Different possibilities to learn from the same task

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 tarea

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.En 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.This research was funded by a grant from the Swedish National Research Council

Single unit counting – An impediment for arithmetic learning

In this paper we direct attention to the single unit counting strategy that is observed to be limiting students’ opportunities to develop their arithmetic skills. We describe what impediments single unit counting may entail when encountering novel subtraction tasks, and how these impediments can be explained. From a sample of 121 interviews of students aged 7-8 we have chosen nine who were using single unit counting as their dominating arithmetic strategy. An analysis based on variation theory reveals that the impediments are related to the students not experiencing numbers as composed units and thereby lack in discerning number relations necessary to handle multi-digit subtraction. Educational implications are discussed grounded in the theoretically driven findings.This work was supported by the Swedish Institute for Educational Research [Grant number 2018-00038]

Research on early childhood mathematics teaching and learning

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Teaching and learning addition and subtraction bridging through ten using a structural approach

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 Swedish Grade 1. A goal was that at the end of Grade 1, students would solve tasks like 15–7= using part-whole relations of numbers. 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, to investigate long-term effects and whether students used a structural approach when solving tasks in Grade 2. Results show that students in the intervention group increased their learning outcomes the most and to a larger extent solved tasks in higher number ranges using a structural approach.This work was supported by the Swedish Institute for Educational Research [Grant number 2018-00038]

What is taught and what is learned. Professional insights gained and shared by teachers of mathematics

The aim of the thesis is to contribute to knowledge about relationships between teaching and learning in school. The framework used in this research, variation theory, states that, to improve student learning, attention must be paid to what is being learned, the capability that is to be improved and the features (critical features) that it is necessary for the learner to discern. The studies reported here focus on the significance of critical features and are based on two ‘learning studies’ in mathematics, one of the density of rational numbers and one of the addition and subtraction of negative numbers. In a learning study, teachers work together with design, analysis and revision of their teaching of a single lesson with the aim of enhancing students’ learning by gaining insight about features that are assumed to be critical and enacting them in their teaching. The question answered in this research is whether the insight gained in the learning studies about critical features can be shared by other teachers and used to enhance other students’ learning. Two studies based on the previous studies were carried out together with a total of eight teachers and sixteen groups of students. Each teacher enacted two lessons with different conditions in terms of the critical features made use of. The lessons were video recorded and analysed with respect to which critical features were enacted in the lessons and what the students learned as indicated in pre and post tests. It is suggested that the critical features were transferable in two regards, in terms of student learning and in terms of a means of communication that could be shared among teachers. It was indicated that the critical features that were enacted in the teaching constrained what it was possible to experience in the classroom and what students learned. What was taught seemed to be reflected in what the students learned. Furthermore, the analysis indicates that it was not sufficient to simply name the critical features to the students; it seems that they must be discerned in order for learning to take place. It was found that the teachers made use of the critical features that were identified by other teachers in a learning study as a means to plan and teach lessons. This suggests that teachers can make use of the notion of critical features in their own teaching to enhance student learning

Learning about the numerator and denominator in teacher-designed lessons

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.Det andra stege

What is made possible to learn when using the variation theory of learning in teaching mathematics?

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

Mechanisms enabling knowledge production in learning study

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.Validerad;2020;Nivå 2;2020-02-27 (alebob)</p