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Inkludering av minoritetsspråklege elevar i den matematiske klasseromsdiskursen
Full text linkSpråkleg mangfald er blitt ein realitet i Norge (Statistisk sentralbyrå, 2023). Dette er noko som lærarar må ta hensyn til i skulen, og då også i matematikkundervisninga. Med kjerneelementa i LK20 har det blitt meir fokus på å kunne kommunisere matematikk. I tillegg til læreplanmåla viser teori og tidlegare forsking at å kommunisere matematikk er ei nyttig kjelde for både matematisk og språkleg utvikling (Chapin et al., 2009; Van de Walle et al., 2015). Likevel viser fleire undersøkingar at dei minoritetsspråklege elevane ofte vert sett til å arbeide med individuelle oppgåver (Grimstad & Myklebust, 2012; Hansson, 2012). Å leie matematiske diskusjonar av høg kvalitet kan vere utfordrande for ein lærar (Wæge, 2015). Dette, i tillegg til at den matematiske diskursen krev forkunnskapar (O’Halloran, 2015), kan kanskje vere årsaker til den store mengda individuelt arbeid som minoritetsspråklege elevar vert sett til i matematikk.
Målet med masteroppgåva er å finne praktiske tiltak som kan hjelpe lærarar å skape ein klasseromsdiskurs i matematikk som er inkluderande for minoritetsspråklege elevar. Interaksjonane skal fremje læring og utvikling hos dei minoritetsspråklege elevane, samt dei andre elevane i klassa. For å finne desse tiltaka har eg, med ei multimodal tilnærming, undersøkt korleis fire ulike minoritetsspråklege elevar vert inkludert i den matematiske klasseromsdiskursen som finn stad i matematikkundervisning. Eg har observert tre ulike matematikkundervisningar med tre ulike lærarar. I etterkant av observasjonane intervjua eg lærarane om inkluderingstiltaka dei brukte og elevane om deira tankar kring tiltaka. Ved å analysere meiningsinnhaldet frå observasjonane og intervjua hadde eg som mål å finne ut kva lærarane og dei minoritetsspråklege elevane tykkjer om inkluderinga i den matematiske klasseromsdiskursen.
Det vart tydeleg gjennom undersøkinga at det er viktig å fremje munnleg matematisk aktivitet for minoritetsspråklege elevar. Omgrepslære vart eit viktig moment for at elevane skal kunne utvikle seg og delta i den matematiske diskursen. Kodeveksling viste seg som eit tiltak med mål om å individualisere og inkludere innanfor omgrepslære. Her vekslar lærarane mellom eit kjent kvardagsspråk, eit vitskapleg matematikkspråk og gestar. Som eit resultat av analysen verka gruppearbeid som ein viktig faktor for deltaking i den matematiske klasseromsdiskursen. Her deltok og trivdes dei minoritetsspråklege elevane, samt utvikla seg språkleg og matematisk. For å skape gruppearbeid og klasseromsdiskusjonar i matematikk som er føremålstenlege for læring, spelar læraren ei viktig rolle. Korleis læraren tilrettelegg for munnleg aktivitet, framhever elevane sine ferdigheiter og responderer på utsagn, er avgjerande for den matematiske utviklinga og den vidare deltakinga i den matematiske klasseromsdiskursen.Linguistic diversity has become a reality in Norway (Statistisk sentralbyrå, 2023). Something that needs to be taken into consideration in school and mathematics education. In the Core Elements of LK20, there is more focus on being able to communicate mathematics. In addition to the curriculum aims, theory and previous research illustrates the importance of communicating in mathematics, where communication can benefit both mathematical and linguistic development (Chapin et al., 2009; Van de Walle et al., 2015). Nevertheless, several surveys show that the minority language speaking pupils often are assigned individual tasks (Grimstad & Myklebust, 2012; Hansson, 2012). Leading high-quality mathematical discussions can be challenging for a teacher (Wæge, 2015). Considering this, and furthermore comprehending that mathematical discourse requires prior knowledge (O’Halloran, 2015) could explain the extensive amount of individudal work that the minority language speaking pupils are required to do in the mathematics classroom.
