Search results

The purpose of this paper is to investigate what might be relevant to younger children’s understanding of a number line and how teaching can be designed according to variation theory to give pupils the opportunity to develop an understanding of the number line as a tool for mathematical thinking.

A Learning Study was conducted by mathematics teachers at Sjöstadsskolan in Stockholm. Variation theory was used as a theory of learning. In order to use the number line as a tool in mathematical thinking, pupils have to discern what is important for this form of representation of the numeral system. Critical aspects from other studies with similar learning objects have been taken into consideration for pre-test design. These aspects supplemented and enhanced the analysis of the results and the results were consistent with earlier learning studies’.

Four critical features, for younger pupils’ understanding of how the number line can be constructed, have been established in the study: the number line can have a different range (e.g. 0-20 or 30-60), correlation between the distance and value, two reference points are needed to place a third number and determine the scale, and the correlation between a part and whole.

Learning Study is emphasized as a powerful and structured model for teacher driven research aiming to develop the praxis. The use of experiences and results from other studies is recommended.

There have been ideological variations in the understanding of student-centered learning (SCL), culminating in varied practices of SCL across the world. The purpose of this paper is to explore the impact of learning study on teachers’ appropriation of the conditions for learning in SCL lessons.

Three secondary school mathematics teachers in Tanzania formed a learning study group, guided by the theory of variation, to share their experience of how to engage learners in experiencing critical aspects of the object of learning. In-depth interviews, records of teachers’ lesson preparation meetings, and students’ tests were tools used to collect data. All of the qualitative data were analyzed using a phenomenographic variation framework and coding strategies. Moreover, a paired sample t-test was used to analyze the students’ pre- and post-test results.

The results show that teachers were able to identify critical aspects of two objects of learning for mathematics and create conditions for engaging learners in experiencing those aspects sequentially and simultaneously. There was strong evidence that the theory of variation as a framework helps teachers to learn effective ways of creating conditions for students to appropriate features of the objects of learning for mathematics as well as developing a new SCL pedagogical framework.

This study suggests that using learning study guided by the theory of variation supports teachers’ appropriation of the conditions for student learning within an SCL framework.

The purpose of this paper is to identify changes in teacher’s pedagogical content knowledge (PCK) as a result of their participation in two learning studies in mathematics.

A group of four mathematics teachers and the researcher of this paper explored two objects of learning related to division with decimal numbers and the straight-line equation in a learning study (LS) project over the course of one year. The data used consist of eight video-recorded lessons, four written student tests, eight student interviews and 14 recorded team meetings. The analysis was based on the variation theory principle about learning (Marton, 2015) and on ontological assumptions within the phenomenographic approach (Marton, 1981).

It was found that teachers’ PCK was changed and refined through the LS process in terms of differences in ways of understanding and discussing critical aspects related to each object of learning. In the paper, the terms “presumed critical aspects” and “explored critical aspects” are used to illustrate this difference. Furthermore, the concepts are viewed in relation to three subdomains of PCK: knowledge of content and curriculum (KCC), knowledge of content and students (KCS) and knowledge of content and teaching (KCT).

Bringing out the difference between “presumed critical aspect” and “explored critical aspects” advances the view that critical aspects are dynamic and emergent in practice – in the interaction between teaching and student learning. The research exposes “explored critical aspects” as a specific form of PCK in which KCC, KCS and KCT are intertwined.

The purpose of this paper is to report a case study of how learning study was incorporated in teacher education programs in Hong Kong. It aims to share the success of the program and to disseminate how pre‐service teachers enhanced their mathematical content knowledge and pedagogical content knowledge by practising learning study. Building on the work, this paper suggests incorporating the framework of learning study, a theory‐guided pedagogical principle, as an integrated subject of mathematics pedagogy and teaching practice in teacher education programs.

In total 32 pre‐service teachers’ learning journals of their reflections of learning processes were analyzed. The analysis of data and reporting of findings are linked tightly to how pre‐service teachers enhanced their mathematical content knowledge and pedagogical content knowledge by practising learning study.

