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  • 1.
    Andersson, Charlotta
    Stockholm University, Faculty of Humanities, Department of Teaching and Learning.
    Relationer eller operationer - två sidor av samma mynt: Elevers utforskande av en del-helhetsmodell som redskap för att urskilja relationer i additiva strukturer2022Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    The aim of the licentiate thesis is to examined a specific structural model in order to support students' exploration of relationships between numbers in additive structures even when negative numbers is included. One intention is that the finding should qualify the teaching and constitute a support when plan and implement teaching regarding equations with additive structure based on algebraic teaching as an alternative to primarily finding a solution with support of rules and procedures. During the study, the students' experiences of the phenomenon relationships between quantities was examined, categorized and described. In the study it was also examined whether and, if so, in what ways a specific structural model was used by students during the work of exploring the structure of equations.

    Students from grades 3, 8 and 9 participated in semi-structured interviews and in research lessons. Three groups of teachers and two researchers planned, conducted and revised the research lessons based on the learning study approach.

    Phenomenography was the theoretical approach for the initial semi-structured interviews. Variation theory and learning activity were the theoretical frameworks for the research lessons where Davydov's curriculum was a source of inspiration.

    In order to achieve the aim, the following research questions were formulated:

    1) What different ways of experiencing the phenomenon of relationships between quantities can be discerned in student interviews?

    2) In what ways do students use a specific structural model to explore equations?

    The first research question is answered in Article 1, which shows that students experience relationships between quantities as something to be calculated or something to be related. The second research question is answered in Article 2, which shows that students used the specific structural model examined in the study as a form to fill in alternatively as a learning model and a tool to identify the part-whole structure between numbers in an equation and to choose appropriate operation to find the unknown number.

    The findings show the possibility, but also the challenge, of introducing an algebraic teaching with a focus on analysis and theoretical reasoning also for students with experiences from an alternative background.

    Download full text (pdf)
    Andersson Charlotta 2022
  • 2.
    Andersson, Charlotta
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Andrén, Sanna
    Eriksson, Helena
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Tuominen, Jane
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Skapa behov av multiplikation2020In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, no 4, p. 11-15Article in journal (Other (popular science, discussion, etc.))
    Abstract [sv]

    Kan multiplikation förstås på något annat sätt än som upprepad addition? Här prövar författarna ett nytt sätt att undervisa om multiplikation. Genom att atbeta med indirekt mätning skapas ett behov av multiplikation.

  • 3.
    Andersson, Charlotta
    et al.
    Stockholm University, Faculty of Humanities, Department of Teaching and Learning.
    Tuominen, Jane
    Stockholm University, Faculty of Humanities, Department of Teaching and Learning.
    En modell som stöd för att utforska ekvationer: [A model to support exploring equations]2022In: LUMAT: International Journal on Math, Science and Technology Education, E-ISSN 2323-7112, Vol. 10, no 1, p. 182-209Article in journal (Refereed)
    Abstract [en]

    The aim of the article is to highlight whether, and if so in what ways, a selected model can constitute support when students explore mathematical structures in equations. The article is based on an empirical research study where students explored mathematical structures with support by the model, which is intended to visualize structures. Teachers and researchers worked in a collaborative and interventional study in iterative processes. A total of 149 students from compulsory school grades 3, 8 and 9 participated in video recorded research lessons based on the research approach learning study. The lessons were designed with inspiration from the framework of learning activity and the students were challenged in a theoretical work. The analysis was performed on the basis of a thematic approach and two qualitatively different core themes were identified: Template respectively Learning model. In the analysis, it emerged that the teaching has to be challenging enough for the students to find the model meaningful. The teaching needs to enable students to discern relationships between all numbers in an equation, where the relationships can be described as a part-whole structure.

    Download full text (pdf)
    En modell som stöd för att utforska ekvationer
  • 4.
    Tuominen, Jane
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Andersson, Charlotta
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Björklund Boistrup, Lisa
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education. Malmö University, Sweden.
    Critical aspects of equations when explored as a part-whole structure2020In: Sustainable mathematics education in a digitalized world: Proceedings of MADIF 12 The twelfth research seminar oft he Swedish Society for Research in Mathematics Education Växjö, January 14–15, 2020 / [ed] Yvonne Liljekvist, Lisa Björklund Boistrup, Johan Häggström, Linda Mattsson, Oduor Olande, Hanna Palmér, Göteborg: Svensk förening för MatematikDidaktisk Forskning - SMDF, 2020, p. 223-233Conference paper (Refereed)
    Abstract [en]

    The aim of this paper is to present critical aspects that were identified when students explored equations as a part-whole structure with negative numbers included. Students in grades 3, 8 and 9 participated in a ”theoretical work”. Learning study was used as a research approach and learning activity theory constituted a guiding principle when designing research lessons. According to the analysis, five critical aspects were identified, regardless of grade. The critical aspects are: there is a relationship between all the numbers in an equation; two parts together equals a whole with the same value; what constitutes the parts and the whole, respectively; the same relationship can be formulated in four different ways; the whole can assume a lower value than the parts.

    Download full text (pdf)
    Critical aspects of equations when explored as a part-whole structure
  • 5.
    Tuominen, Jane
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Andersson, Charlotta
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Björklund Boistrup, Lisa
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Eriksson, Inger
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Relate before calculate: Students’ ways of experiencing relationships between quantities2018In: Didactica Mathematicae, ISSN 0208-8916, Vol. 40, p. 5-33Article in journal (Refereed)
    Abstract [en]

    The aim of this article is to contribute with findings concerning students’ ways of experiencing general mathematical structures and, in particular, relationships in additive structures.When students discern relationships in additive structures, it may lead to positive consequences for students’ future ability to perform calculations in addition and subtraction tasks. In the study, semi-structured interviews were conducted with students in grades 3, 8, and 9. An illustration showing a set of different quantities was the starting point during the interviews, together with an opening question regarding how the diverse quantities could be equalised. After the students’ discussions, they were asked if this could be described mathematically using written symbols. The students’ expressions concerning the phenomenon “relationships between quantities” were analyzed using phenomenography as an analytical tool. According to phenomenography, there are a limited number of ways in which a phenomenon can be experienced. Further, it is not about exploring how many individuals hold a specific experience that is of interest. In the case of this article, it is about capturing qualitatively different ways of experiencing the phenomenon relationships between quantities. Despite no specific numbers being presented, many students attributed specific numbers and values when expressing relationships between quantities. The students expressed general mathematical structures only to a limited extent and, in those cases, mostly only after encouragement from the interviewer. Following the phenomenographical analysis, the students’ ways of experiencing “relationships between quantities” are: as something that has to be calculated, or as something that has to be related. The first of these was most common in all grades. In this study, one critical aspect was identified, namely, that quantities are related to each other, additively. Instead of introducing mathematics with a focus on answer-oriented tasks, it is essential to introduce mathematics based on general structures such as additive structures. Even if the students are not familiar with such a mathematical “culture”, it is worth it. This was confirmed in our study.

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