My purpose with this master’s thesis is to find practical implications that may aid teachers in creating classroom discourse in mathematics which is inclusive for minority language speaking students. The interactions will promote learning and development both for these pupils, and also for other students in the class. In order to find these measures, I have, with a multimodal approach, examined how four different minority language speaking students are included in the mathematical classroom discours that occurs in mathematics education. I have observed three different mathematics lessons with three different teachers. Afterwards, I interviewed the teachers about the measures of inclusion they employed, and furthermore interviewed the pupils about their attitudes towards these.
Through my study, the importance of promoting oral mathematical activity for the minority-language pupils became evident. Concept learning became an important element for the pupils to be able to develop and participate in the mathematical discourse. Code-switching proved to be a measure with the intention of individualizing and including within concept learning. The teachers alternated between a familiar everyday language, a scientific mathematical language, and gestures. As a result of the analysis, group work appeared to be an important factor for participation in the mathematical classroom discourse. In group work the minority-language students participated and thrived, as well as developed linguistically and mathematically. In order to achieve group work and classroom discussions in mathematics that are appropriate for learning the teacher plays an important role. It is evident that the manner the teacher organizes oral activities, emphasise the students skills, and respond to statements is crucial for the mathematical development and further participation in the mathematical classroom discourse
Kommunikasjonens rolle i matematikkundervisning
Full text linkI denne studien har hovedfokuset vært på kommunikasjon i matematikkundervisning, både verbalt og ikke-verbalt. Det har blitt undersøkt hvilken rolle lærerens kommunikasjon har for engasjement og deltakelse blant elevene. Studien har blitt gjennomført kvalitativt ved å observere og ta videoopptak av tre matematikktimer på 9. trinn, i deres undervisning om geometri. Lærerens kommunikasjonsvalg har blitt analysert og diskutert i sammenheng med elevenes deltakelse og engasjement. Jeg har i denne studien fokusert på elevenes visuelle deltakelse og engasjement, altså når tydelige tegn til deltakelse og engasjement vises. Bruk av tavle og konkreter i undervisning står også sentralt som en av lærerens valg av kommunikasjon.
I et klasserom er det et bredt spekter av personligheter og interesser. Ulike faktorer og forutsetninger er også redegjort for, da flere ytre faktorer kan ha en innvirkning på elevenes deltakelse og engasjement. Blant annet diskuteres interesser, tidsbruk og elevenes evne til å holde fokus over tid.
Funnene indikerer at lærerens kommunikasjon tydelig spiller en rolle for elevenes deltakelse og engasjement, men i ulik grad. Elevene ser ut til å speile og implementere lærerens forklaringer i egne forklaringer, både verbalt og ikke-verbalt. I tillegg virker elevene mer engasjerte når det er mer å følge med på. Lærerens aktive bruk av tavle, konkreter og tydelige gestikuleringer fanger ofte oppmerksomheten til elevene, som i flere tilfeller fører til økt deltakelse. Funnene diskuteres opp mot læreplanen og dens fokus på kommunikasjon i skolen og matematikkundervisning.The main focus of this study has been on communication in mathematics teaching, both verbally and non-verbally. The role of the teacher's communication for engagement and participation among the pupils has been researched. This study adopted a qualitative approach, where I have been observing and video recorded three mathematics lessons, in their normally and naturally occurring lessons where the focus of teaching was about geometry, at a 9th grade in Norway. The teacher's communication choices have been analysed and discussed in the context of pupils' participation and engagement. In this study, I have focused on the pupils’ visual participation and engagement, when clear signs of participation and engagement are shown. The use of blackboards and manipulatives in the teaching was also perceived to be as central as one of the teacher's choices of communication.
In a classroom, there is a wide range of personalities and interests. Different factors and prerequisites are also accounted for, as several external factors may have an impact on the pupils’ participation and engagement. Among other things, their interests, time frame and the pupils’ ability to stay focused over time are also discussed.
