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.

Developing mathematics education contributing to equity in preschool class

In the article, we present and discuss preschool class students’ ability to see and use number structures to determine number and how this ability develops after participating in interventions during one academic year. A total of 361 students who participated in the interventions or in regular preschool class teaching were interviewed early in the autumn term and at the end of the preschool class year. One task containing a spatial pattern in the interview material forms the basis for identifying how students experience and use number structures, quantitatively and qualitatively. The analysis is further based on a variation theory view of learning, where the way of experiencing a phenomenon, in this case numbers and structures as they appear in a figure arranged in a spatial pattern, has significance for what the student can do with numbers, e.g., how to determine numbers. In particular, implications of the results for developing mathematics teaching in preschool class and students continued arithmetic learning are discussed.

In this paper, we present what 361 preschool class students discern of number structures in a task containing a spatial pattern. The analysis, grounded in the variation theory of learning, focuses on the students’ discernment of composite units as parts of a larger whole. To discern a task as a structure of parts and whole is presumed to be significant for what the student can do with numbers, e.g. to determine the exact number of objects in a spatial pattern. The results reveal four different ways in which students discern structures, each one with implications for the students’ abilities to determine the number of the pattern. The results induce implications for the development of mathematics teaching in preschool classes and also raise further questions for research about students’ arithmetic learning.

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 presentation we share experiences of having implemented a French teaching program in a Swedish grade 1 classroom for the duration of one school year. The French teaching program, Arithmetic and Comprehension at Elementary School (ACE) takes a structural and relational approach to teaching and learning numbers in early grades. The aim of this presentation is to describe the implementation process of the ACE-program in a Swedish context and to discuss how principles of variation theory can develop the program. Following the structure and the principle of the program we found that the program, with small adjustments, is possible to implement in a Swedish grade 1. We suggest that the program may be further developed by adding principles of variation theory.

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

Developing the teaching of numbers and number relations in preschool class

The purpose of this article is to contribute knowledge about how an intervention, focused on teaching numbers and number relations in preschool class, can develop teaching. In the intervention, we employed a practice-oriented model based on planning, teaching, evaluating, and reflecting on teaching in an iterative process, in collaboration with colleagues and researchers. We carried out this intervention over four years, during which we identified key components of the model that were essential for the development of mathematics teaching. We also changed these components based on the purpose and local context. A key component is the reflection documents that were developed and how they contributed to reflection on teaching and learning. Another key component is the research-based activities that we refined. The knowledge product includes a teaching guide with documented and designed teaching activities with reflections on the teacher's enactment of the mathematical activities as well as the teaching model.

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.

Not all students in early grades develop efficient strategies for solving subtraction tasks. In this paper, we examine subtraction teaching in the 1–20 number range. We analyzed two first-grade lessons addressing similar subtraction tasks, using variation theory to identify what aspects of the content were foregrounded in the teaching. The analysis showed that both lessons supported the discernment of aspects of subtraction. However, the learning opportunities differed depending on what aspects the students were enabled to discern, the recurrence and variation of the elicited aspects, and how the teachers made the aspects visible by means of representations. The findings highlight how teaching can make it more likely for students to experience aspects of learning necessary to solve subtraction tasks such as 13–5 = _ . Additionally, the findings show how subtraction can be taught to enhance strategies using number relations that are also useful in higher number ranges.

This paper examines differences in how three Grade 3 South African teachers responded to students’ incorrect answers in whole class teaching of the part-whole relationship in additive missing number problems. Nine video recorded lessons, taught by three teachers, were analysed, with attention paid to teaching episodes containing incorrect students’ answers. The variation theoretical analyses indicated differences in the ways teachers responded to incorrect answers. We argue that different ways of responding to incorrect answers may provide different learning possibilities.

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.

In this article, we present a coding framework based on simultaneity and connections. The coding focuses on microlevel attention to three aspects of simultaneity and connections: between representations, within examples, and between examples. Criteria for coding that we viewed as mathematically important within part-whole additive relations instruction were developed. Teachers’ use of multiple representations within an example, attention to part-whole relations within examples, and relations between multiple examples were identified, with teachers’ linking actions in speech or gestures pointing to connections between these. In this article, the coding framework is detailed and exemplified in the context of a structural approach to part-whole teaching in six South African grade 3 lessons. The coding framework enabled us to see fine-grained differences in teachers’ handling of part-whole relations related to simultaneity of, and connections between, representations and examples as well as within examples. We went on to explore the associations between the simultaneity and connections seen through the coding framework in sections of teaching and students’ responses on worksheets following each teaching section.

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. 

