedish preschool teachers worked closely with a research team, planning, enacting and analyzing activities. In this paper I describe how nine pre-school teachers, across 67 video-recorded films, handled one of these activities, called the ‘snake game’ with their groups of 5-year-old children. Using analysis based on variation theory, the results point to differences in the enactment of the ‘snakegame’ in terms of if and how the teachers foregrounded the structural aspects of numbers embedded in the activity.

In this paper, differences in the implementation of a number activity called the snake game are studied. Nine Swedish preschool teachers worked in collaboration with a research team, enacting the same activity with their groups of 5-year-old children over a 3-month period. Variation theory forms the basis for the analysis of 67 videorecorded enactments. The results suggest that an activity such as the snake game can bring various aspects of numbers to the fore through differences in enactment. The activity became mathematically richer when the teacher compared children’s different finger patterns and used systematically varied examples of number relations. This study’s results contribute knowledge about characteristics of teaching that foregrounds numbers’ part-whole relations.

In this thesis, results from four empirical studies and a re-analysis are synthesized with what can constitute a structural approach to teaching and learning additive part-whole relations among learners aged four to eight years. In line with a structural approach to additive relations, the relations of parts and whole are in focus from the outset and are seen as the basis for addition and subtraction (Davydov 1982; Neuman, 1987). This approach was introduced by the researches in two intervention studies across different contexts. The researches collaborated with teachers in planning part-whole activities, teachers teaching them in their own settings, and then reflecting on them together with the research team. The empirical material consists of video-recorded lessons (Grade 3), small-group teaching (preschool) and individual video-recorded task-based learner interviews (with preschoolers). The teaching episodes and interviews were analyzed on a micro-level, using analytical tools and concepts from variation theory (Marton, 2015). To deepen the knowledge, a re-analysis was also conducted with the purpose of identifying qualitative differences in teachers’ enactments of mathematical ideas and principles associated with a structural approach to additive relations.

Looking at the articles and the re-analysis, the results suggest that, for learning, it matters which representations are offered to the children. Some representations seem to facilitate the discernment of the parts and whole, and their relations. The results suggest that it matters which examples are offered. A systematic sequence of examples has the potential to bring to the fore relations between different part-whole examples, which offer the children opportunity to learn mathematical principles such as commutativity. Furthermore, the results indicate that what is made possible to learn about additive part-whole relations is associated with what aspects are opened up as dimensions of variation (Marton, 2015). Foremost, though, the results reveal the importance of making connections to highlight number relations and key features associated with the structural approach to additive relations. The results suggest that how variation is offered, and whether and how the teacher explicitly (verbally and gesturally) draws attention to relations, ideas and aspects, is crucial for the learning of additive part-whole relations. Moreover, through the separate articles and the re-analysis, the outcomes indicate that the structural approach to additive part-whole relations and conjectures from variation theory are possible to implement in different contexts and for different ages of children.

The aim of this presentation is to discuss a variation theoretical analysis of teaching conducted in two research projects involving teachers in preschool and primary grades. The same mathematical ideas possible to emphasize in teaching number relation tasks were scrutinized in purpose to describe differences in learning opportunities being offered. Video-recorded teaching episodes were analyzed on a micro-level. Aspects that were opened up as dimensions of variation up (or not) associated with the focused mathematical idea were identified in each teaching episode. Also, how the teachers used connections (linking gestures and talk) to draw the learners’ attention to target relations in the learning situation were analyzed. The analysis points to differences between an enactment where the dimension was opened up and the mathematical idea was made perceptually visible for the learners and an enactment where the teacher also explicitly draw the learners’ attention to the specific idea by the use of connections. The results and the way of expanding a variation theoretical analysis might contribute to a discussion on how to use tools and theoretical concepts from variation theory.

