Angelika Kullberg

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.

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.

Can findings from learning studies be shared by others

Purpose – The purpose of this study is to explore whether the insights gained by teachers in a learning study can be shared by others and used to enhance other students’ learning.Design/methodology/approach – An exploratory study analysed how four Mathematics teachers implemented the knowledge gained about student learning from a learning study about decimal numbers in their teaching. The teachers enacted two lessons with different conditions in terms of the knowledge identified about student learning in a total of eight classes in the fifth and sixth grades.Findings – The findings indicate that, when the teachers enacted the knowledge in their lessons, students’ learning improved significantly as compared to when only some parts of this knowledge were enacted.Practical implications – This suggests that teachers can produce knowledge that is also usable for other teachers and that enables better possibilities for student learning.Originality/value – The study provides insights as to what extent findings f...

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.

Students’ open dimensions of variation

Purpose – The purpose of this paper is to focus on how the content of a single lesson that dealt with the addition and subtraction of negative numbers was handled in the interaction between students and a teacher in the 7th grade. The study is a follow‐up of a learning study and can be described as a teaching experiment. In total, four critical features for students’ learning were identified in the learning study.Design/methodology/approach – In the new study, the teacher deliberately did not bring up all the critical features of the object of learning in one lesson. The aim was to see how different critical features made a difference for student learning.Findings – The results show that a particular students questions had an impact on the critical features that were brought up and hence on what it is possible to learn. In this lesson it was the students that opened some of the dimensions of variation for learning negative numbers.Originality/value – The paper suggests that being aware of critical featur...

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).”