Ulla Runesson

Beyond discourse and interaction. Variation : a critical aspect for teaching and learning mathematics

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.

What is it Possible to Learn? On Variation as a Necessary Condition for Learning

A recent development of phenomenography, variation theory, is illustrated using a video recorded case of learning. From a variation theory perspective, to learn is to be aware of critical aspects of what is learned. The way we experience or understand something depends on what aspects we are aware of and can discern simultaneously. The possibility for the learner to discern and focus on these aspects is critical for learning. But we can only discern an aspect if we experience a variation in that aspect. Thus the possibility of experiencing variation in critical aspects is a necessary condition for learning. Variation theory is proposed to be a powerful means for describing and revealing conditions critical for learning in a pedagogical setting.

Sharing and developing knowledge products from Learning Study

Purpose – It has been proposed that lesson study creates joint and sharable knowledge products in terms of lesson plans that could be used and developed by other teachers in other school contexts (Morris and Hiebert). The purpose of this paper is to report on a study with the aim of examining how such a knowledge product produced in a Hong Kong school could be communicated and appropriated by a group of Swedish teachers.Design/methodology/approach – A report from a successful Learning Study – a revised version of lesson study, based on a theoretical framework of learning – was brought to a group of three teachers. On the basis of this report, they planned and conducted two lessons in their own classes. The analysis draws on data from three of six video recorded lessons and was framed within a variation theory perspective. The focus of the analysis was on the object of learning.Findings – It was found that the Swedish teachers used the documented Hong Kong lessons as a resource. They adapted the insights g...

Views of Learning, Teaching Practices and Conceptions of Problem Solving in Science

In this chapter we first describe university science teachers differing ways of understanding what it takes for their students to solve problems. Using the phenomenographic approach described in the chapter by Martin et al., this volume, we have found significant variation in how teachers understand student problem solving situations in science courses. Some of the teachers conceive the problem as obvious or unproblematic to the student and in these cases problem solving is experienced as a process of application. Others focus on the meaning of the problem and consider the interpretation of this to be important. In these cases the problem solving process is experienced as understanding and making sense of the problem. This variation has been constituted from interviews with seventeen teachers of first year science students. In the same interviews the teachers were asked questions about their conceptions of teaching, their conceptions of learning and their approaches to teaching. We found that most teachers who conceived of teaching as transmitting knowledge to students also conceive the problem as unproblematic to the student. On the other hand, those teachers who conceived of teaching as helping students develop or change their conceptions also saw that students’ understanding of the meaning of the problem is not experienced as obvious or unproblematic. In a more detailed analysis of three of the cases we found that these experiences of problem solving are embedded in more general views of learning — three very different views of what it takes for students to solve problems are embedded in three very different views of learning. Thus the chapter describes relations between teachers’ conceptions of teaching, their views of learning and their conceptions of the classroom practice of problem solving in tertiary science courses.

Sensitivity to Student Learning: A Possible Way to Enhance Teachers’ and Students’ Learning?

To calculate with negative numbers seems to be one of the hardest topics in mathematics courses for 13–14 year old students to learn. Our experience is that this very often is taught by ‘telling the rules,’ and even adults when given a task like ‘ - 3 - (-5) =’ try to remember some kind of ‘trick’ to solve the problem. How could this be taught for understanding? What is necessary to be aware of in order to understand? What is critical for learning this? Questions like these were addressed by a group of teachers who collaboratively used a systematic approach to investigate what it takes to learn to subtract negative numbers in grade seven and eight in a Swedish compulsory school. In this approach, student learning and their understanding of that which was taught was the main concern and in the forefront of the teachers’ attention. In an iterative process of planning, observing and revising a single lesson, they in depth analysed and inquired the progress of student learning before, in and after the lesson. In this process the teachers tried to learn about the particular difficulties their students had for learning what was intended. This was done from an analysis of students’ written tests and from a close observation of video recorded lessons. In this observation, they tried to understand students’ learning outcomes in the light of how the content taught was handled in the lessons. Through the process they gained some insights from which they successively could change their teaching in a way that had an effect on students’ learning. We also demonstrate how reflection on teaching and the learning from teaching can take its point of departure from sensitivity to students’ understanding and perceptions of that which is to be learnt.

Pedagogical and learning theories and the improvement and development of lesson and learning studies

PurposeIt 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) ...

What is to be Learned? Teachers’ Collective Inquiry into the Object of Learning

ABSTRACT 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.

Teaching one thing at a time or several things together? – teachers changing their way of handling the object of learning by being engaged in a theory-based professional learning community in mathematics and science

Abstract 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.

Beyond the borders of the local: How “instructional products” from learning study can be shared and enhance student learning

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 ...

Learning to Teach with Variation

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).”