Ur abstraktet: SATSA står för Strukturell Ansats i undervisning Som grund för hållbart Aritmetik lärande och är ett projekt finansierat av Vetenskapsrådet. I SATSA riktas uppmärksamheten mot kvaliteten i undervisning i förskoleklass (6-åringar) med avseende på hur tal, antal och räknestrategier behandlas som innehåll för lärande. 

Ett utvecklingsprojekt startades 2019 på initiativ av Vaggeryds kommun. Då kommunens lektor under en längre tid har varit knuten till den praktiknära forskningsmiljön PUF och forskargruppen i matematik, vid Högskolan för Lärande och Kommunikation, och det beviljades ULF-medel från Göteborgsnoden, startades projektet VA–MER? Vid uppstarten formulerades syfte och mål för projekten utifrån kommunens behov. Målet med projektet var bland annat att genom kollegialt lärande och med stöd av forskare gemensamt bygga ett fundament i matematikundervisningen i förskoleklass med stöd av aktuell forskning, i syfte att utveckla elevers taluppfattning.

Projektet följer ett forskningsbaserat upplägg. Alla lärare som deltar i projektet planerar och prövar undervisningsaktiviteter. Därefter reflekterar de över dessa tillsammans med kollegor och lärosätets forskare/ lärare. Projektet går nu in på sitt tredje läsår. Syftet med projektet har delvis reviderats. Utifrån förra läsårets utvärdering och elevresultat har en övergripande forskningsfråga formulerats, liksom frågor kopplade till lokala behov på respektive enhet. Ett ännu större fokus läggs nu på elevers lärande och på hur undervisning möjliggör för alla elever att utveckla sin förmåga att resonera om tal.

I nuläget deltar två lärare/forskare från Högskolan för lärarande och kommunikation och tolv lärare i förskoleklass (från sju enheter). Några lärare har avslutat sin medverkan av naturliga skäl, medan andra lärare har tillkommit. Lärarna och forskarna träffas kontinuerligt, för att planera och diskutera matematikundervisning tillsammans. Deltagarnas olika erfarenheter av att ha undervisat i förskoleklass, den geografiska spridningen i kommunen, mixen av små och stora enheter, samt lärosätets lärare med forskning inom området och lång erfarenhet av undervisning utgör en god grund för skolutveckling.

Unikt med projektet är att alla kommunens lärare i förskoleklass deltar. Samordnare är kommunens lektor i matematikdidaktik, som har 20% av sin tjänst för arbete med bland annat skolutveckling och för att vara en länk mellan akademin och skolan. Hon har även det organisatoriska ansvaret och samverkar kontinuerligt med forskare, rektorer och verksamhetschef. Det finns täta kontaktvägar när problem uppstår.

Ur ett huvudmannaperspektiv är en kontinuerlig och systematisk fortbildning, grundat på analyser av mål, styrdokument och elevers måluppfyllelse samt aktuell forskning avgörande för att utveckla undervisning. Vi tror att VA–MER? kan stärka likvärdigheten i Vaggeryd kommuns förskoleklassverksamhet samt förskoleklassens pedagogiska uppdrag, vilket i sin tur kommer attbidra till elevernas högre måluppfyllelse.

Lärosätets upplevelse av VA–MER? är att det tar tid att bygga relationer mellan lärare och lärare och forskare för att kunna se varandra som resurser och föra diskussioner på lika villkor. Utifrån ett lärosätesperspektiv ställs man inför utmaningar att få modellen att fungera, stötta de som är ensamma på sin enhet, få alla att våga pröva nya aktiviteter, filma och utmana de som deltagit under två läsår till att utveckla sin undervisning ytterligare.

Vi tänker att framgångsfaktorer för projektet VA MER är: långsiktigheten, - förändring tar tid, likvärdigheten - alla kommunens enheter deltar, den goda framförhållningen - tid för träffar schemaläggs i läsårsplaneringen, avgränsningen – matematik och fokus på taluppfattning samt att projektidén är väl känd - projektet är allas angelägenhet (verksamhetschefens, utvecklingsledarens, rektorernas och lärarnas och lärosätets).

