I artikeln presenteras och diskuteras förskoleklasselevers förmåga att se och använda talstrukturer för att bestämma antal, samt hur förmågan utvecklas efter att de deltagit i interventioner under ett läsår. Totalt intervjuades 361 elever som deltagit i interventionerna alternativt i vanlig förskoleklassundervisning. Intervjuerna gjordes under tidig hösttermin samt vid förskoleklassårets slut. En specifik uppgift innehållande ett spatialt mönster i intervjumaterialet utgör underlag för att synliggöra hur eleverna erfar och använder talstrukturer, både kvantitativt och kvalitativt. Analysen tar vidare avstamp i en variationsteoretisk syn på lärande, där sättet att erfara ett fenomen, i detta fall tal och strukturer så som de framträder i en figur ordnad i ett spatialt mönster, har betydelse för vad eleven kan göra med tal, till exempel på vilket sätt man kan bestämma antal. Särskilt diskuteras vilka implikationer resultaten har för utvecklingen av matematikundervisning i förskoleklass och elevers fortsatta aritmetiklärande.

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.

Learning to calculate with natural numbers by structuring seems promising but how this can be taught in a sustainable manner remains an open question. An eight-month-long intervention based on the idea of using a structural approach to addition and subtraction, and particularly bridging through ten, was implemented in four Swedish first-grade classes. One goal was that by the end of first grade, students would be able to solve tasks such as subtracting 8 from 15 by using part-whole number relations. In this paper, we report on learning outcomes from task-based interviews with intervention and control groups before, immediately after, and one year after the intervention, in order to investigate long-term effects and whether students used a structural approach when solving tasks in a higher number range in the second grade. In comparison to controls, students in the intervention group showed higher increases in their learning outcomes. Moreover, the intervention group used a structural approach to a larger extent when solving tasks in higher number ranges, whereas students in the control group more commonly used single-unit counting to solve such tasks. These findings have implications both for teaching and for research on students' development of arithmetic skills.

Under hösten 2021 gjordes en kartläggning av matematikundervisningen i 95 förskoleklasser i syfte att bidra med kunskap om vad kvalitet i undervisning kan innebära och vad elever ges möjligheter att lära om tal, tals egenskaper och dess användning. Analysverktyget "Mediating Primary Mathematics" användes för att identifiera skillnader i undervisningens kvalitet, det vill säga hur ett matematiskt innehåll behandlades och medierades i undervisningen. Resultatet visar på en variation i hur innehållet medieras och därmed vad elever ges möjligheter att lära. Goda undervisningsexempel har observerats, samtidigt som utvecklingsområden identifierats. Artefakter förekommer ofta, men används inte alltid på sätt som synliggör matematiska samband och innebörder. Lösningsmetoder får sällan stå i centrum för undersökning, jämförelse och värdering i undervisningen. Elevinspel bekräftas ofta, samtidigt som det blir tydligt att undervisningen sällan erbjuder utveckling och bearbetning av dessa.

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.

Purpose: The purpose of this paper is to explore whether and how principles from variation theory can contribute to the planning of teaching and learning beyond learning study.

Design/methodology/approach: We study whether and how principles from variation theory contributed to a group of teachers’ planning of teaching and learning about decimal numbers in Grades 4 to 7 working in Subject Didactic Groups – a collaborative arrangement suited to daily teaching. A theoretical thematic analysis approach was used when analyzing eight audio-recorded meetings and written documents.

Findings: The study shows that variation theory principles contributed to the teachers’ planning of teaching and learning. Two themes were identified: the theory contributed to the teachers being able to (1) specify what their students needed to learn and (2) design tasks that they anticipated would afford the opportunity to learn what was identified as being necessary to learn.

Originality/value: The paper demonstrates how variation theory can contribute to teachers’ planning of teaching and learning when used in a collaborative arrangement other than learning study. This leads into a discussion about variation theory being used separately from learning study and the benefits and limitations this other collaborative arrangement can have for gaining knowledge of what is to be learned and taught.

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.

The decomposition of numbers when solving subtraction tasks is regarded as more powerful than counting-based strategies. Still, many students fail to solve subtraction tasks despite using decomposition. To shed light upon this issue, we take a variation theoretical perspective (Marton, 2015) seeing learning as a function of discerning critical aspects and their relations of the object of learning. In this paper, we focus on what number relations students see in a three-digit subtraction task, and how they see them. We analyzed interview data from 55 second-grade students who used decomposition strategies to solve 204 − 193 =. The variation theory of learning was used to analyze what number relations the students experienced and how they experienced them, aiming to explain why they made errors even though they used presumably powerful strategies in their problem-solving. The findings show that students who simultaneously experienced within-number relations and between-number relations when solving the task succeeded in solving it, whereas those who did not do this failed. These findings have importance for understanding what students need to discern in order to be able to solve subtraction tasks in a proficient way. 

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. 

The teaching of mathematics cannot be thought of without considering the use of examples. The examples that teachers use during a lesson and how they mediate the example set is critical to what opportunities for learning are opened up during the lesson. In this article, we explore how a teacher mediates an example set with focus particularly on what is varied and what remains the same. The case that we draw on is taken from a larger learning study conducted in Grade 10 mathematics classes and the lesson that is used in this article was the last lesson in the learning study cycle. We use variation theory, specifically how the sequencing, pairing and juxtaposing of examples provides learners with opportunities to discern the critical aspect of the object of learning. We analyse the teacher's mediation of the example set on a micro level, as this enables us to illuminate and develop our argument, while simultaneously offering a detailed example of mathematics teaching. We argue that it is the systematic and deliberate structuring of variation within an example set in the midst of invariance coupled with the teacher's mediation of both planned and learner-generated examples that is critical for opening opportunities to learn.

The use of variation theory as a tool for designing lessons has been widely applied and received a lot of attention e.g. in mathematics education. More limited attention has been given to the use of variation theory as a means to analyze teaching and learning. This theoretical paper addresses the connection between the afforded and experienced variation when analyzing teaching and learning with a variation theory framework. Variation theory is in this case used as a tool for analyzing the relationship between teaching and learning, exploring the enacted, and lived object of learning. The connection between what is afforded in teaching and what students learn, can in this way be described in commensurable terms. With examples from a study about students’ learning of number relations in mathematics in first grade, we highlight this connection by illustrating differences identified in the teachers enactment of the same task, and what the students’ learned.

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.