# Classroom Research on Mathematics and Language: Seeing Learners and Teachers Differently

Section I: Theorising the complexity of language in mathematics teaching and learningDeveloping a perspective on multiplicity in the study of language in mathematics classroomsIntroductionMultilingualism and multiculturalismMultimodalityMultiple levels of analysisDefining and operationalising ‘language’Defining and operationalising ‘mathematics classroom’Language as the object of study or as the medium of investigationConclusions/ways forwardNotesReferencesLanguage, paralinguistic phenomena and the (same-old) mathematics registerIntroductionMidstream #1: Four anecdotesMidstream #2: Somewhat picky transcriptionsMidstream #3: Engaging with the purported ‘multimodal’ registerMidstream #4: Communication does not only involve languageUniversity lecturer gesturesGrade 3 students working on the iPad App TouchTimesIn conclusionAcknowledgementsNotesReferencesBewitched by language: Questions on language for mathematics education researchIntroductionTheorizing the relation between language and mathematicsResearching the relation between language and mathematicsTheme 1: Linguistic mechanisms that generate mathematical objectsTheme 2: The role of language in the historical emergence of mathematical discoursesTheme 3: Linguistic relativity of mathematicsResearching the relation between language and mathematics learningTheme 4: Linguistic changes in the process of learning mathematicsTheme 5: Linguistic gaps in the classroomTheme 6: Dialogic engagement as a protection from falling into linguistic gapsResearching the relation between language and (mathematics) teachingTheme 7: Teacher’s mathematical languageTheme 8: The language in which the teacher speaks about learners and learningTheme 9: The language in which the teacher speaks about her professional practicesConcluding remarksNotesReferencesLearners’ language in mathematics classrooms: What we know and what we need to knowIntroductionWhat we knowMultiple theoretical frameworksLearners’ language in mathematics classrooms is complexLearners’ mathematical language is much more than precise wordsDeficit views of bilingual learners’ language miss their strengthsImplications for instructionWhat we need to knowDetails of how learners’ multiple languages provide resources for learning mathematicsDetails of learners’ informal language practicesBroadening what is meant by participationDetails of how learners learn to participate in formal language practicesReferencesContent and context specificity matter in the ‘how’ of language-responsive mathematics teacher professional developmentIntroductionThe WMCS and its MDI frameworkExplanatory talk through naming and legitimating and extensions into a teaching frameworkWhy, what and how of mediating language-responsive teaching in WMCS PDPrinciple 1: Distinguish the modelling of word use and mathematical justifications from their mediation as language-specific instructional practicesPrinciple 2: Select/construct mediating tasks where: (a) the learner task and teacher task are clearly distinguished; (b) it is rooted in records of teaching/learning practices and focused on a specific curriculum-level mathematics; and (c) the mathematicPrinciple 3: Elicit and externalize teachers’ spontaneous explanatory talk for collective reflection and interrogationPrinciple 4: Communicate explicit evaluative criteria for improving the quality of word use and mathematical justifications in classroom talkReflection/discussion and conclusionAcknowledgementsNotesReferencesConceptualising and researching mathematics classrooms as sites of communicationIntroductionTheoretical resourcesExample: Multilingual classrooms or second - language learnersCommunication as making functional choicesPedagogic discourse: Participation, social relations and identitiesPosing questionsResearching classrooms as sites of communication: Analytic approachesExample 1: What kinds of mathematics?Example 2: Classroom assessmentEpisode 1a: An answerEpisode 2: A format for answeringEpisode 3: A behaviour or way of feeling or beingFurther directionsAcknowledgementsNotesReferencesSection II: Opening spaces of learning with mathematics classroom research on languageThe role of mathematical vocabulary in moving from the particular to the general with visual representationsIntroductionExtract 1. Whole-class discussion of Figure 7.1 in Tamsin’s lessonLearning the vocabulary of mathematicsThe particular and the generalStructures of interactionMethodThe case of Tamsin and the area of a triangleExtract 2. Simon and Tamsin co-construct a triangle inscribed in a rectangleThe case of Talia and the moment of a forceExtract 3. Sacha and Talia co-modelling the action of a wrenchExtract 4. Simeon offers an alternative way of conceptualising the length of ADExtract 5. Stefan explains the particular case and Shannon offers a new methodDiscussionReferencesMathematics through play: The influence of adult intervention on young children’s shifts between play and mathematical discoursesIntroductionPlay and mathematicsTheoretical underpinningResearch method‘Spontaneous’ play with minimal adult interventionAdult intervention during playFurther reflections on ‘more’Discussion and conclusionReferencesMultilingual mathematics learning from a dialogic-translanguaging perspectiveIntroductionTranslanguaging in multilingual mathematics education researchExploring a dialogic stance in translanguagingDialogic translanguaging for multilingual mathematics learningDialogic translanguaging of diverse meanings for “baixar”Episode 1: “Going down does not mean one by one”Dialogic translanguaging of diverse meanings for “sobras”Episode 2: “I don’t think leftovers means that”Concluding remarksReferencesQuality dimensions for activation and participation in language-responsive mathematics classroomsIntroductionDeveloping a theoretical framework in four dimensions and the supply and use perspectiveSupply-use model for framing the research overviewIntroducing four quality dimensions of mathematics classroom interactionTalk-related dimensionConceptual dimensionDiscursive dimensionLexical dimensionIllustrative case for disentangling four dimensions in two perspectivesOutlook: Operationalizing the dimensions for video-ratingOperationalizing teachers’ intended activation (supply perspective)Operationalizing students’ participation (use perspective)Next steps towards a quantitative video-rating studyAcknowledgementsReferencesReal-world contexts in the mathematics classroom and their impact on the pupils’ language and mathematical learningIntroductionFundamental theoretical considerationsThe dichotomy of academic and everyday languageDiscourseLearning within discourses, learning within interactionReal-world contexts in mathematics educationStudy designMain goals of the study and research questionsThe fundamental methodological assumptions and methods of the studyInitial resultsScene 1. Class discussion at the start of the lesson: What does Croco like to eat?Scene 2. Assistance during individual work on the tasks in the workbook: Do you need help, Nabil?Concluding remarks about the impact of (narrative) real-world contexts for language use and learning opportunitiesReferencesPreservice teachers learning from teaching mathematics in multilingual classroomsIntroductionTheoretical perspectivesMethodologyResultsEpisode 1: Students develop measurements unitsModalitiesCommunicationPotential for learningAvailable identitiesEpisode 2: The students “who struggle a bit” and language issuesModalitiesCommunicationLearning potentialAvailable identitiesDiscussions and concluding remarksReferences