Bron mellan Vaggeryds kommun och Högskolan för Lärande och Kommunikation där ULF-projekt bedrivs har blivit mer stabil. Samarbetet fortsätter att utvecklas beroende av den täta kontakt som finns etablerat mellan lärosätets forskningsmiljö, lektor och verksamhetschef i samverkande kommun.

I konferenspresentationen delges erfarenheter från projektet utifrån såväl ett huvudmanna- som från ett lärosätesperspektiv.

I forskningsprojektet FASETT (VR-UVK 2014-1791) undersöker vi 5-åringars taluppfattning och aritmetikfärdigheter och möjligheterna att genom målorienterade processer stötta barn i att utveckla framgångsrika strategier för aritmetisk problemlösning. I två kommuner har vi tillsammans med förskollärare prövat ut aktiviteter i syfte att utveckla särskilda förmågor som förmodas vara nödvändiga för framgångsrik problemlösning i aritmetik, såsom att urskilja tals del-helhets-struktur, fingertal samt representationer av tal. Barnens (N= 65) taluppfattning och hur deras förskollärare arbetat med förmågorna på ett systematiskt sätt presenteras i föreläsningen, med tyngdpunkt på vad som görs möjligt att lära i aktiviteterna och effekter för barnens utveckling av talförståelse och aritmetikfärdigheter. Aktiviteter som prövats ut i projektet är teoretiskt grundade i Variationsteorin (Marton, 2015). I projektet uppmuntrar vi användning av fingrarna som redskap för att strukturera tal och operera med tal. Flera av aktiviteterna påminner om bekanta lekar och spel från förskolan, till exempel tärningsspel, ”10-masken” och enklare räknesagor. I föreläsningen presenterar vi en fördjupad analys av aktiviteten ”10-masken”, hur förskollärarna iscensatt aktiviteten på olika sätt och vilken betydelse det har haft för barnens lärandemöjligheter. ”10-masken” består av ett snöre med tio pärlor, där ett antal pärlor göms i handen och resten förblir synliga. Uppgiften är att ta reda på hur många pärlor som är gömda. Detta är inte en ny aktivitet i förskolan. Däremot, indikerar vår analys, spelar det roll hur en aktivitet med så många matematiska idéer introduceras och tas om hand av förskolläraren i mötet med barnen. Baserat på resultatet kan vi se att ”10-masken” erbjuder en potential av viktiga matematiska idéer, som är av stor pedagogisk betydelse för barns matematiklärande, men det spelar en avgörande roll hur förskolläraren gör i aktiviteten.

This paper reports on an eight months long intervention program with eight five-year-olds in Swedish preschool. Four main activities were designed to enable the children to discern part-part-whole relations of the first ten numbers. The aim of this paper is to present how progress in children’s arithmetical skills are associated with the activities they have encountered in the intervention program. Learning outcomes based on pre-, post- and delayed interviews show that the participating children made distinct progress in the way they experience numbers, with long-term effects on their arithmetic skills. In this paper we discuss the analysis of what was taught and what was learnt incommensurable terms.

In this paper one specific task in a series of tasks focusing on part-part-whole relations of the ten first natural numbers and finger patterns for structure number relations, is presented and discussed. The tasks were designed, planned and enacted in an intervention program conducted in Swedish preschool during an eight-month period. In the program nine preschool teachers worked in close collaboration with the research team in planning how to enact tasks with their 5-year-old children, in an iterative process. The specific task, called the ‘snake game’, consists of five or ten beads on a string, some of the beads where to be hidden and the children would find out the hidden part by using structured finger patterns. The task was designed in accordance with the variation theory assumptions that certain aspects need to be discerned as dimensions of variations. The aim of the paper is to examine which dimensions of variations that were opened up by the teachers and what was made possible for the children to learn from the enactment of the ‘snake game task’. The data set includes 67 video observations from the teacher’s enactment of the task. The results suggest that what seems to be a ‘non-complex task’ (five/ ten beads on a string) offers rich mathematical experiences and has potential to bring fore important aspects of numbers and number relations. However, depending on which dimensions of variations that were opened up reveals different learning possibilities.

