In this article we report on one of four schools involved in a research project aiming to develop knowledge about the way in which teaching knowledge is conveyed with the help of information and communication technology. A particular interest was issues of fundamental values. Two classes in Grade 9 of Swedish compulsory school were studied, by observations and interviews, when seeking information about international conflicts on the Internet. The results show that most students searched for and collected information with a focus on mainly one dimension: the temporal dimension of the conflict in question. How this focus on chronologically ordered historical events affected the students' ability to develop the capacity for individual standpoints on questions concerning ethics, morals, equality, and democracy by way of studying conflicts is discussed.

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.

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. 

In this paper, we explore how learning experience can best be described and relate to the teaching enactment so as to inform teaching practices in specific contexts. Two lessons dealing with the same topic in Primary 4 in Hong Kong schools were videotaped and a post-lesson diagnostic worksheet was given to the students. The aim of the study was to identify differences between the two lessons in what was made possible for learning on the topic, and to relate those differences to students’ perception and outcomes in learning. The data collected were analysed from the theoretical assumption that “variation” in the “object of learning” is essential to creating learning opportunities in the classroom. The results showed a critical difference in the way the teachers handled the object of learning. This was in turn found to have contributed to the opening of different “patterns of variation and invariance” in and thus, different possibilities for learning the object of learning. This difference was also reflected in the students' report of their perception and outcomes in learning.

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.

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.

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.

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.

In this paper, we report how 5-year-olds’ arithmetic skills developed through participation in an 8-month-long intervention. The intervention program aimed to enhance the children’s ways of experiencing numbers’ part-part-whole relations as a basis for arithmetic skills and was built on principles from the variation theory of learning. The report is based on an analysis of assessments with 103 children (intervention group n = 65 and control group n = 38) before and after the intervention and a follow-up assessment 1 year after the intervention. Our findings show that the learning outcomes of the intervention group were significantly higher compared to those of the control group after the intervention and that differences between the groups remained even 1 year after the intervention. In particular, the results show that children participating in the intervention group learned to recognize and use part-part-whole relations in novel arithmetic tasks.

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. 

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.

Within the phenomenographic tradition the object of learning depicts the capability that is to be learned by the learner (Marton &Booth, 1997). The object of learning can be defined by its critical aspects, since they are seen as necessary for the learner to discern in order to learn. The aim of this paper is to discuss the nature of the object of learning by investigating how its meaning can change as it is explored by teachers. We analyzed seven recorded meetings in which four teachers and a researcher discussed the nature of the object of learning while they were planning, analyzing and revising a lesson. We found that the meaning of the critical aspects identified changed for the teachers due to the discussion and analysis of the lessons and thereby the meaning of the object of learning changed also. From at first being defined, they later become refined and specified as the teachers acquired deeper understanding of the object of learning. Distinctions were made to separate out what was of significance for the object of learning and what is not (the objects external horizon). Furthermore, an exploration by the teachers was made of how different aspects relate to each other (the objects internal horizon). The findings indicate that qualitative differences in teachers’ experience of the object of learning emerge through the collaborative investigation.

This paper investigates how teachers explore the object of learning in a learning study in mathematics. The object of learning depicts the capability that is learned by the learner. For each object of learning there are critical aspects that the learner needs to discern. The aim of the paper is to describe the meanings that the critical aspects have for the teachers at different stages in the learning study process. The study is a part of a lager study in Sweden investigating teachers’ learning from learning studies (LGK-project). In this paper we report on the analysis of seven collaborative meetings, with four teachers and a researcher, from the point of view of how the critical aspects and object of learning are discussed using the framework of variation theory. The object of learning was that students in the 7th grade would understand that in a division, with a denominator between 0 and 1, the quotient becomes larger than the numerator. The study shows that the meaning of the critical aspects, identified by the teachers, changes for the teachers due to the discussion and analysis of the lessons. From at first being defined, they later become refined and more explicit as the teachers get deeper understanding of the object of learning. Furthermore, student learning is enhanced, most likely, by the changes made in the teaching due to the teachers’ deeper understanding of the object of learning.

Within the phenomenographic research tradition, the object of learning depicts the capability that is to be learned by the learner. It has been argued that the object of learning cannot be fully known in advance since what is to be learned depends on the learners as well as on the content taught. The object of learning and its nature needs to be explored. In this paper, we analyze how a group of teachers collaboratively investigated an object of learning when they planned, enacted, analysed, and revised a mathematical task. We describe distinctions made by the group in the inquiry into teaching and learning, and how delimitations and distinctions made transformed the teaching and meaning of the object of learning.

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.

This study concerns pupils’ experience of unit and non-unit fractions of a discrete quantity during specially designed lessons. The aim was to explore pupils’ understanding of operations such as b/c of a in lessons where the teachers were aware of some pupils’ difficulties beforehand and what needed special attention. Five classes were involved in the study and 10 video-recorded lessons and written pre- and post-tests were analysed. Even though the lessons were designed for learning how to operate with both unit and non-unit fractions, we found that more pupils could solve items with unit fractions than with non-unit fractions. We found that few pupils in this study had difficulties with equal partitioning. Instead, it seemed difficult for some pupils to understand the role of the numerator and denominator and to differentiate between the amount of parts and the amount of objects in each part, and some pupils did not differentiate between the numbers of units and the amount of objects within a unit. This study identified some critical aspects that the pupils need to discern in order to learn how to operate with unit and non-unit fractions of a discrete quantity.

