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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.
    Fred, Jenny
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Mikhail, Hiba
    Nyman, Martin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Staffansson, Boel
    ”Då är det två C!” – Elever i åk 1 undersöker algebraiska uttryck2018In: Lärarnas forskningskonferens 30 oktober 2018: Abstracts, 2018, p. 51-51Conference paper (Other academic)
    Abstract [sv]

    Presentationen syftar till att fördjupa diskussionen kring hur en undervisning kan designas för att elever så tidigt som i årskurs 1 kan ges möjlighet att resonera kring algebraiska uttryck. 

    Forskare i det matedidaktiska fältet (se t.ex. Kaput, 2008; Kieran, 2006; Lins & Kaput, 2004; Stacy & Chick, 2004; Radford, 2010, 2014) belyser att det kan finnas en problematik när undervisningen i algebra endast introduceras i en aritmetisk tradition där numeriska lösningar och rätt svar fokuseras. Detta då elever fastnar i numeriska lösningar i stället för generella resonemang och slutsatser (se t.ex. Davydov, 2008; Kaput, 2008; Kieran, 2006; Lins & Kaput, 2004; Stacy & Chick, 2004; Radford, 2010, 2014). Att kunna resonera i algebraiska termer är ett av de sätt som matematik kommuniceras på. Vidare är det ett av de sätt som matematisk kunskap etableras som sann. Att elever redan i de tidigare åldrarna ges möjlighet att utveckla en förmåga att föra och följa algebraiskt utformade resonemang blir därför en viktig målsättning för undervisningen. Dock behövs det kunskap om hur en undervisning som möjliggör för elever att utforska algebraiska situationer där sådana resonemang inkluderas kan arrangeras. Samt hur det är möjligt att genomföra så tidigt som i årskurs 1.

    Under höstterminen 2017 och vårterminen 2018 har en forskargrupp bestående av undervisande lärare, forskarstuderande lärare, forskarutbildade lärare samt vetenskapliga ledare genomfört en studie, där vi i en kollaborativ process har utforskat vilka typer av uppgifter och klassrumsdiskussioner som skapar förutsättningar för utveckling och kvalificering av algebraisk resonemangsförmågan hos elever i årskurs 1.

    Learning study har valts som ansats och metod då den erbjuder ett “maskineri” för att närgånget kunna studera relationen mellan undervisning och elevers lärande. Vidare bygger studien på Davydovs (1962, 2008) arbete avseende design av uppgifter och klassrumskommunikationen samt analys av data.

    Inledande analyser antyder att vissa typer av uppgifter och klassrumskommunikation inbjuder till att eleverna prövar relationen mellan ett uttryck och dess representation i form av cuisenaire-stavar samt inbjuder till att resonera om vilka av dessa relationer som fungerar eller ej. 

    Presentationen fokuserar på lärarens interaktion med eleverna under slutfasen av en lektion samt hur lärarens val av kommunikation och sätt att styra det gemensamma arbetet driver processen framåt.

  • 5.
    Fred, Jenny
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Nyman, Martin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Finding new ways of informing iteration between lessons in a learning study: A case of algebraic reasoning2018Conference paper (Refereed)
    Abstract [en]

    The issue for this paper is lesson iteration informed by students’ responses to lesson design seen from the theoretical standpoint of learning activity in a learning study on a lesson framing algebraic reasoning.

    A body of research (e.g.; Kieran, 2006; Lins & Kaput, 2004; Radford, 2010) suggests that there can be a problem if algebra is introduced from an arithmetic standpoint focusing on finding singelnumerical solutions to tasks.This can lead to students getting stuck in the arithmetical interpretation of algebraically problems and prevent them from using algebra as a tool for general mathematical understanding (e.g. Davydov, 2008; Kieran, 2006; Lins & Kaput, 2004; Radford, 2010).

    Learning study can be understood as a lesson study with a focus on a specific object of learning and in combination with an explicit theory of learning (Elliot, 2012). Marton (2015) argues for any functional theory of learning but commonly variation theory is the theory used. During the last years learning activity (Davydov, 2008) has been tested as an alternative and/or complementary theoretical design tool when designing teaching in learning studies (Eriksson, 2017). One design principle when trying to establish a learning activity is that the teacher has to create a situation inviting students to theoretical work (Davydov, 2008). This situation needs to include fundamental aspects of the theoretical knowledge as well as evoke students’ curiosity and urge them to engage in a theoretical work (Eriksson & Jansson, 2017). One difficult act of balance for the teacher is to stage a situation that evokes the curiosity needed for the students to create their own problem/learning task instead of simply solving a teacher assignment.

