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  • 1.
    Eriksson, Inger
    et al.
    Stockholms universitet, Humanistiska fakulteten, Institutionen för de humanistiska och samhällsvetenskapliga ämnenas didaktik.
    Wettergren, Sanna
    Stockholms universitet, Humanistiska fakulteten, Institutionen för de humanistiska och samhällsvetenskapliga ämnenas didaktik.
    Nordin, Anna-Karin
    Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.
    Fred, Jenny
    Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.
    Fermsjö, Roger
    Stockholms universitet, Humanistiska fakulteten, Institutionen för de humanistiska och samhällsvetenskapliga ämnenas didaktik.
    Nyman, Martin
    Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.
    Gerholm, Verner
    Stockholms universitet, Humanistiska fakulteten, Institutionen för de humanistiska och samhällsvetenskapliga ämnenas didaktik.
    Same critical aspects regardless of age – indicating lack of experiences of algebraic expressions2019Konferensbidrag (Refereegranskat)
    Abstract [en]

    The issue for this paper is to discuss what can explain that students in different grades seem to experience a phenomenon in more or less the same way and thus, need to discern the same critical aspects in teaching. Data comes from a three-year-long learning study project conducted in four different Swedish schools, in K–9: preschool, grade 4, 6 and 9 respectively.  

    Understanding what students yet need to learn in relation to a specific content is of importance if to organize a teaching situation where students can work in a zone of proximal development (Vygotsky, 1986). Within the theory of variation the concept of critical aspects is regarded as bearing such information for teachers to use when designing teaching (Marton, 2005; Runesson, 2013).

    In the research project we used phenomenography as a theoretical framework in search for critical aspects. Phenomenography is a research approach aiming to understand qualitatively different ways of experiencing a phenomenon. A basic assumption is that we, on the basis of what we have experienced in life, what situations and problems we encountered, experience a phenomenon in a specific way (Marton, 1981; Eriksson, 1999). But the ways we experience a phenomenon do not vary very much. This is believed to be because the contexts and activities do not vary at all. A phenomenographical analysis tends to result in a limited but qualitatively different way of experiencing a phenomenon (Eriksson, 1999). Critical aspects can be identified when comparing different ways of experiencing a phenomenon.

    The research question addressed in this paper is what possible explanations can be found that students in different grades seem to experience a phenomenon in more or less the same way and thus, need to discern the same critical aspects in teaching.

    In order to identify critical aspects groups of students from each grade were interviewed when presented algebraic expressions and some possible answers. The interviews were transcribed and phenomenographically analysed.

    The results indicates that regardless of earlier schooling the students experienced the phenomenon of algebraic expressions in a similar way and thus same critical aspects were identified as needed to discern for the students in the four grades. Based on the analysis three different critical aspects were identified. Students from preschool class and grade 4 were interviewed during the first project year and the students from Grade 6 and 10 the second project year. The analysis from the first project year resulted in critical aspects common for the students and that was not so surprising since we knew that these students had not yet experienced algebraic expressions. However almost the same critical aspects were identified in the analysis of the interviewed students in Grade 6 and 9 during second project year. This was more of a surprise since the students had met algebraic expressions in school.

    The discussion will focus on these findings in relation to what possible experiences student may have of discerning aspects of algebraic expressions in relation to the content of mathematical education.

  • 2.
    Fermsjö, Roger
    Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.
    Rekonstruktion av logaritmer med tallinjer som medierande redskap2014Licentiatavhandling, monografi (Övrigt vetenskapligt)
    Abstract [en]

    The aim of the research reported in this licentiate thesis was to create an environment that could support students’ learning about logarithms. To develop such a learning environment, Davydov’s ‘learning activity’ was used as a theoretical framework for the design. A new tool was created, that was used by the students to unfold and single out some of the unique properties of logarithms when solving different learning tasks. The construction of the model was inspired by Napiers original idea from 1614, i.e. exactly 400 years ago, by using two number lines; one arithmetic (i.e. based on addition) and one geometric (i.e. based on multiplication).