Mapping of Preschool Class Mathematics Teaching about Numbers - Qualitative Differences and Learning Opportunities

During the autumn of 2021, an observational study was conducted in 95 Swedish preschool classes with the aim of mapping and describing the quality of teaching. The framework "Mediating Primary Mathematics" was utilized to identify and assess variations in the quality of teaching about numbers, their properties, and usage, that is, how the mathematical content was treated and mediated in the teaching situation. The result revealed variations in how the mathematical content was mediated, which impacted the learning opportunities provided to students. While several examples of high-quality mathematics teaching were observed, areas for improvement were also identified. Artifacts were frequently used in teaching, although their usage did not consistently emphasize mathematical relationships and meanings. Furthermore, teaching rarely emphasized comparison and evaluation of different solution methods. Student contributions were often acknowledged, but it became evident that their input was rarely elaborated upon.

Ur abstraktet: Sedan 2019 har forskningsgruppen i matematik, vid Högskolan för Lärande och Kommunikation, Jönköping University, ett väl upparbetat samarbete med Vaggeryds kommun. Forskare och kommunens lektor har under tre år utarbetat ett hållbart samverkansavtal (ULF) där utveckling av matematikundervisning i förskoleklass stått i fokus. I det så kallade VA-MER projektet har forskare tillsammans med lärare i förskoleklass i en iterativ process utvecklat och systematiskt utprövat ett antal vetenskapligt förankrade undervisningsaktiviteter som ska främja elevers taluppfattning och problemlösningsförmåga. Det har resulterat i ett antal dokumenterade undervisningsaktiviteter som är tänkta att fortsättningsvis vara ett stöd i matematikundervisningen för såväl verksamma som nyanställda lärare i kommunens förskoleklasser. En fråga som har väckts under projektets gång är hur de grundläggande matematiska idéer ska förvaltas och utvecklas i årskurs 1. 

The aim of the symposium is to present and discuss the notion of “critical aspects” as a key concept for developing teaching and thus students’ learning. The concept “critical aspects” is central to phenomenography and variation theory, pinpointing what aspects students need to discern in order to develop their understanding of the object of learning in focus for the teaching. In four different presentations, with a focus on different age groups and subjects, the notion of critical aspects is discussed. In the first presentation the authors discuss how critical aspects can be used by teachers in primary school as a point of departure in mathematics teaching – in tasks as well as activities. In the second presentation the authors discuss how teachers can help kindergarteners to discern critical aspects through reading a picture book. In the third presentation the author presents a study focusing on primary teachers’ understanding of “growth mindset” and how critical aspects of this concept can be used as a foundation to inform teacher education. The final presentation discusses how teachers and teacher students understand the notion of critical aspects and what needs to be focused upon in teacher training in order for teachers/teacher students to grasp this concept in a qualified way. The symposium has scientific relevance as it opens up for different perspectives of a central concept in phenomenography and variation theory. It also has educational relevance, as the notion of critical aspects can be understood as foundational for developing teaching.

Children’s ways of handling numbers in arithmetic tasks has been studied extensively, providing us with insights about strategies for solving tasks and development of arithmetic skills (Carpenter & Moser, 1982). Children’s ability to decompose numbers is one important part of development, since the ability allows children to apply different strategies when solving tasks (Hunting, 2003). Children’s different ways of encountering numbers in simple tasks may give a comprehensive understanding of the challenges in learning to use numbers in proficient ways. When experiencing part-part-whole relations of numbers, the child needs to consider the parts and the whole simultaneously. This paper reports on an analysis of 103 individual interviews with 5-year-old children on two occasions during their last year in preschool in Sweden. We report on the analysis of one particular task illustrating children’s experience of numbers when partitioning seven hidden marbles into two parts. The specific research question was: What different ways of experiencing numbers by 5-year-old children were exposed in a partitioning task?

Variation theory (Marton, 2015) was used to analyse children’s ways of experiencing numbers and what aspects were critical to discern in order to solve the task. Variation theory emanates from more than thirty years of phenomenographic research, investigating different ways in which the same phenomena can be experienced. We found that children experienced numbers in six different ways: as number words, as names, as extents, as countables, as structure, or as known number facts. Our study shows that those ways of experiencing numbers that are foregrounding the cardinal, ordinal and the parts and whole simultaneously end up in plausible answers and the children initiate ways to handle the task in powerful ways. Consequently, if the children experience either the cardinal (e.g. numbers as extent) or the ordinal (e.g. numbers as names) they are not able to decompose the whole and thereby solve the task.

References

Carpenter, T. P., & Moser, J. M. (1982). The development of addition and subtraction problem-solving skills. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective. Hillsdale, NY: Lawrence Erlbaum.

Hunting, R. P. (2003). Part-whole number knowledge in preschool children. Journal of Mathematical Behavior, 22(3), 217-235.

