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.

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. 

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 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.

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.

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 present how experiences gained from a theory-informed lesson study – learning study (LrS) – in regard to a specific learning goal can be shared and used by other teachers in new contexts.

Design/methodology/approach: A group of teachers worked together in a cyclic, iterative process of planning, evaluating and revising teaching. The aim was to provide possibilities for grade 2 and 3 students to become familiar with negative numbers. The teacher group came to the conclusion that the students needed to be able to differentiate some aspects of negative numbers. The conjecture was put to the test in a follow-up study (FS) with five new teachers and eight classes. One lesson was taught based on the empirical findings in the LrS.

Findings: The results suggest that teachers’ collaborative work has possibilities to produce knowledge about critical aspects of learning that can be communicated and adopted in new contexts. The teachers in the FS were able to make sense of the results from LrS and incorporate the critical aspects in their teaching in a way that enhanced students’ learning.

Originality/value: It is demonstrated that teacher collaboration in LrS can create knowledge that goes beyond the border of the local context. 

The aim of the paper is to demonstrate how a theory-informed lesson study can be a form of practice-based research where knowledge is generated within the process of teachers’ actions. Learning study shares features with lesson study, such as the iterative design, the teacher driven approach and with attention to student learning, but is guided by a theoretical framework. The dominant theory has been variation theory. In learning study, the focus is the object of learning and what must be learned to make the object of learning one’s own. A learning study about learning and teaching negative numbers to young pupils (age 8-9) in a Swedish context is used as an example. Our proposal is in resonance with Morris’ and Hiebert’s (2011) suggestion; that lesson study is a system that can generate instructional products that are sharable and open for improvement by other actors. The ‘instructional product’ from learning study is a theoretical description of the object of learning, how it is constituted and can be taught. In the learning study reported, three teachers worked in collaboration to identify the critical aspects for realizing the existence of negative numbers. The critical aspects emerged and were successively specified in the process and as a result of a thorough analysis of data on pupils’ learning and the lessons. 

In this paper, we present how experiences gained from a theory informed Lesson study in regards to a specific learning goal can be shared and used by other teachers in new contexts. A group of teachers worked together in a cyclic, iterative process of planning, evaluating and revising teaching. The aim was to provide possibilities for grade 2 and 3 students to become familiar with negative numbers, thus to extend the number range from N to Z. Based on theoretical tenets about learning and empirical findings in the study the teacher group draw the conclusion that the pupils needed to be able to differentiate some aspects of negative numbers. The conjecture was put to the test in a follow-up study with five new teachers and eight classes. One lesson was taught based on the empirical finding in the Lesson study. When learning gains from pre- to post-test in these classes were compared to those in the Lesson study, similarities were found. It is suggested that accumulate evidence about ‘what must be learned’ across different classroom settings can be gained through a theory informed and goal oriented Lesson study.

In this article a theory of learning, variation theory, is presented in order to address the ‘provocation’ created by the relationship between theoretical positions and findings in research and what these do and do not disclose. I demonstrate how this can be used as an analytical tool for studying classroom learning in mathematics by juxtaposing an analysis of the same data made from other theoretical positions with that from a variation theoretical perspective. By this means, I demonstrate that the inclusion of what is learned, i.e., the object of learning, is significant for understanding classroom learning. Further, variation theory is suggested as a complement to other theoretical perspectives due to its power to reveal constraints on what it is possible to learn in mathematics classrooms.

The Learning study could be considered as a form of teacher research aiming at understanding and improving practice and building theory. It is focused on the development of teaching-learning units, guided by a theoretical framework –variation theory and aims to accomplish the learning of specific objects of learning. The knowledge product is designed to help the professionals to solve problems with teaching and learning they encounter in their everyday practice. Hence, it is more connected to professionals’ tasks than to professionals. Just like the Lesson Study it creates joint and shareable knowledge products in terms of lesson plans which could serve as a resource for other teachers. A significant feature of the Learning Study is however, its possibility to theoretically describe the results of the specification process regarding objects of learning within different subject matter areas (Carlgren, 2012). In the presentations we will deal with issues about the knowledge products of the Learning study as well as Learning study as teacher research.