    The research approach used was learning study where teachers and researcher worked collaboratively in an iterative process to refine the research lesson. The study was conducted in six groups with six teachers in upper secondary school in a major city in Sweden. The sample comprised about 150 students and data were collected by filming lessons and by interviews with some of the students. The data were analysed using an analytic framework derived from ‘learning activity’ and the results show what supports, but also what does not support, the creation of an environment for supporting students’ learning of logarithms.

    The results from the study are related to former research regarding instrumental/procedural vis-à-vis relational/conceptual understanding and also about research about students’ ‘errors and misconceptions’. It is argued that the formal definition of logarithms, y = 10x <-> x = lgy (y > 0), should not be used to introduce the concept, instead a new way is proposed. One conclusion is that it is possible to reconstruct logarithms without using the definition as a tool. The results from the analysed lessons show how students looked for ways to solve learning tasks using the new tool. The definition and the identities regarding logarithms appear as bi-products of the students learning activity. When analysing students actions, they rarely over-generalised mathematical rules, e.g. used the distributive law, or separated log-expressions, e.g. adding log expressions part by part, that seemed to be an issue according to former research.

  • 3.
    Fermsjö, Roger
    et al.
    Stockholms universitet, Humanistiska fakulteten, Institutionen för de humanistiska och samhällsvetenskapliga ämnenas didaktik.
    Gerholm, Verner
    Stockholms universitet, Humanistiska fakulteten, Institutionen för de humanistiska och samhällsvetenskapliga ämnenas didaktik.
    Buchberger, Helena
    Nilsson, Birgitta
    Algebraic reasoning in upper secondary school2019Konferensbidrag (Refereegranskat)
    Abstract [en]

    The participants (n = 85) in this study were 16-17 years old and in their first year of upper secondary school in Sweden. In total, three lessons were recorded and transcribed, all with the same teacher. The first two lessons were conducted in a class of the Social Science Programme, i.e. a university-preparatory programme, and the third lesson in a class of the Building and Construction Programme, i.e. a vocational programme. The difference between the classes could be understood from their grade averages from compulsory school, 265 and 278 for the former two classes and 200 for the latter, where 340 is the highest possible score. The classes were chosen for different reasons, such as if there would be any differences according to former grades or choice of education. The intention was to use the same learning tasks in all classes, but small changes were made according to the analyses made after each lesson.

    Learning Study has been used in several studies, initially to test the Variation Theory, and is therefore also the most commonly used theory. It originates in the iterative process used in Lesson Study, and the methodology for Learning Study could be described by this iterative process:

    1. Find an object of learning
    2. Make a pre-test
    3. Design the lesson
    4. Implement the lesson
    5. Make a post-test
    6. Analyse the lesson
    7. Revise the lesson
    8. Return to 4

    Other theories have been used in former Learning Studies, e.g. Learning Activity (LA). This study also uses LA as a theoretical framework and the research questions are:

    • What kind of tasks and discussions in the classroom could lead to the development of algebraic reasoning?
    • What learning actions, i.e. algebraic reasoning, could be shown orally, bodily or symbolically?
    • What qualities in the reasoning could be seen? 

    In order to analyse the lessons, the LA framework was used. Aspects to examine and questions to ask were, for example:

    • What is the learning task, and what is it possible for the student to understand of the task?
    • What is intended with the lesson, and what are the students trying to do?
    • What is known/unknown for the students?

    An important part of LA is reflection, e.g. considering goals, motives, means of action by oneself and others, etc. Important questions were thus:

    • How are solutions handled (both by the teacher and by classmates)
    • How are solutions evaluated in class?
    • What kind of reflections are made?

    The preliminary results from the study show that the mathematical tool could be helpful, if used, in developing algebraic reasoning. Its usefulness could depend either, or both, on the way in which the tool was introduced and the way in which the learning tasks were presented and handled in class. Analyses in relation to the LA framework also raised questions about the motive of the tasks, e.g. reducing variables. The importance of this in a mathematical sense is clear, but the task did not enable the students to see this.

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