Marton, F. (2015). Necessary conditions of learning. New York: Routledge.

This presentation aims to illustrate how the integration of theory and teaching experiences can support pre-service teachers to generate knowledge for teaching mathematics. The pre-service teachers took part in a 5-week mathematics education course in a teacher education program in Sweden, in which a theory driven lesson study model entitled learning study were established to deepen the teachers’ awareness about the relationship between instruction and student learning. The course design consisted of two intervention cycles in which the pre-service teachers used variation theory as well as their teaching experiences as tools for lesson design and redesign. To enable us to identify if and how theory and experiences drawn from practice were employed and realized, we collected data from the pre-service teachers’ individual written reports at the end of the course. The unit of analysis from the written reports were 64 redesigned mathematical tasks and associated reflections about the reason for redesign. We found five different categories of how tasks were modified: expanding tasks, making tasks more explicit, making tasks less explicit, bringing metaphors and representations to the foreground, and creating new tasks. The findings could contribute to reflections and discussions about course design in teacher education and in what way the integration of theory and practice can be regarded as an important key point for pre-service teachers’ professional and sustainable learning. 

The theory-practice divide in teacher education is commonly viewed as there aretwo separate entities – theory and practice. However, in practice-based researchapproaches, theory is commonly integrated with existing practical knowledge withthe aim to deepen teachers’ knowledge about practice or to create new knowledge. In this study, we examine 30 pre-service teachers taking part in a 5-week course in ateacher education program in Sweden, in which an action-research approach termed Learning study was used to deepen the pre-service teachers’ thinking and reasoning about mathematics teaching in order to develop primary student learning. Variation theory was used as a tool to support the pre-service teachers’ reflections on how different ways of structuring the mathematical content are related to student learning outcomes. This research aims to illustrate how the integration of theory and teaching experiences from the 5-week mathematics education course supported pre-service teachers’ generation of knowledge about teaching and learning mathematics. In this study, we regard mathematical tasks created by the pre-service teachers and usedin the lessons as generated knowledge about the practice of teaching. Data were collected during the course and consist of written reports about task refinements inthe pre-service teachers’ lessons. We identified five different ways of re-designing the tasks: expanding tasks, making tasks more explicit, making tasks less explicit,bringing metaphors and representations to the foreground, and creating new tasks.

I den här studien har vi undersökt på vilka sätt 41 grundlärarstudenter, med inriktning mot årskurs 4‒6, designade och förändrade matematikuppgifter för att utveckla kunskap om undervisning och skolelevers lärande av specifika ämnesinnehåll. Under höstterminen 2018 deltog lärarstudenterna i en fem-veckors matematikdidaktikkurs inom ramen för grundlärarprogrammet på Högskolan för lärande och kommunikation, Jönköping University. Kursen är till stor del uppbyggd utifrån learning study‒konceptet (Marton & Pang, 2003), vilket kort kan beskrivas som ett kollegialt, iterativt och intervenerande utvecklingsarbete med inslag av ämnesdidaktiskforskning (Runesson Kempe, 2016). I den aktuella kursen innebar detta bland annat att arbetet utgick från studenternas egna frågor om undervisning och lärande men i samråd med lärare från kursens samverkansskolor. Samverkan med skolor i regionen är en avgörande faktor för kursen eftersom det möjliggör för studenterna att undersöka elevers förståelse av något ämnesinnehåll och utifrån det forma och genomföra undervisning. Vidare var studenterna i kursen indelade i grupper (3‒4 studenter per grupp), i vilka man systematiskt planerade, genomförde och analyserade undervisning i två cykler. I detta arbete användes ämnesdidaktisk forskning och variationsteorin (Marton & Pang, 2003) som redskap för att komma åt vad skoleleverna behöver få syn på för att utveckla kunskap i förhållande till specifika ämnesinnehåll. Studiens data samlades in under kursen och består av lärarstudenternas skriftliga reflektioner om de uppgifter som de designade och prövade i lektionerna och om hur och varför dessa förändrades eller borde förändras. Syftet med studien är att identifiera och beskriva de olika sätt lärarstudenterna förändrade matematikuppgifterna utifrån ett variationsteoretiskt perspektiv. I presentationen visas några sådana exempel.

In this paper analysis of Grade 3 mathematics teaching in South Africa shows evidence of associations between teaching and learning outcomes in an adapted learning study. The intervention dealt with partitioning and part-part-whole relations, taking a structural approach within tasks and representations. Our analysis of this teaching emphasizes simultaneity of examples, and connections within and across examples and representations. This analysis indicated differences in enactment of a jointly planned lesson that related to different patterns of learning outcomes between the three classes. Episodes of teaching containing work with representations marked by connections and simultaneity closed gaps in learning outcomes seen in the pre-test.