I forskningsprojektet FASETT[1] flätas teori och praktik samman. Projektet syftar till att generera kunskap om barns tidiga taluppfattning, utifrån delvis andra perspektiv än de dominerande inom fältet, och se hur en pedagogisk verksamhet i samverkan med forskare kan bidra till barns utveckling av aritmetikfärdigheter.

I projektet arbetade nio förskollärare och 65 femåringar på fem förskolenheter i ett tätt samarbete med en grupp forskare under en åttamånadersperiod. Deltagarna träffades kontinuerligt, diskuterade och fördjupade sig i aritmetiken med fokus på aktiviteter gällande de tio första talens del-del-helhetsrelationer. Aktiviteterna var till viss del välkända, men bearbetade utifrån tidigare forskningsresultat och variationsteorin (Neuman, 1987; Marton, 2015). Utgångspunkten och reflektionerna vid gruppträffarna var intervjuer av barnens olika sätt att uppfatta tal och lösa enklare aritmetikproblem samt lärarnas iscensättande av de planerade aktiviteterna. Genom att gemensamt diskutera de filmade aktiviteterna kunde aktiviteterna förfinas och förskollärarnas didaktik utvecklas för att möta barnens behov. 

Pågående analyser visar att designen av projektet möjliggjort för förskollärare och forskare att i kollaboration implementera ett alternativt teoretiskt underbyggt sätt att utveckla barns taluppfattning och förmåga att   lösa enklare aritmetikproblem genom att exempelvis använda sig av fingrarna som redskap för att strukturera talrelationer och inte enbart räkna ’ett till ett’.

Analysen av de barnintervjuer som gjordes innan aktiviteterna introduceras och de barnintervjuer som gjordes efter forskningsprojektets slut visar att förskollärarnas målorienterade processer med största sannolikhet haft effekter på barnens aritmetiska förmågor. De preliminära resultaten indikerar att valet av att fokusera på ett fåtal aktiviteter möjliggjorde för en djupare reflektion kring teoretiska antaganden och vad barn behöver få syn på för att lära sig om tal och talrelationer.

[1] ”Förmågan Att Sinnligt Erfara de Tio första Talen som nödvändig grund för aritmetiska färdigheter”, finansierat av Vetenskapsrådet 2015-2018

De senaste åren har en rad förändringar gällande förskoleklassen genomförts. Den 1 juli 2016 fick förskoleklassen en egen del i läroplanen och från och med läsåret 2018/19 blev förskoleklassen obligatorisk och omfattas nu av skolplikten. Det förändrade uppdraget i förskoleklassen samt det obligatoriska kartläggningsmaterialet, har ökat lärares behov av att möta uppdraget. Kartläggningsmaterialet i matematiskt tänkande ska hjälpa lärare att tidigt identifiera elever som visar en indikation på att inte nå de kunskapskrav som sen ska uppnås i årskurs 3 i grundskolan. Däremot saknas direkta implikationer för undervisning som stimulerar alla elevers möjlighet att utveckla sina förmågor i matematik.

I ett nystartat samverkansprojekt mellan Vaggeryds kommun och Högskolan för Lärande och Kommunikation kommer kommunens alla lärare i förskoleklass att arbeta tillsammans med forskare tillhörande forskargruppen i matematik (MER-gruppen), i syfte att stärka likvärdighet i utbildningen inom Vaggeryd kommun.

Syftet med projektet är att se hur en pedagogisk verksamhet i samverkan med forskare kan bidra till att lärarna:

• utvecklar förmågan att diskutera ämnesinnehåll i matematikundervisningen,
• utvecklar förmågan att utifrån kartläggningsmaterialet diskutera didaktiska implikationerna för undervisning,
• gemensamt bygger ett fundament i matematikundervisningen, med stöd av aktuell forskning, i syfte att utveckla alla elevers förmågor i matematik.

