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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.

PurposeThis study determines which aspects of the intended object of learning (planned by teachers during the first phase of a learning study) is made discernible from a learners' perspective. In a learning study, the intended, enacted, and lived object of learning are considered. This study focuses on the learning material used by teachers while designing a lesson.Design/methodology/approachIn many learning studies, variation theory is used to design lessons, which predicts difficulties in and possibilities for student learning. The data consisted of a lesson part – instruction through a video-recorded dance choreography – employed to enhance primary school (in a Swedish context, grade 4) students' dancing skills in the subject of Physical Education and Health. The choreography comprised five different sequences, where a variation occurred when the subsequent (new) sequence was applied to the previous movement pattern. The sequences acted as building blocks, where the students' transitions from one movement pattern to another were logical and distinguishable.FindingsThe results of this study show in what way an analysis of learning material, based on variation theory, can help teachers take into account the level of complexity of the object of learning. The results also identify which parts of a lesson design can be predicted to present a higher degree of challenge and by that more difficult to grasp, especially for students with different educational needs.Originality/valueLessons may be designed based on theoretical assumptions to ensure effective classroom learning and provide guidance to teachers based on student needs.

Active engagement does not necessarily produce meaningful mathematical learning. It is not uncommon that in mathematics classrooms students seemed to be engaged in-class activities, be happy with what has been done, and everybody enjoyed the lessons, but the assessment revealed that students only learned some factual knowledge on the surface failing to apply the learned knowledge to solve related problems. What is the missing part of the puzzle? Students learn through experiencing the available content. Active engagement does not produce meaningful mathematical learning if some essence of content is missed. If we want our students to learn structured mathematics and be able to apply what have learned to a new situation or context, we must make this kind of learning content available in a mathematics classroom. transfer of learning will never happen without intentionally designed content. “What” needs to be learned should be always the starting and ending point of our teaching/learning design. This paper discusses the existing issue and proposes some pedagogical strategies to promote deep learning.

Purpose 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. Design/methodology/approach 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. Findings 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. Originality/value 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.

Learning path recommender systems are emerging. Given the popularity of ontology/knowledge-based systems in adaptive learning, this work reviews learning path in ontology-based recommender systems. The review covers recommendation trends, ontology use, recommendation process, recommendation technique, contributing factors, and recommender evaluations. A total of 12,972 articles published between 2010 and 2020 were identified in the initial search across five major databases, and 9 of them are considered in this work. Currently, student model, learning objects, learning activities, and external environment are contributing factors for recommending learning object sequence. We also found that the current trend for LP recommendations process is semi-dynamic and dynamic. Semi-dynamic learning path are started by a pre-set path, while dynamic learning path is flexible from the first step and intended for personal use. The recommendation process itself has four phases: predelivery of the first learning object, current learning object delivery, learning object postdelivery, and predelivery of the next learning object. The current recommendation technique collaborates ontology and several techniques, such as Bayesian networks, data mining, and other artificial intelligence technique. To evaluate performance, learning path recommender systems use real students, control groups in parallel or sequential experiments, and student satisfaction surveys. Ontology could work with knowledge representation instruments, educational psychology, and evolutionary computation to create a future dynamic learning path in adaptive learning environment.

Although learner-centered instruction (LCI) is at the heart of many curriculum innovations in the world, its implementation has generally lacked the kind prominence it deserves. The current study explores how learning study can develop teachers’ professional capabilities to practice LCI. Using interviews, lesson preparatory meetings, teachers’ reflective journals, and classroom observations, the study employed the phenomenographic variation framework and coding strategies to analyze the resultant data. The results demonstrate that the teachers’ understanding of LCI morphed from considering it as a methodological orientation to an object of learning. Therefore, engaging teachers in learning study may bring out valuable pedagogical innovations essential in boosting both the job performance and quality of teaching and learning outcomes.

Computing and computers are introduced in school as important examples of technology, sometimes as a subject matter of their own, and sometimes they are used as tools for other subjects. All in all, one might even say that learning about computing and computers is part of learning about technology. Lately, many countries have implemented programming in their curricula as a means to address society’s dependence on, and need for programming knowledge and code. Programming is a fairly new school subject without educational traditions and, due to the rapid technological development, in constant change. This means that most programming teachers must decide for themselves what and how to teach. In this study, programming teachers’ teaching is studied. With the aim of exploring the connection/possible gap between teacher’s intentions and the teacher’s instructional practice , an expansion of the conceptual apparatus of phenomenography and variation theory is tested. In the article, phenomenography and variation theory and the suggested supplementary theoretical tool (Georg Henrik von Wright’s model of logic of events) are briefly presented and then deployed upon one selected case. Findings reveal that teachers’ intentions (reflected in their actions) include an emphasis (of teachers’ side) on the importance of balancing theory and practice, using different learning strategies, encouraging learning by trial-and-error and fostering collaboration between students for a deeper understanding of concepts. In conclusion, logic of events interpretations proves to be useful as a complementary tool to the conceptual apparatus of phenomenography.

PurposeIn this paper, knowledge concerning the meaning of knowing the object of learning is developed using learning study as a research approach. The purpose of this paper is to show how the object of learning, in this case, playing with groove in rhythmic sequences, can be analysed and recognised using phenomenography.Design/methodology/approachThe paper reports on a learning study conducted in three different fifth grade elementary school classes. Seven music teachers were involved in preparing the pre-test, designing three research lessons and analysing the material.FindingsThe findings show that several aspects of the object of learning must be discerned by the learner in order to experience the chosen object of learning and develop in the learning situation. The phenomenographic analysis contributed to refining the meaning of the object of learning.Originality/valueIn a learning study, knowledge concerning the meaning of the object of learning is generated. The development and specification of this knowledge will be empirically revealed through the analysis process. This paper will contribute to the discussion concerning what must be known in order to develop a specific capability in music education, namely, the capability to feel the groove.

This work presents the “Realistic Dynamics Of Simulated Operations” (DROpS), the name given to the dynamics using the “dropper” device as an object of teaching and learning. The objective is to present alternatives for professors teaching content related to simulation of discrete events to graduate students in production engineering. The aim is to enable students to develop skills related to data collection, modeling, statistical analysis, and interpretation of results. This dynamic has been developed and applied to the students by placing them in a situation analogous to a real industry, where various concepts related to computer simulation were discussed, allowing the students to put these concepts into practice in an interactive manner, thus facilitating learning

Purpose: The purpose of this paper is to contribute to the understanding of the processes that make teachers learn in a collaborative arrangement similar to lesson study (LS) and learning study (LearS). The teachers in this collaboration wanted to enhance teaching and student learning (grades 4-7) about decimal numbers. Design/methodology/approach: The analysis is based on data from five teachers' collaborative work in an adaptive arrangement of LS and LearS called subject didactic groups. Data consist of eight audio recordings of the teachers' meetings as well as written and photographic documentation of the meetings. The analysis was carried out through the lens of expansive learning within an activity system (Engeström, 1987). This entailed a focus on contradictions between teachers' ways of thinking and acting when individually and collaboratively developing their teaching, on the solutions to the conflicts produced by the teachers, and on how these challenged the teachers' ideas about what the students need to learn. Findings: The authors identified contradictions between formative and summative assessment, exams and stressed students, prevailing norms about teaching and the theoretical tool used for planning and analyzing lessons and student learning, and the local curriculum and time constraints. The solutions to the conflicts were the driving force for developing new and more qualitative knowledge about what the students need to learn. Originality/value: This paper gives explicit examples of contradictions and solutions that can challenge and drive teachers to expand their learning in an adaptive form of LS and LearS suited to daily teaching.