The results indicate that the teacher's communication clearly plays a role in pupils' participation and engagement, but to varying degrees. The pupils seem to mirror and implement the teacher's explanations in their own explanations, both verbally and non-verbally. In addition, the pupils seem to be more engaged when there is more to pay attention to. The teacher's active use of the blackboard, manipulatives and gestures often captures pupils’ attention, which in several cases leads to increased participation. The findings are discussed in relation to the curriculum and its focus on communication in school and mathematics teaching
Jeg bare ser på spørsmålet og så popper det opp regnemetoder i hodet
Full text linkMatematisk fleksibilitet har blitt et viktig læringsmål i skolen de seneste årene. Fleksibilitet defineres som å kunne flere ulike løsningsstrategier, samt å kunne bytte mellom og velge en hensiktsmessig strategi for en oppgave/problem – gitt at man tar hensyn til konteksten som oppgaven blir gitt i, og hva eleven som skal løse oppgaven selv kan. Det kommer stadig mer forskning på feltet, men lite av forskningen er gjort i norske kontekster. Gjennom denne studien har jeg fokusert på evnerike elevers matematiske fleksibilitet innen brøk, da evnerike elever også er lite forsket på i en norsk kontekst. Målet med studien er å gi lærere bedre innsikt i evnerike elevers fleksibilitet, samt hvordan lærere kan tilrettelegge for slike elever. Studien er en kvalitativ studie, hvor to hele klasser løste 5 brøkoppgaver. Elevene skulle løse oppgavene med så mange strategier de klarte. Ut fra svarene elevene gav og en samtale jeg hadde med læreren deres, ble det plukket ut 5 elever som kan betegnes som evnerike. Dagen etter oppgaveløsningen, ble de evnerike elevene intervjuet angående løsningsstrategiene deres. Intervjuene ble filmet slik at man ordrett fikk det de sa, og slik at man kunne se på kroppsspråket til elevene. Etter innsamlingen av datamaterialet, begynte transkripsjonsprosessen og den systematiske analysen. Det ble gjort en tematisk analyse. Resultatene av analysen fikk frem at de evnerike elevene kan betegnes som fleksible innen brøk. I tillegg ble det tydelig at de fleste elevene brukte og foretrakk å løse oppgavene ved formell notasjon. Samtidig viste resultatene at mange av elevene brukte mentale strategier for å løse brøkoppgavene. Til tross for fellestrekket med at elevene var evnerike, kom det tydelig frem at elevene var forskjellig når det gjaldt om de likte å løse samme oppgave flere ganger eller ikke.
Nøkkelord: matematisk fleksibilitet, evnerike elever, tilpasset opplæring, brøkMathematical flexibility has become an important learning goal in schools the recent years. Flexibility is defined as the knowledge of multiple strategies, while being able to switch between and apply a meaningful strategy to a problem – given the context the problem is given, and the students personal knowledge and preference. Flexibility has received increasing research attention in the field, but little of the research has been done in a Norwegian context. Throughout this study, I have focused on gifted and talented students' mathematical flexibility in fractions, as gifted and talented students also need more research in the Norwegian context. The aim of the study is to envision teachers with better insight on the flexibility of gifted and talented students so they can give an inclusive education. The study is a qualitative study were two classes solved 5 fraction problems. The students solved the problems using as many strategies they knew. Given the answers they gave and a conversation I had with the teacher, I selected 5 students that were perceived to be gifted and talented. The following day I interviewed the gifted and talented students regarding their problem solving strategies. The interviews were filmed so I could hear verbatim what they said, and see their body language. Afterwards the transcription prosess and the systematic analysis began. A thematic analysis was made. The results indicate that the gifted and talented students can be described as flexible in dealing with fractions. In addition, it became clear that most students used and preferred to solve the tasks using formal notation. At the same time, the results illustrated that many of the students used mental strategies to solve the fraction problems. Despite the common feature that the students were gifted and talented, it became clear that the students differed when it came to whether they enjoyed repeatedly solving the same problem or not.
