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  • 1.
    Eriksson, Inger
    et al.
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Fred, Jenny
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Nordin, Anna-Karin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Nyman, Martin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Wettergren, Sanna
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Enabling students’ participation in algebraic reasoning with ‘learning models’ as tools for theoretical work2019In: NOFA7 Abstracts, 2019Conference paper (Refereed)
    Abstract [en]

    The issue for this paper is to discuss conditions for students’ participation in theoretical classroom discussions. In order to develop students’ ability to reason and solve problems algebraically Lins and Kaput (2004) argue that it is important to introduce algebra early in mathematics education. In line with this, researchers working within the sociocultural tradition argue that this may start as early as in first grade (Lins & Kaput, 2004). The so-called Davydov programme (e.g. Schmittau, 2005) is referred to as a promising alternative type of teaching, enhancing young students’ capability to reason algebraically. Central to the Davydov programme is the concept of ’learning model’. Such models (not to be confused with mathematical models) are used to enable students’ elaboration of mathematical phenomena. The characteristics of a learning model is that it materialises central theoretical aspects of a content. Further, it creates specific conditions necessary for students to understand and elaborate on each other’s arguments. However, there are many demanding issues related to the design of such learning models that needs to be further developed. In this paper we address the following research question: What in students’ classroom discussions can be taken as signs of emergent ability to reason algebraically? In this paper we use data from two learning studies conducted in 2017–2018. One in grade 1 (age 7) with three iteratively designed and revised lessons, and one in grade 5 (age 11) with four iteratively designed and revised lessons. We focus especially on how students use learning models as tools to enforce and maintain a collective reasoning indifferent communicative situations. To identify possible algebraic reasoning we have analyzed the communicative situations by drawing upon Toulmin’s (2003) model of argumentation with claim, data, warrant and backing. The tentative results indicate that the ability to work with the learning models not only facilitated an individual student to clarify and substantiate his or her arguments, but also enabled other students to follow and elaborate on the reasoning. This paper also contributes with examples of tasks and especially how to set the tasks in motion to enable students’ theoretical work.

  • 2.
    Eriksson, Inger
    et al.
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Wettergren, Sanna
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Fred, Jenny
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Nordin, Anna-Karin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Nyman, Martin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Tambour, Torbjörn
    Stockholm University, Faculty of Science, Department of Mathematics.
    Materialisering av algebraiska uttryck i helklassdiskussioner med lärandemodeller som medierande redskap i årskurs 1 och 52019In: Nordisk matematikkdidaktikk, NOMAD: [Nordic Studies in Mathematics Education], ISSN 1104-2176, Vol. 24, no 3-4, p. 81-106Article in journal (Refereed)
    Abstract [en]

    The aim for this article, which draws upon on data from a design research project based on Davydov’s principles of learning activity, is to discuss which functions learning models can have to promote students’ collective discussions on algebraic expressions. The data is comprised of videotaped lessons in Grade 1 and 5 respectively. The analysis focuses on conditions for qualifying whole-class discussions and the functions learning models can have for the students’ collective exploration of mathematical structures and relationships in algebraic expressions. The result indicates that learning models as mediating tools enable the students to conduct creative and reflective discussions on algebraic expressions and their components.

  • 3.
    Eriksson, Inger
    et al.
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Wettergren, Sanna
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Nordin, Anna-Karin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Fred, Jenny
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Fermsjö, Roger
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Nyman, Martin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Gerholm, Verner
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Same critical aspects regardless of age – indicating lack of experiences of algebraic expressions2019Conference paper (Refereed)
    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.

  • 4.
    Nordin, Anna-Karin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Matematiska argument i helklassdiskussioner: En studie av elevers och lärares multimodala kommunikation i matematik i åk 3-52016Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This study aimed at investigating and analysing the communication occurring during whole class discussions, with a specific focus on the nature of the mathematical arguments. The investigation was a qualitative case study where the communication during eight whole class discussions in grade 3-5 were analysed.