    In this project the data comes from a learning study in year 1 (age 7) with three lesson iterations. Tentative results indicate that small adjustments in lesson design made on the basis of learning activity generates both a higher degree of student reasoning and on a shift from numerical/arithmetical to algebraical conceptual tools. In the presentation we will provide examples of how learning activity has worked as a tool to improve the staging of a situation inviting the students to engage in algebraic reasoning.

  • 6.
    Fred, Jenny
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Nyman, Martin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Andersson, Carina
    Andersson, Lars
    Björklund, Jenny
    Hur möjliggör helklassdiskussioner urskiljandet av algebraiska komponenters funktion?2019In: Book of abstracts: Lärarnas forskningskonferens 2019, 2019, p. 43-44Conference paper (Other academic)
    Abstract [sv]

    Presentationen syftar till att diskutera hur undervisning kan skapa förutsättningar för att elever engageras i kreativa och reflektiva resonemang om olika komponenters funktion och relation i algebraiska uttryck. 

    Under senare år har intresset för klassrumskommunikationen ökat och därmed även intresset att studera utvecklingen av elevers förmåga att argumentera och delta i matematiska diskussioner i klassrummet (Larsson, 2015; Lithner, 2008). Mot bakgrund av detta uppstår frågor som (1) vilka redskap kan användas i klassrumskommunikationen för att främja innehållsrika, kreativa och reflektiva klassrumsdiskussioner samt (2) hur kan redskapen användas för att främja detta. Lärandeverksamhetsteori (Davydov, 2008) tillhandahåller fyra principer om hur undervisning,uppgifter och klassrumskommunikation kan utformas för att skapa förutsättningar för elever att engageras i ett teoretiskt arbete: problemsituationer, lärandemodeller, motsättningar och kollektiva reflektioner. Idén med en problemsituation är att den ska få eleverna att engageras i ett teoretiskt arbete gällande det avsedda kunskapsinnehållet. Lärandemodeller fungerar i det arbetet som ett nödvändigt visualiserande redskap för eleverna, såväl i utforskandet av innehållsliga aspekter som i de kollektiva reflektiva diskussionerna (Gorbov & Chudinova, 2000). Motsättningar fungerar som en drivkraft som upprätthåller det teoretiska arbetet (Davydov, 2008). Ide kollektiva reflektionerna uppmanas – och utmanas – eleverna till att försöka förklara andras i ljuset av sina egna förklaringar. De utmanas därmed i sitt teoretiska tänkande (Zuckerman, 2004).

    Presentationen grundar sig i en del av ett mer omfattande forskningsprojekt med det övergripande syftet att utforska hur undervisningen, i termer av uppgifter och arbetssätt, kan utformas och utvecklas för att bidra till att eleverna utvecklar en algebraisk resonemangsförmåga. I presentationen kommer följande fråga att fokuseras: Hur kan principerna för lärandeverksamhet fungera somredskap för att möjliggöra för att elever engageras i kreativa och reflektiva resonemang om komponenternas funktion och relation i algebraiska uttryck?

    I forskningsprojektet har learning study (Marton, 2015) använts som ansats för dataproduktion. Data som ligger till grund för presentationen består av fyra transkriberade videoinspelade forskningslektioner genomförda i årskurs 7. Tentativa resultat indikerar att en problemsituation kan iscensättas med hjälp av noggrant utvalda kombinationer av en välkänd figur, såsom en rektangel, kopplad till ett algebraiskt uttryck. Därigenom skapas en komplex problemsituation som bjuder in eleverna till ett teoretiskt arbete. Familjariteten med de ingående delarna i problemsituationen skapar en avspänd öppenhet som i kombination med väl avvägda "provokationer" från läraren, möjliggör för eleverna att ifrågasätta triviala förståelser för algebraiska begrepp och därmed kvalificera diskussionen och nå en djupare förståelse.