Key words: mathematical flexibility, gifted and talented students, inclusive education, fractio
Deictic gestures as amplifiers in conveying aspects of mathematics register
No full textInternational audienceThis paper investigates communication (verbal and nonverbal) in the bilingual (Farsi-English) mathematics classroom. The study examines the communicative repertoire that interlocutors employed in lessons enabling them to construct meaning and mediate understanding. That is, the ways in which language and gesture can be seen as resources in supporting and conveying mathematical ideas is described. It appeared that the pointing gestures that were produced by the classroom teachers served as a mnemonic device to help remembering the key mathematics register
Making multi-modal mathematical meaning in multilingual classrooms
No full textThis thesis investigates communication (verbal and nonverbal) in a bilingual (Farsi-English) complementary school mathematics’ classroom. The study examines gestures were used as a resource for teaching mathematics in a bilingual setting, enabling intercolutors to construct meaning and mediate understanding. That is, the ways in which language and gesture can be seen as resources in supporting and conveying mathematical ideas is described. I investigated a number of verbal and nonverbal resources and show how these are culturally and socially shaped. I also explored how modes of communication are employed in creating mathematical meaning in a bilingual classroom context. A multimodality framework was adopted to analyse data which included audio and video recordings, observations and interviews with teachers and pupils. I found that gestures were employed to convey aspects of the mathematical register and how these were used to amplify what interlocutors were expressing verbally. Furthermore, I identified that different languages activated a different conceptual understanding of the same mathematical concept which was reflected through the students’ and teachers’ gestures
El problema con la brecha generacional en la educación: una aproximación bidireccional
No full textMe gustaría comenzar este comentario editorial con una cita que leí: “La juventud de hoy está mimada desde la raíz. Son malos, impíos y perezosos. Nunca serán como los jóvenes de antes y nunca podrán aferrarse a nuestros valores”.
Esta cita no proviene de un artículo que leí en un periódico conservador publicado hace 2 días, ni es parte de un discurso político de un candidato presidencial chileno de derecha. Esta oración se muestra en una piedra babilónica, la que se ha escrito hace más de 3000 años. A lo largo del tiempo, ha habido un fenómeno conocido como la brecha generacional. Este fenómeno es un término popular utilizado para describir las diferencias en las normas culturales entre los miembros de una generación más joven y sus mayores
Dois peixes movendo-se em seus mares : como se mostra a linguagem corpórea de professores que ensinam equações matemáticas?
Full text linkBackground: “Culture hides much more than it reveals and, strangely enough, what it hides, it hides more effectively from its own participants” (Hall, 1959, p. 39). This quote corresponds well to a Persian proverb, also a well-known aphorism that has been widely cited in many ethnographic articles: “a fish will be the last to discover water.” Being immersed in water, surrounded by it, makes it invisible and almost impossible to perceive. In other words, we often do not know our interactional behaviour as mathematics teachers when we perform it in our usual and localised professional practice. Objective: To discuss mathematics teacher’s body language when teaching equations and thus perceive this language in terms of possible fruitful educational action when teaching equations in the classroom. Design: Qualitative methodology. Data collection and analysis: Based on theoretical references that deal with body language, corporeality, and perception, we analysed individually and comparatively the classes of two mathematics teachers who taught equations in Birmingham (United Kingdom) and Rolante (Brazil). Thus, particularly attentive to mathematical culture in the classroom and analysing the localised gestures in the teachers’ teaching of equations and the non-verbal behaviour, we can understand mathematics teaching through body movement, which often goes unnoticed. Results: We understand from the results of this research that perceiving the body language of mathematics teachers, which is produced with speech, gives us indications of the materialisation of the meanings attributed to the equation and how this will possibly affect the very constitution of the student’s mathematical knowledge, in terms of possible meanings attributed to each gesture. Conclusions: We consider that knowing the body language can favour the teacher’s