    Three types of arguments, wich are functional in the communication and convey different aspects of mathematics, were identified in the study. The types are (a) argument conveying a solution to a task/ a problem (b) argument conveying conceptual properties, and (c) argument conveying a mathematical relationship. The arguments types explain why an answer to a task is correct (type a), illuminate properties of a mathematical object (b), and clarify a mathematical relationship (c).

    The findings also reveal that arguments may be expressed through the use of a broad range of communicative resources, such as spoken language, written language, symbols, drawings, the use of manipulatives, and gestures. This highlights the importance of taking into account more than speech when construing arguments/reasoning communicated in mathematics classroom.

    The study also points to the importance of paying attention to arguments/reasoning that are created during other occasions than during task work or problem solving, and that arguments can enable the discerning of mathematical aspects for learners.

  • 5.
    Nordin, Anna-Karin
    et al.
    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.
    A framework for identifying mathematical arguments as supported claims created in day-to-day classroom interactions2018In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 51, p. 15-27Article in journal (Refereed)
    Abstract [en]

    This article addresses how to distinguish mathematical arguments created during whole class discussions in grades 3-5 in Sweden, while taking a broad range of communicational resources, such as speech, drawings and symbols, into account. We present a step-by-step framework of how to systematically reconstruct mathematical arguments. The framework is developed drawing on Toulmin's model of argumentation and a multimodal approach. When giving account for the framework, we show how various communicational resources convey the mathematical meaning of the arguments created. The framework can be used for further research investigating interaction in classroom settings, for teacher students as a basis for reflection during practicum periods, as well as a lens for teachers in identifying informal and formal mathematical arguments in day-to-day communication in the mathematics classroom.

  • 6.
    Nordin, Anna-Karin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Sträng, Cecilia
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    TEACHERS' COLLABORATIVE LEARNING AND STUDENTS' OPPORTUNITIES TO PARTICIPATE IN MATHEMATICAL REASONING2013In: Proceedings of the seventh international mathematics education and society conference, vols 1 and 2 / [ed] Berger, M; Brodie, K; Frith, V; LeRoux, K, 2013, p. 172-175Conference paper (Refereed)
    Abstract [en]

    The aim of this presentation is to briefly describe two ongoing studies, linked to each other. One study will be focusing on teachers' perception of their teaching before and after participating in school development projects. The other study focuses on teachers' moves in the classroom and the opportunities they provide for students to participate in mathematical reasoning. The two school development projects where empirical data is collected are the same for both studies. One project is a small project conducted by the authors of this paper and the other project is a national project, although we are just looking at a very small part of it. The national project is aiming to improve student achievement in Sweden through teacher development and will be available for all teachers in primary and secondary school.

  • 7.
    Nordin, Anna-Karin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Wettergren, Sanna
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    A learning study: exploring algebraic expressions through “learning models” in Grade 52018Conference paper (Refereed)
    Abstract [en]

    The issue of the paper is what aspects must be built into a assignment so it can serve as a mediating tool enabling students to explore algebraic expressions.

    Algebra has a unique position in mathematics as it is found in all mathematical areas. The introduction of algebra is in most cases arithmetic in its basis (eg. Kieran, 2006; Lins & Kaput, 2004; Radford, 2010). According to Greer (2008) teaching in algebra needs to create the conditions for students to develop abilities such as algebraic reasoning, making algebraic generalizations and using algebraic representations rather than learning a number of procedures. 

    The data used in this paper is from a learning study in grade 5 (11 year old students) comprising three research lessons focusing on algebraic reasoning. Learning activity was used as a theoretical framework, both when designing and analyzing the research lessons. Variation theory (Marton, 2015) is commonly used as a theoretical framework in learning studies but in this case we have chosen to use learning activity (Davydov, 2008).