Erfarenheter från ett nyligen avslutat forskningsprojekt där lärare och forskare i tätt samarbete genomförde och reflekterade över undervisningsaktiviteter, visar att detta verkade vara en framgångsrik modell för att utveckla undervisning. Därför väljs ett liknande upplägg i detta samverkansprojekt.

I den inledande fasen av projektet träffas lärargruppen och forskarna en gång per månad. Träffarna förläggs på kommunens skolor. De påbörjade samtalen mellan deltagarna i samverkansprojektet visar att det finns ett stort behov av att diskutera matematikundervisning och att utveckla ett målinriktat arbete som främjar alla elevers möjligheter att lära matematik.

In this study, we examine how 30 pre-service teachers designed and modified mathematical tasks to enhance primary students’ learning. The pre-service teachers took part in a 5-week course in a teacher education program in Sweden in which a theory-based lesson study model entitled learning study were established to deepen the teachers’ awareness about the relationship between instruction and student learning. The learning study course design consisted of two intervention cycles in which the pre- 17 SIG 9 Phenomenography and Variation Theory service used variation theory as a tool for lesson design and re-design. The aim of this study is to explore in what ways the pre-service teachers modify mathematical tasks when employing variation theory during the 5-week mathematics education course in which LS cycles were incorporated. Data were collected at the end of the course and consist of written reports about task refinements based on their reflections about the students’ performance during the lessons. We identified five different ways of task modifications: expanding tasks, making tasks more explicit, making tasks more implicit, bringing metaphors and representations to the foreground, and creating new tasks. These categories might become a complementary tool to variation theory, for reflecting about task design and redesign.

Teaching addition and subtraction using 10 as a benchmark is seen as a powerful strategy for advancing pupils’ learning of arithmetic skills when solving tasks like 13-5. Using 10 as a benchmark for solving the task entails for example that the pupils need to be aware of that 5 can be partitioned into 3 and 2, and that the task may be solved in two steps, 13-3=10, and 10-2=8. In this study two lessons in two grade 1 classes, taught by different teachers, are analyzed on a finegrained level. Our aim is to exemplify and discuss how 10 as a benchmark is used in teaching subtraction bridging through ten and what that may imply for pupils’ learning. Our research question is; What different opportunities to learn ‘10 as a benchmark’ are offered in two lessons of subtraction in the number range of 1 to 20? Variation theory (Marton, 2015) is the theoretical framework used for analysis. The analysis shows how different opportunities to learn subtraction bridging through ten are offered in the lessons by the aspects of the content brought to the fore for the pupils to experience.

In this article, we present aspects of teaching that draw attention to connections – both within and between examples – in order to explore the potential objects of learning that are brought into being in the classroom space and thus what is made available to learn. Our focus is on exploring differences in teaching over time, in the context of learning study style development activity of additive relation problems in three Grade 3 classes in South Africa. In a context where highly-localised and fragmented instruction has been noted, this study reports on the nature and extent of changes in connections in instruction over time. The application of a coding framework focused on simultaneity and connections in teaching points to a richer range of structural relationships within examples, and more connecting work between examples in the second year in comparison to the first year.

Communication and reasoning in mathematics is described as important for student learning. Hence, teachers have a central role to invite students to engage in reasoning and collective problem solving. One common way to promote this is for the teacher to ask questions. However, there is limited knowledge of how teachers make use of the input provided by students on the asked questions in their mathematics teaching to support further learning and mathematical inquiry. In the present study, we investigated qualitative differences in how preschool class teachers responded to and incorporated 6-year old students’ input in teaching about numbers and arithmetic. The data gathered for analysis consisted of fieldnotes collected through observations of 145 mathematics teaching episodes in 95 classes. To make it possible to map the qualitative different ways teachers responded to and incorporated students’ input in their teaching, the Mediating Primary Mathematics framework was used as an analytical tool. The results show that teachers responded to and incorporated student input in different ways in their teaching. In almost 2/3 of the teaching episodes, teachers stopped at only briefly confirming the input given as right or wrong, or just gave generally encouraging responses to the student. In just under 1/3 of the teaching episodes, teachers took advantage of and incorporated student input in their teaching to advance and verify their mathematical reasoning. This highlights that teachers may develop their ways of responding to and elaborating on students’ input in teaching, which could improve students’ opportunities for learning mathematics.