The variation theory of learning emphasizes variation as a necessary condition for learners to be able to discern new aspects of an object of learning. In a substantial number of studies, the theory has been used to analyze teaching and students’ learning in classrooms. In mathematics education, variation theory has also been used to explore variation in sets of instructional examples. For example, it has been reported how teachers, by using variation and invariance within and between examples, can help learners to engage with mathematical structure. In this paper, we describe the variation theory of learning, its underlying principles, and how it might be appropriated by teachers. We illustrate this by an analysis of one teacher’s teaching before and after he participated in three lesson studies based on variation theory. Both the theory and the empirical illustration focus on ‘what is made possible to learn’ in different learning situations. We show that in the two analyzed lessons, different things were made possible to learn.

Twelve lower secondary schoolteachers in mathematics and science were asked to teach a topic of their choice during a lesson that was video-recorded. We were able to analyse 10 of the cases and we found that all of them were similar in one respect: concepts and principles were introduced one at a time, each one followed by examples of the concept or principle in question, apparently to highlight its essential meaning. All the teachers participated in three modified lesson studies with three cycles in four different groups during three semesters. The modified lesson studies were built on a theoretical idea supported by a large number of recent studies. The theory states that new meanings (of concepts and principles, for instance) are learned through engaging with instances of contrasting concepts and principles. The core idea is that new meanings derive from differences, not from sameness. After the three modified lesson studies, the teachers were asked to once again teach the same topic as in the recorded lessons before the lesson studies. The new lessons were also recorded and the analysis showed that there was one thing in common in all cases: all of the 10 teachers dealt with the relevant concepts and principles in relation to each other (i.e. simultaneously) and not one at a time. By thus bringing out the differences between them, their meaning was made possible to grasp for the students. The study lends support to the conjecture that the modified lesson study is a powerful tool for enabling teachers to structure the content of their teaching in accordance with a principle that is more powerful in making learning possible, even if this contradicts their taken-for-granted practice.

In this paper we focus on variation of the design and the implementation of a specific task during three mathematics lessonsin the 8thgrade in a learning study (Marton & Tsui, 2004; Runesson, 2008). The theme of the lesson was division, with a denominator between 0 and 1. The teachers wanted their students to understand that when dividing with a denominator less than 1, the quotient is larger than the numerator. Four teachers collaboratively planned, analyzed and revised three lessons in a cyclic process. The study shows that the implementation of the task changed between the lessons. Although the same task wasused in the lessons, the way it was enacted provided different possibilities to learn.

In this chapter we focus on variation of the design and the implementation of a specific task during three mathematics lesson in the 8th grade in a learning study (Marton & Tsui, 2004; Runesson, 2008). The theme of the lesson was division, with a denominator between 0 and 1. The teachers wanted their students to understand that when dividing with a denominator <1, the quotient is larger than the numerator. Four teachers collaboratively planned, analyzed and revised three lessons in a cyclic process. The study shows that the implementation of the task changed between the lessons. Although the same task wasused in the lessons the way it was enacted provided different possibilities to learn.

Purpose: The purpose of this paper is to add to the discussion about practitioner research in schools – by addressing mechanisms and systematic strategies based on theory in a research model, which enables the creation of knowledge products that enhance student learning and are sharable between teachers.

Design/methodology/approach: The research question is the following: Can a specific form of teachers’ research produce practice-based knowledge relevant beyond the borders of the local school context? This question is addressed through empirical examples from previously published papers on learning studies in natural sciences, mathematics and language.

Findings: This paper promotes the view that teachers in learning studies can create practical public knowledge relevant beyond their local context. The authors suggest that learning studies and variation theory can offer teachers mechanisms to create such public knowledge.

Originality/value: The paper proposes that teachers’ collaboration in professional learning communities, as in a learning study, not only has the capacity to increase students’ and teachers’ learning, but it can also be used to create practical public knowledge. 

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.

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.

In Swedish research on reading instruction, it has been emphasized that students already in middle school (grades 4-6) need to develop literary competence and receive guidance on how they can process a text in order to develop in-depth reading comprehension. The aim of the article is to contribute with knowledge regarding how teaching that aims to develop the ability to draw conclusions about a fictional protagonist’s character traits can be designed. A specific activity focusing literature education that was developed within the framework of a Learning study in grades 4 and 5 was studied in more detail. The purpose of the teaching unit was that the students would develop their abilities to draw conclusions about a protagonist’s character traits in a fictional text. 15 hours of video-recorded lessons, where the same story was used in all cycles, were analyzed using variation theory. The analysis shows that the different cycles offered different learning opportunities depending on how the teacher gave instructions to the learning activity, how the task was formulated and designed, if the text and the protagonist’s character traits came into focus and related to each other, and which aspects were opened up as dimensions of variation.