    Learning model is a key concept in Davydov’s theory of learning activity. In learning activity the aim is to give students opportunity to gain access to theoretical content that is built into eg. algebraic expressions. A learning model can be iconic (sketches or drawings) or symbolic (eg. algebraic expressions). “Models are a particular kind of abstraction, where the visually perceived and represented connections and relations of the material or semiotic elements reinforce the essential relations of the object” (Davydov, 2008, p. 95). Further a learning model in learning activity aims to enhance students’ collective discussions and reflections.

    In the paper we will exemplify how we analyzed the students’ work (video recorded) and iteratively developed a learning model. Furthermore, we will discuss how to set a learning model in motion in communicative situations during the research lessons.

  • 8.
    Nordin, Anna-Karin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Wettergren, Sanna
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Utformande av matematiska problemsituationer med lärandeverksamhet som teoretiskt ramverk2019In: Book of abstracts: Lärarnas forskningskonferens 2019, 2019Conference paper (Refereed)
    Abstract [sv]

    Algebra innehar en naturlig särställning inom matematiken eftersom den återfinns i samtliga matematiska områden. Såväl generella resonemang i aritmetiken, bevisföring inom talteori som geometriska formler för area och volym använder algebra. Om elever har goda kunskaper i algebra underlättar det således för dem att lyckas med matematikstudier. Utveckling av algebraiskt tänkande bygger på att undervisningen utgår från det generella, och de grundläggande och teoretiska sambanden. Mot denna bakgrund behöver en undervisning skapa förutsättningar för elever att utveckla förmågor som att resonera algebraiskt, att göra algebraiska generaliseringar samt att använda algebraiska representationer snarare än på att lära ut ett antal procedurer (Greer, 2008; Kaput, 1999; Usiskin, 1988).

    Syftet med föreliggande presentation är att diskutera de teoretiska och metodologiska utgångspunkterna för problemsituationer utformade med fokus på elevers utforskande av algebraiska uttryck. Data kommer från ett treårigt forskningsprojekt där fyra learning studies (med vardera 3-4 lektioner) genomförts i årskurs 1, 5, och 7 samt i gymnasieskolans åk 1. Som en ingång till respektive learning study genomfördes en fenomenografisk studie (Marton, 2015) där kritiska aspekter identifierades för elevgrupperna. 

    Studien har genomförts med lärandeverksamhet (Davydov, 2008) som teoretiskt ramverk. En lärandeverksamhet i Davydovs mening karakteriseras av att elever introduceras till en problemsituation som innefattar sådana teoretiska aspekter de behöver urskilja. I föreliggande studie var det generella strukturer och relationer i algebraiska uttryck relaterat till de kritiska aspekterna som eleverna behövde bearbeta. Problemsituationen måste uppfattas som meningsfull av eleverna men där elevernas nuvarande kunskaper i viss utsträckning är otillräckliga. Vidare behöver eleverna några medierade redskap, lärandemodeller, som kan hjälpa dem att bearbeta det identifierade problemet. 

    För att skapa, uppnå eller etablera en lärandeverksamhet måste ett teoretiskt arbete genomföras kollektivt av eleverna i relation till en specifik problemsituation (Eriksson, 2017).

    Som Zuckerman (2004) beskriver måste de kollektiva reflektionerna i en lärandeverksamhet organiseras så att elever bland annat kan ta andras perspektiv. Således ska eleverna ges möjlighet att reflektera över andras bidrag, inklusive andras användning av lärandemodeller. Exempelvis kan eleverna bjudas in i en problemsituation genom att läraren presenterar vad fiktiva elever har svarat. Kollektiva reflektioner i lärandeverksamhet har i detta avseende en avgörande funktion i att utveckla elevernas förståelse av algebraiska uttryck. Därför måste planering för kollektiva reflektioner också betraktas som en del av utformandet av problemsituationerna. 