The potential for student mathematics learning lies both in the teacher ability to ask questions and to follow up and incorporate student input into the teaching of a specific content (Murata, 2015). Swedish students are expected to engage in reasoning and collective problem solving in highly communicative teaching practice. To improve these learning situations, it is important to understand how teachers are making use of student input in teaching. In this study we seek to map and understand how teachers in preschool classes respond to and incorporate student input in mathematics teaching.

This paper reports on findings from a study focusing on mathematics teaching in preschool classes in Sweden (6-yearolds). The data consist of 145 observations (from 95 individual teachers) of mathematics teaching relating to whole numbers. The data for analysis consist of fieldnotes and was collected during fall 2021. The MPM-framework “Mediating Primary Mathematics” (Venkat & Askew, 2018) was used as an analytical tool to identify the ways teachers in preschool classes respond to and incorporate student input in their mathematics teaching. Following the four levels in the framework, the results show that in 61,4% of the observations, teachers give students very little opportunity to contribute with input beyond short responses, merely confirming or giving generally encouraging responses to the student. This is to becompared to 29,7% of the observations, where teachers take advantage of student input by incorporating these into the teaching situation to advance or verify students’ mathematical reasoning. In the third largest group (8.3 % of the observations), the teachers pulled back or made no evaluation of the student input. Only in one case (0.7% of the observations), the teacher took advantage of the student input and both advanced and explained it further to support learner progression.

The results raise questions about how teachers’ ways of responding to and elaborating on students’ input might influence students’ opportunities for learning about numbers. In particular, when teachers advance, verify, and explain student input, significant connections and justifications for solution methods are highlighted. 

This study aims to investigate the information that can be collected about counting

strategies when tracking a person’s eye-movements when he/she is doing a mental

calculation on a given expression. Eye-tracking is based on the study of what a person look

at and the eye-movements between fixations. This study relies on the eye-mind hypothesis

which claims that there is a connection between eye movements and the cognitive

processes. The result is based on data from two participants and three tasks. The tasks

consisted of mathematical expressions with addition and subtraction that the participants

solved with mental arithmetic. The data from eye-movements is of such quality that it is

possible to see indications on which counting strategies the participants are using and the

results are in-line with previous research. The data also show which number the participant

focus most on and how they begin to solve the task. On one task both participant gave an

incorrect answer and the eye-movement showed where the participants made a mistake.

Using this sort of eye-tracking investigation has both its limitations and its possibilities.

Syftet med den här studien är att sammanställa och problematisera forskning kring hur eye-tracking som metod kan användas för att synliggöra elevers förståelse för matematik. Eye-tracking är en metod där människors ögonrörelser studeras för att undersöka den visuella uppmärksamheten utifrån fixeringar och sackader. En eye-tracker är en apparat som används för att mäta ögonrörelser.  Studiernas undersökningar utgår från en hypotes om att det finns ett samband mellan ögonrörelser och kognitiva processer. Sammanställningen har sitt ursprung i en litteraturstudie där artiklar har funnits genom sökningar i olika databaser. Tolv vetenskapliga studier valdes ut baserat på inklusions- och exklusionskriterier, och analyserades i detalj. Dessa studier pekar på att eye-tracking kan visa skillnader mellan vilka strategier eleverna använder, skillnader mellan hög- och lågpresterande elever samt skillnader i kognitiv belastning. Studierna visar också att det finns flera möjligheter med eye-tracking som metod men lyfter också fram begränsningar. En möjlighet är att eye-tracking i vissa fall är mer objektiv än andra metoder och en annan är att eye-tracking kan visa strategier som eleven själv inte kan verbalisera. Begränsningar är att den kognitiva uppmärksamheten inte alltid stämmer överens med deltagarens visuella fokus samt att resultatet kan vara svårt att urskilja vid emotionella reaktioner.  Eye-tracking jämförs ofta med olika sorters intervjuer och flera studier förespråkar att kombinera eye-tracking med andra metoder för att samla in kompletterande data. I slutet av texten lyfts de etiska dilemman fram som kan uppstå vid eye-tracking.

