This paper describes an approach to teaching that enhances pupils’ learning in mathematics. The model described – Learning study – involves teachers and researchers cooperating in an iterative process, gathering data about teaching and pupils’ learning, analysing the data, planning and revising their teaching. A particular theoretical framework was used as a guiding principle when designing and analysing learning. The goal was to identify aspects critical for learning the angle concept. It is demonstrated how the teachers were able to identify the critical aspects and change the teaching in a way that promoted pupils’ learning. What these critical aspects may entail and what teachers and researchers can learn from a Learning study is discussed.

The aim of this paper is to illustrate how the topic of fractions can be taught differently by making a comparison between two cultures. We have studied mathematics teaching in classrooms in Hong Kong and Sweden. One of our basic assumptions is that the way in which the content is taught in a classroom has an important implication for what students may possibly learn. With reference to the framework of Variation Theory, two different spaces of learning are delineated. The Hong Kong lesson demonstrated a pattern of many juxtaposed variations, whereas the Swedish lessons presented a pattern of sequential and wide spreading character.

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.

PurposeThe 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/approachThe 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.FindingsThis 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/valueThe 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 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.

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.

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.

Let us imagine two different grade 6 classrooms where the aim of the lesson is the same; to calculate examples such as ¾ of 12 = 9. When introducing this, in one of the classrooms, the teacher demonstrates a method for computing; "divide the integer (12) by the denominator (4) and multiply the quotient (3) by the numerator (3)."

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.

Purpose - The purpose of this paper is to discuss two theoretical frameworks, Pirie and Kieren's work (Pirie and Kieren, 1994) and variation theory of learning (Marton, 2015) in relation to lesson/learning study and mathematics teaching and learning.Design/methodology/approach - The point of departure is the article: "Folding back and growing mathematical understanding: a longitudinal study of learning" (Martin and Towers, 2016) where it is demonstrated how Pirie and Kieren's work (1994) and particularly the notion "folding-back" can be used as the theoretical framework in lesson/learning study. By dealing with similar arrangements and different theories, the two frameworks are contrasted.Findings - It is suggested that the theory appropriated must be in resonance with the aim and focus of the study the theoretical perspective taken since it has implications for what becomes the focus of the process and subsequently the results of lesson/learning study.Originality/value - This paper contributes to the discussion about how a more theory-informed lesson study and a broader theoretically framed learning study would improve and change the scope and progress of the two.

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.

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.

Purpose– It has been suggested that, if pedagogical and learning theories are integrated into lesson and learning study, a systematic construction of pedagogical knowledge is possible (Elliott, 2012). In this Special Issue, it is reported how theory and theoretical concepts can add value to lesson and learning study. The purpose of this paper is to introduce the Special Issue and explore the above concepts.Design/methodology/approach– This paper presents the Special Issue papers thematically and the main issues are discussed.Findings– Together the papers suggest that pedagogical theories and theorizing practice may contribute to the improvement of teachers’ practical knowledge and knowledge about teachers’ professional tasks and objects. Furthermore, some theories and theoretical concepts hitherto under-exploited in lesson and learning study are presented and discussed from the point of view how these might improve the quality of the studies.Originality/value– As a total, this collection of papers bring out issues about the role of pedagogical and learning theories and how these could inform lesson and learning study.

In this paper we focus on variation of the design and the implementation of a specific task during three mathematics lessons 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 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 was used in the lessons, the way it was enacted provided different possibilities to learn.Diferentes posibilidades para aprender con una misma tareaEn este artículo nos centramos en la variación del diseño y la implementación de una tarea específica durante tres sesiones de clase de matemáticas en octavo grado en un estudio de aprendizaje (Marton y Tsui, 2004; Runesson, 2008). El tema de las clases fue la división con un divisor entre 0 y 1. Los profesores querían que sus estudiantes entendieran que, cuando se divide por un divisor menor que 1, el cociente es mayor que el numerador. Cuatro profesores colaboraron, en un proceso cíclico, en la planificación, análisis y revisión de las tres sesiones de clase. El estudio muestra que la implementación de la tarea cambió entre las sesiones. A pesar de utilizarse la misma tarea en las sesiones, la manera en que se implementó proporcionó diferentes posibilidades para aprender.Handle: http://hdl.handle.net/10481/31597Nº de citas en WOS (2017): 4 (Citas de 2º orden, 2)Nº de citas en SCOPUS (2017): 4 (Citas de 2º orden, 2)