    I presentationen kommer vi att exemplifiera och diskutera hur problemsituationer utformades och utvecklades iterativt i tre forskningslektioner med hjälp av principerna för lärandeverksamhet.

  • 9.
    Wettergren, Sanna
    et al.
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Fred, Jenny
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Nordin, Anna-Karin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Vad i elevernas resonemang om algebraiska uttryck indikerar urskiljande av kritiska aspekter?2019In: Book of abstracts: Lärarnas forskningskonferens 2019, 2019, p. 60-61Conference paper (Refereed)
    Abstract [sv]

    Syftet med denna presentation är att exemplifiera och diskutera elevers utforskande av algebraiska uttryck i helklassdiskussioner.

    I kursplanen för matematik (Skolverket, 2017) anges algebra som centralt innehåll inom samtliga stadier i grundskolan. Samtidigt framstår algebra som ett utmanade kunskapsområde i undervisningen, vilket flera forskare inom fältet early algebra beskriver kan bero på att algebra vanligen introduceras på en aritmetisk grund (se t.ex. Lins & Kaput 2004). Vidare framhålls att undervisningen i algebra behöver skapa förutsättningar för elever att utveckla förmågor som att resonera algebraiskt, att göra algebraiska generaliseringar samt att använda algebraiska representationer snarare än på att lära ut ett antal procedurer (Stacy & Chick, 2004; Radford, 2010).

    Presentationen bygger på data från ett så kallat ramprojekt inom Stockholm Teaching & Learning Studies (STLS) där verksamma lärare och koordinatorer medverkat. Ett ämnesdidaktiskt ramprojekt tar sin utgångspunkt i en övergripande forskningsfråga som relateras till en specifik förmåga i kurs- och ämnesplanerna. Inom ramprojekt utformas och prövas olika sätt att utveckla undervisningen med sikte på att öka förutsättningarna för elevernas lärande. Ramprojektet har en kollaborativ ansats vilket innebär att koordinatorer och medverkande lärare tillsammans utformar uppgifter och forskningslektioner. I föreliggande ramprojekt utforskades “Förmågan att kunna föra och följa algebraiska resonemang” i årskurserna 2, 3, 4 och 5. 

    I projektet har learning study (Marton, 2015) använts som forskningsansats. I respektive årskurs genomfördes två till tre iterationer. I iterationerna identifierades tre kritiska aspekter gällande förmågan att kunna resonera kring algebraiska uttryck utan att bestämma värdet på ingående variabler: 1) att kunna urskilja att ett uttryck består av olika komponenter som har olika funktioner, exempelvis att a, b och c i uttrycket c + a = b är variabler, att + är en operator och att = uttrycker en relation (vilken information som finns i ett uttryck) siffror är konstanter 2) att kunna urskilja att en och samma variabel i ett uttryck har samma värde och 3) att kunna urskilja att värdet på en variabel i ett uttryck bestäms relationellt.

    Datamaterial för föreliggande presentation utgörs av tio videoinspelade forskningslektioner i matematik från samtliga nämnda årskurser, samt transkriptioner av dessa lektioner. I analysarbetet har vi utgått från ett ramverk (Nordin & Boistrup, 2018) som bygger på Toulmins (2003) argumentationsmodell för att identifiera möjliga resonemang. Ytterligare analys har genomförts gällande hur urskiljandet av de kritiska aspekterna kom till uttryck i elevernas resonemang.

    Under presentationen kommer vi att exemplifiera vad i elevernas resonemang om algebraiska uttryck som utgör indikationer på att de urskiljer identifierade kritiska aspekter.

  • 10.
    Wettergren, Sanna
    et al.
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Nordin, Anna-Karin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Designing tasks in a learning study with learning activity as a framework2019Conference paper (Refereed)
    Abstract [en]

    The issue for this paper is to discuss the theoretical and methodological foundations of tasks constructed and used in a learning study with a focus on students' exploration of algebraic expressions. Data comes from a research project conducted in a Grade 10 (first year of upper secondary school) in a Swedish school comprising three research lessons.