The 32 pre‐service teachers noted that the learning study subject fostered their understanding of relationship between theory and practice and their understanding of transforming knowledge into action. In particular, they came to understand that knowledge of pupils and content involves a particular mathematical idea or procedure and familiarity with students’ prior knowledge and misconceptions. They also reported that they understood better what mathematics pedagogy content knowledge means and what components it includes.

The suggestions of incorporating the framework of learning study in teacher education programs is supported and manifested by the positive feedback and comments of the 32 pre‐service teachers who underwent the entire learning process of learning study in Hong Kong. The findings demonstrate how pre‐service teachers’ mathematical content knowledge and pedagogical content knowledge were enhanced by practising learning study.

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.

This paper presents the Special Issue papers thematically and the main issues are discussed.

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.

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.

The purpose of this paper is to report how the use of a direct and an inverse task gave students an opportunity to discern the structure of a distributive law that they could apply to expand or to factorise algebraic expressions. The paper discusses a teaching sequence (of about 40 minutes) in a Grade 8 (14-15 year-olds) algebra class and it focusses on how the use of inverse tasks opens the dimension of variation. For instance, if the distributive law has always been used as a(b+c), factorising the expressions ab+ac means opening up the dimension of distributive law. The analysis showed that two central processes, transformation and variation, improved communication in the classroom.

The data used come from a longitudinal study conducted in Sweden. The methodology is grounded in educational design research. Two secondary school teachers conducted a lesson with variation theory as a guiding principle, supervised by a researcher. The relationship between teaching and learning was analysed in the enacted object of learning. The critical aspect for students’ learning was identified by asking questions to probe the students’ understanding.

The use of a direct and an inverse task gave the students an understanding of the structure of the distributive law that they could apply to expand or to factorise algebraic expressions. The teacher opened up a dimension of variation by similarity that gave the students the opportunity to discern the commonality in direct and inverse tasks as well as to relate the direct and inverse tasks to each other. Without an identification of similarity that might help students to compare underlying meanings, or to match one representation to another, students may not experience variations because there is not concordance among the relationships between the representations.

Teachers can produce new knowledge as well as communicate successfully in the classroom when creating teaching activities that promote the discernment of similarity and difference that might help students to compare underlying meanings, or to match one representation to another.

The study represents an example of research which has the aim of improving teachers’ practices by using research results from mathematics education whilst keeping in mind that learning must be improved.

The central educational problem is to have students make sense of sophisticated ways of reflecting on the general laws used in mathematics in relation to the algebraic ways of acting and reflecting. Variation theory sees learning as the ability to discern different features or aspects of what is being learned. It postulates that the conception one forms about the object of learning is related to the aspects of the object one notices and focusses upon.

The commutative law for algebraic generalisations is not characterised by the use of notations but, rather, by the way the general is dealt with. Algebraic generalisations entail: the grasping of a commonality related to the discernment of whole-parts relationships, the generalisation of this commonality to two types of transformations: treatments and conversions, and the formation of direct and inverse tasks that allows one to discern the relationship between the whole, the parts, the relations between the parts, the transformation between the parts and the relationship between the parts and the whole.

The purpose of this paper is to investigate the relationship between a student’s study time and the learning outcome from a perspective that will correspond to Bloom’s (1956) taxonomy and how teaching can be developed using variation theory.

The author designed a learning study using an experiment with three different classes of students. The experimental research question was “Does a student’s study time (massed or distributed spacing) have an impact on the learning process?”

Results indicated that students in the “strictly supervised study time” grouping improved on their learning outcomes more than those in the “not strictly supervised study time” equivalent and those in the control group. It is important for students to manage their own learning activities and follow a regular study routine to improve their learning outcomes.

This study used undergraduate students at a university in Ghana, and its findings may not necessarily be applicable to other populations. One other limitation was that the author did not control for the lecturer’s expectations and how these may have influenced students’ learning outcomes. Another potential limitation was that total quality management was the only subject area used for this study.