  • 7.
    Nyman, Martin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Primary school children's expressed beliefs towards mathematics2017Conference paper (Other academic)
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  • 8.
    Nyman, Martin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Fred, Jenny
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Class discussions enabling discerning algebraic properties - developed in learning study iteration2019Conference paper (Refereed)
    Abstract [en]

    The issue for this paper is to discuss how a lesson can be structured to enable students to engage in creative and reflective discussions about the function for and relation between components in algebraic expressions.

    In recent years, interest in the communicative elements of mathematics education has increased internationally, and then with a special focus on the development of students' mathematical thinking and their ability to reason, argue and participate in mathematical classroom discussions (Kieran, 2001; Larsson, 2015; Lithner, 2008 ; Radford & Barwell, 2016). Based on the above, questions can be raised about which tools are used and how they can be used to promote content-rich, creative and reflective classroom discussions.

    The theory of learning activity (Davydov, 2008) provides four principles on how teaching, tasks and classroom communication can be designed to enable students engage in a theoretical work: (1) the creation of problem situations, (2) the creation and establishment of learning models, (3) the creation or advancement of contradictions and (4) joint reflexive action. The function of a problem situation is to challenge students to be active in a theoretical work where the processing of the problem gives the students the opportunity to work with aspects of the knowledge content that they have not yet distinguished. Learning models should enable the students to theoretically explore the abstract (general) of a given object and further serving as a tool for classroom communication and reflective discussions (Gorbov & Chudinova, 2000). Contradictions are historically developed tensions and the idea of contradictions can be used in the design of problem situations to challenge students to engage in a theoretical work (Davydov, 2008: Zuckerman, 2003). The idea of collective reflections is also central to learning activity, because the students are challenged in their own theoretical thinking by trying to explain someone else’s thoughts and putting it in relation to their own (Zuckerman, 2004).

    The aim of this study is to examine how mathematics education can be designed to develop students algebraic thinking regarding discussing algebraic expressions. The research question addressed in this paper is: how can learning activity work as a tool for informing iteration between lessons in a learning study regarding enabling students to discern algebraic properties using reflective whole class discussions?

    In this project the data comes from a learning study in year 7 (age 13) with four lesson iterations. Learning study (Marton & Tsui, 2004; Marton, 2015) has been used as an approach for data production. The data consists of video recordings from four research lessons and transcriptions of those.

    Tentative result indicate that a problem situation can be successfully staged using combinations of well known conceptual parts, such as a rectangle and an algebraic expression, and thus creating a complex situation enabling students to engage in a theoretical work. The implicit familiarity of the conceptual parts in these combinations, in combination with finely tuned “provocations” from the teacher, forces the students to question trivial understanding of algebraic concepts, thus qualifying the discussion and potentially reaching a deeper understanding.

  • 9.
    Nyman, Martin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Fred, Jenny
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Andersson, Carina
    Andersson, Lars
    Björklund, Jenny
    ”Nä, jag tänkte nog fel ...” - Elevers agens vid algebraiska resonemang i helklass2020Conference paper (Other academic)
    Abstract [sv]

    Vad krävs för att undervisningen ska skapa förutsättningar för elever att engageras i kreativa och reflektiva resonemang om algebraiska uttryck? Davydovs lärandeverksamhetsteori tillhandahåller principer som kan användas av lärare för att designa en undervisning som möjliggör för elever att delta i ett teoretiskt och kvalificerande arbete. Presentationen kommer att diskutera dessa faktorer samt ge exempel på hur uppgifter kan utformas och iscensättas för att möjliggöra detta.

  • 10.
    Nyman, Martin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Sumpter, Lovisa
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    The issue of ‘proudliness’: Primary students’ motivation towards mathematics2019In: LUMAT: Luonnontieteiden, matematiikan ja teknologian opetuksen tutkimus ja käytäntö, ISSN 2323-7104, E-ISSN 2323-7112, Vol. 7, no 2, p. 80-96Article in journal (Refereed)
    Abstract [en]

    In this paper, we study year 2 and year 5 students’ expressed motivations for doing mathematics. The responses were analysed using thematic analysis; first with a deductive approach using themes from previous research, and then an additional inductive analysis searching for new themes. The results show that the children express both intrinsic motivation (cognitive-oriented and emotional-oriented), as well as extrinsic motivation (including outward and compensation). Two new categories of cognitive intrinsic motivation were found—normative and personal. The results also indicated an interplay not only between the different categories but also within categories, signalling that expressed motivation is double-layered. Some implications are discussed.

    Download full text (pdf)
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