    As part of the learning study a phenomenographic study (Marton, 2015) was conducted, identifying following critical aspects: 1) to discern what constitutes an information-bearing unit and that it can consist of several components, e.g. an expression within parenthesis 2) to discern how the information in a situation can be represented by variables and expressions and 3) to discern that a component of an expression can be expressed in various ways with the information given, e.g. y can be replaced by 300 - x.

    Whereas variation theory (Marton, 2015) is commonly used as a theoretical framework in learning studies, this study employs learning activity (Davydov, 2008), when designing and analyzing the research lessons. A learning activity in Davydov’s sense is characterized by introducing students to a problem situation comprising such theoretical aspects that they need to discern. The problem situation needs to be perceived as meaningful but where the students’ current knowledge is to some extent insufficient. Further, the students should be provided with some mediating tools, learning models, that can help them elaborate on the identified problem.

    When constructing a problem situation an analysis of the content of the learning object can be of help in finding a possible problem that needs to be transformed into a learning task. Thus, tasks in a learning activity aims to give students opportunities to gain access to theoretical content that is built into a specific knowledge e.g. algebraic expressions. In order to create, achieve or establish a learning activity, a theoretical work needs to be done collectively by the students in relation to a specific problem situation (Eriksson, 2017).

    As Zuckerman (2004) describes, the reflection process in a learning activity needs to be organized so that the students among other things can take others’ perspective. Thus, the students are to be given opportunities to reflect on others’ contributions including others’ use of learning models e.g. inviting students into a fictional situation by presenting what fictitious students have answered. Collective reflections have, in this sense, a crucial function in a learning activity developing students’ understandings of algebraic expressions. Therefore, planning for collective reflections must also be considered as part of the task design.

    In the paper we will exemplify and discuss how two tasks were designed and developed iteratively in three research lessons utilizing the principles of learning activity.

  • 11.
    Wettergren, Sanna
    et al.
    Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.
    Nordin, Anna-Karin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Lärandemodell som ett redskap vid resonemang om algebraiska uttryck2018In: Lärarnas forskningskonferens 30 oktober 2018: Abstracts, 2018Conference paper (Refereed)
    Abstract [sv]

    Algebra innehar en naturlig särställning inom matematiken eftersom den återfinns i samtliga matematiska områden. Såväl generella resonemang i aritmetiken, bevisföring inom talteori som geometriska formler för area och volym använder algebra. De uppgifter som vanligtvis används i introduktionen av algebra är i de flesta fall aritmetiska i sin grund (Gravemeijer, 2002; Kieran, 2006; Lins & Kaput, 2004; Radford, 2010). Greer (2008), Kaput (1999) och Usiskin (1988) pekar dock på att undervisningen i algebra behöver skapa förutsättningar för elever att utveckla förmågor som att resonera algebraiskt, att göra algebraiska generaliseringar samt att använda algebraiska representationer snarare än på att lära ut ett antal procedurer. 

    Övergripande syfte med studien är att undersöka och analysera hur undervisningen, i termer av uppgifter och arbetssätt, kan utformas så att elever ges rikliga möjligheter att utveckla förmågan att föra och följa algebraiska resonemang. Studien genomfördes i form av en learning study i grundskolans årskurs 5. I forskningslektionerna introducerades en lärandemodell (Davydov, 2008) med syfte att inbjuda eleverna i ett utforskande kring algebraiska uttryck. I en lärandeverksamhet är arbetet med lärandemodeller som medierande redskap centralt för att elever ska ges möjligheter att få tillgång till ett teoretiskt kunskapsinnehåll (Eriksson, 2017). I presentationen ger vi exempel på hur lärandemodellen kan fungera som redskap när elever ska utforska algebraiska uttryck. Vidare kommer vi att diskutera vad det kan ha för implikationer gällande utformning av undervisningen.

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