teaching, i.e., metaphorically, knowing the sea can favour the fish to swim.Contexto: “A cultura esconde mais do que revela e, por mais estranho que pareça se esconder, esconde-se mais efetivamente de seus próprios participantes” (Hall, 1959, p. 39). Esta citação se enquadra muito bem com um provérbio persa, também, um aforismo bem conhecido que tem sido bastante citado em muitos artigos etnográficos, o qual se apresenta como “um peixe é a última criatura a descobrir a água”. Estar imerso na água, cercado por ela, torna invisível e quase impossível percebê-la. Em outras palavras, muitas vezes desconhecemos nosso comportamento interacional, enquanto professores de matemática, quando o realizamos em nossa prática profissional habitual e localizada. Objetivo: Discutir como se mostra a linguagem corpórea do professor de matemática ao ensinar equações e, assim, perceber essa linguagem em termos de ação educacional profícua ao se ensinar equações em sala de aula, por exemplo. Metodologia: Metodologia qualitativa. Coleta de dados e análise: A partir de referenciais teóricos que tratam de linguagem corpórea, corporeidade e percepção, analisamos as aulas de dois professores de matemática que ensinavam equações, em Birmingham (Reino Unido) e em Rolante (Brasil), individualmente e comparativamente. Assim, prestando atenção especial à cultura matemática em sala de aula e analisando os gestos localmente situados no ensino de equações e o comportamento não-verbal dos professores, podemos compreender o ensino de matemática por meio do movimento do corpo, o qual muitas vezes passa despercebido. Resultados: Compreendemos com os resultados dessa pesquisa que perceber a linguagem corpórea dos professores de matemática, a qual é produzida-coma-fala, nos dá indicativos da materialização do sentidos atribuídos à equação e como isso possivelmente afetará a própria constituição do conhecimento matemático do estudante, em termos de possíveis sentidos atribuídos a cada gesto. Conclusões: Consideramos que conhecer a linguagem corpórea pode favorecer o próprio ensinar do professor, ou seja, metaforicamente, conhecer o mar pode favorecer o peixe a nadar
The understanding of abstract concepts: a perspective from distributed models of conceptual representation
Full text linkThis theoretical article looks at the process of understanding abstract concepts from the perspective of distributed models of conceptual representation. According to these models, meanings of concepts are essentially componential; that is, the meaning of any concept is represented by small units of meaning, which are called semantic features. Based on these models, this article suggests that scope of interpretation and range of associated semantic features are two key differences between abstract and concrete concepts. While abstract concepts are interpreted within wide scopes and in association with large sets of semantic features, concrete concepts are interpreted within narrower scopes and in association with relatively smaller sets of semantic features. Extending the discussion into the metaphoric understanding of abstract concepts in terms of concrete concepts, it is suggested that when an abstract concept is processed, the activation of low-level sub-features may take place in a variety of ways.publishedVersio
Making visible “the invisible”: Can mathematics embedded in work practices enable critical questioning?
No full textThis short report explores and unpacks mathematical knowledge embedded in work practices (which often remains invisible or sedimented) and how this ‘invisible’ knowledge-form underpins sociocultural resources and issues. In so doing, this report argues that the middle graders’ awareness of embedded mathematical knowledge in work practices not only creates opportunities and affordances for furthering mathematics learning but in the process, this awareness builds potential for enabling the doers to see the “invisible,” leading to a possibility of questioning the complex societal meaning of justice, fair distribution, welfare and access. This possibility of questioning plays a dual relationship of cause and effect with doers’ (read learners’) foregrounds (see Skovsmose, 2014) and creates a platform which enables and empowers them for critical questioning. This broad platform is referred to as “landscapes of investigation” (Skovsmose, 2001). There is not much research in the literature which uses instances of income-generating work practice as examples of landscapes of investigation for exemplifying out-of-school mathematical knowledge of children immersed in such work-contexts and also as possible tools for manifesting and investigating societal issues. In this report, we use examples of out-of-school work contexts and the community’s rich cultural knowledge resource for responding to this gap in the literature
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