The objective of the study was to use the motivated strategies for learning questionnaire (MSLQ) approach by Pintrich et al. (1991) to determine the relationship between a student’s study time and the impact on their learning outcomes. The results imply that students must take more active roles in their learning by having regular study time.

Currently, to the best of the author’s knowledge, there are not many experiment-based research studies on a student’s study time using the MSLQ approach by Pintrich et al. (1991). This study contributes to the existing literature by examining how a student’s study time (massed or distributed spacing) has an impact on the learning outcome as a lesson and learning study.

The purpose of this paper is to contribute to the theoretical reflection on learning study as a research approach. The focus is on description and reflection on the methodology of learning study as paedeutic research. This research is for, not on, teachers, i.e. research into problems and challenges faced by teachers in their professional practice. Learning study as paedeutic research is about the content and processes of formation/Bildung in relation to specific learning objects. Its focus is on subject-specific ways of knowing as well as how such knowing is enabled through teaching.

The point of departure is a perspective on research approaches as practices, i.e. as activities with certain aims as well as ways of “making” knowledge. Based on a description of the knowledge machinery in a learning study, i.e. those mechanisms that together generate new knowledge, the knowledge claims that can be made are discussed together with the theoretical underpinning of the arguments. The knowledge machinery is described in relation to how it is organised around the delimitation and analysis of an object of Learning as well as designing and evaluating ways to make the critical aspects of this object of learning visible. As an epistemological underpinning, some aspects of pragmatic philosophical thinking regarding the relationship between theory and practice are outlined. Based on that the research process may be described as a development of means-ends relationships – from unconscious empirical relationships to conscious staging of internal and theoretical relations. Abduction is an important tool for this meaning-making.

Learning study can be described as a particularistic, theory-building research approach concerning the knowing of specific learning objects as well as how they can be taught and learnt. The knowledge that is generated in learning study is theoretical and describes aspects of the teaching and learning of specific objects of learning. The research process can be described in terms of specification where practice is gradually supplied with a more differentiated meaning. A learning study is organised around a specific object of learning that functions as an open and unfolding object of knowledge. It combines a practice-based development of theory with a theory-based development of practice.

The development of the thinking about learning study as research for, rather than on, teachers is paedeutical research. A contribution to reflection on the knowledge machinery and knowledge claims of such research.

The purpose of this paper is to pull together articles published in the journal over the last five years in terms of their relevance to the main themes and issues that have shaped its development.

The idea is that the paper enables teachers and researchers in schools and higher education institutions to access a substantial and unique collection of lesson and learning studies, and articles about the theoretical and methodological issues they raise, in the context of a rapidly globalising phenomenon.

The paper sets a framework for evaluating the growth and development of lesson studies as a form of practice-based teacher research.

This paper, authored by the founding chief editor of the IJLLS, establishes conceptual links between the theory and practice of lesson study and the wider field of practice-based pedagogical research.

The purpose of this paper is to report a case study that qualitatively describes and analyses teachers’ discussions when planning and evaluating lessons under supervision, and what is described is the focus on their reasoning, the activities decided upon and how they decided to assess the pupils’ learning outcomes when transforming formal curriculum objectives for English as a foreign language (EFL) into classroom instruction. The effect of this transformation, expressed in different ways of designing lessons, is evaluated by assessing the pupils’ learning outcomes.

The content analysis is based on variation theory. An iteratively designed method, the learning study, is used and data from five teacher meetings in this iterative process is analysed.

The analysis shows that the areas the teachers focus on initially are: implementation (activities); content; and pupils’ knowledge, prioritised in this order, even if they partly seem to be intertwined and handled as a whole, with the three different areas being put in the foreground or background during the learning-study process. In the later part of the process, the perspective has changed to a focus on: pupils’ knowledge; content; and implementation (activities).

The findings in this paper suggest to teachers a way to implement a formal curriculum in a local context. The authors also argue for the importance of collective work in this process.