Denna presentation beskriver en pågående studie om hur associativitet kan uppfattas och hanteras i matematikundervisning. Studien syftar till att beskriva elevers olika sätt att förstå associativitet och utifrån dessa designa lämpliga undervisningsinslag för att främja elevernas algebraiska förståelse. Vissa delresultat beskrivs.

Previous research indicates that manipulatives, like base-ten blocks, not necessarily strengthen students’ understanding of numerical place-value and the decimal numeral system. This study takes its starting point in the hypothesis that to create functional teaching situations with base-ten blocks, it is necessary to first know students’ prior understanding of such manipulatives. Therefore, here we present a phenomenographic analysis of students’ understanding how such manipulative material can be used to represent multidigit numbers. The data was collected from individual interviews with 58 students in grade 1 (6-7 years old).We identify six qualitatively different categories of students’ understanding, and, based on these, suggest implications for the design of teaching situations.

Learning study is an adapted version of lesson study developed in Hong Kong and Sweden. It has commonalities with lesson study but is framed within a specific pedagogical learning theory – variation theory. Central in variation theory is the object of learning and what is critical for students’ learning. Hence, as with lesson study, it is a collective and iterative work where teachers explore how they can make the object of learning available to students, but what characterises learning study is the use of a specific learning theory. In this process, special attention is paid to the critical aspects of the object of learning. We argue that to identify the aspects that are critical, the aspects need to be verified and refined in classrooms. In this chapter, we demonstrate how teachers gain knowledge about such critical aspects. Particularly, we show how these critical aspects cannot be extracted only from the mathematical content or the students pre-understanding alone, but evolve during the learning study cycles. For this we use a learning study about the mathematical topic of rate of change in grade 9 in Sweden as an illustration. We describe how an analysis of how students solved tasks in pre- and post-test and during the lessons, as well as how the mathematical content was presented in lessons, helped the teachers identify what was critical for learning to understand and express the rate of change for a dynamic situation.

The upper secondary school's transition to digital education in spring 2020 showed Swedish schools'preparedness to redirect to distance learning (The Swedish Schools Inspectorate, 2020; The SwedishNational Agency for Education, 2020; ifous, 2020). The processes during the transition revealed thattechnology not only occupied the physical space and replaced the walls of the classroom, but also tookover every day learning practices. In this situation the teachers´ “cultural capital” (Bourdieu 2011) regarding technology became crucial to develop teaching that matched a context that wascharacterized by a high degree of digitalization.

In order to create in-depth knowledge of the conditions and practices that facilitated the transition,a data set has been constructed based on upper secondary school teachers’ stories about theirpersonal experiences of the transitional period. In total, the analysis is based on 52 stories. A surveywith partly open questions regarding the transition to distance education, answered by 1745 students,functions as a “mirroring” tool to these stories. The narrative approach (Riessman, 1993)methodologically serves to access qualitative aspects and different dimensions of practice in change.Theoretically, Kemmis (2019) describes the importance of paying attention to practices in motion andto provide for and support practices in the making, since human development and the individual'slearning are based on constantly changing practices.

The data material has been collected in two Swedish regional upper secondary school settings andhave been analyzed thematically based on the principle of researcher triangulation (Cohen et al, 2011;Lincoln & Guba, 1986). The results reveal three overarching thematic practices, collaborative,communicative, and creative, as crucial in the development work with digital and distance learning.The results offer indications that if these three practices work together satisfactorily for the teachers,the quality of the teaching, and, consequently, of the learning, will also develop. 

Uppdragsutbildning bedrivs vid många lärosäten som en lite undanskymd verksamhet vid sidan om grunduppdragen utbildning och forskning. Vid Högskolan för lärande och kommunikation, Jönköping University har uppdragsutbildning under många år fungerat som en sorts fönster ut mot de verksamheter vi utbildar våra studenter för. Lärarutbildare som håller i våra uppdragsutbildningar vittnar om möten med yrkesverksamma som stimulerar och leder till reflektioner som inte är möjliga med studenter som ännu saknar denna erfarenhet. Detta ger även en fördjupad förtrogenhet med det som våra studenter kommer möta efter sin utbildning hos oss. I den här boken delar kollegor vid högskolan med sig av de erfarenheter de fått i olika möten med yrkesverksamma. Detta är något vi tar med oss in i våra utbildningar och kan ses som en kunskapsproduktion i termer av högskolans beprövade erfarenhet och vetenskapliga grund.

In this study, the purpose is to analyze how growing geometric patterns occur in mathematics textbooks. The study proceeds from the variation theory and its conceptual apparatus. Therefore, the main focus is how the contextual variation occur in mathematics textbooks. It will also be an analyze of other variation parts, in order to see how contextual parts and other variation parts affects students’ opportunities of learning.  

The study proceeds from a qualitative method where a content analysis is used in the analysis of mathematics textbooks. In the analysis, four different teaching materials are chosen, all from sixth grade. These gave the opportunity to find many different variations in the mathematics textbooks.

The analysis of mathematics textbooks led to following result. Analyzing contextual variation parts and other variation parts, showed a great variation in all the mathematics textbooks. However, variation differed in how it occurred and how it gave opportunities for students to learn. 

Based on the analysis, the results from it can supply how variation were used in the mathematics textbooks. To use the mathematics textbooks as single resources is an opportunity for teachers. On the contrary, it is important that the teaching uses a content with variation in order to develop students’ knowledge.

This thesis comprises three original papers about learning rate of change in school mathematics. The overall aim of the thesis is to contribute with knowledge in this particular area of research. Within this aim, each of the original papers contribute with its own perspective. The theoretical framework used in the thesis is variation theory of learning (Marton & Booth, 1997; Marton, 2015), by which learning is seen as experiencing a phenomenon in a new way. This theoretical point of departure is reflected in the research question of the thesis: What is critical to discern to use and express rate as a measure of the relation of changes in two quantities? The empirical study was conducted as a learning study (Pang & Marton, 2003). A learning study is an iterative, interventional research arrangement in which teachers and researchers collaboratively explore a specific ability, the object of learning, worthwhile for the students to learn. The object of learning related to this thesis, ‘to express the quantitative rate of change of a linear relation’, was explored in a series of three research lessons at a secondary school. Data consists of students’ responses to written pre- and post-tests, and lesson videos. Some data have been analysed during the on-going empirical study and some data have been analysed after it was concluded. Principles from variation theory have been used as tools for analysis throughout the study. Main results of Paper I demonstrate how two critical aspects are identified and revised through the process of learning study. In Paper II the results indicate that qualitatively different questions in a task may affect students’ ways to relate changes in two quantities. The results of Paper III suggest how different perspectives of slope may promote homogeneity as an aspect of rate. Results also comprise six critical aspects of the object of learning, four of which was identified by revisiting the results of Paper II. In summary, the critical aspects also specify the meaning of a covariational perspective of rate. Results are discussed in relation to previous educational research about rate of change, covariation of quantities and students’ conceptions of rate and slope. Further research directions are suggested.