Abstract
[English]: In this article, languaging processes in mathematics education will be reflected from a theoretical and methodological viewpoint. Language is not just a tool for language learning: It is a highly complex medium for transporting meaning. It plays a key role in explaining and fostering as well as reconstructing and interpreting cognitive processes – not only in mathematics education, but due to the abstract nature of mathematical objects in a particularly important way. Thus, language as a mediational dimension is essential in the learners’ as well as researchers’ processes of understanding, be it in interpretational processes or in verbalized, deictical, either explicit or implicit explanations and actions. In this article, this dual-sided perspective will be explained by giving insights into language-related processes of interpreting and understanding mathematical relations in Substantial Learning Environments (SLE’s) as well as relational strategies such as the ‘Auxiliary Task.’
[Bahasa]: Artikel ini membahas proses berbahasa dalam pendidikan matematika dari perspektif teoritis dan metodologi. Bahasa bukan saja sebuah alat untuk pembelajaran bahasa tetapi juga sebuah medium yang sangat kompleks untuk menyampaikan makna. Bahasa memainkan peran penting untuk menjelaskan dan memelihara serta mengembangkan dan memaknai proses kognitif - tidak hanya dalam pendidikan matematika, tetapi karena karakateristik objek matematika yang abstrak sehingga peran bahasa diperlukan. Oleh sebab itu, bahasa sebagai sebuah dimensi mediasional sangat penting dalam proses pemahaman siswa dan peneliti baik itu dalam proses interpretasi atau dalam penjelasan dan tindakan yang diverbalkan, deiktis, baik eksplisit maupun implisit. Dalam artikel ini, perspektif dua sisi ini akan dijelaskan dengan memberikan wawasan ke dalam proses yang berhubungan dengan bahasa untuk menafsirkan dan memahami hubungan matematika dalam konteks Linkungan Belajar Substansial (Substantial Learning Environments, SLE) serta strategi relasional seperti Tugas Tambahan (Auxiliary Task).
Downloads
References
- Aebli, H. (1973). Introduction. In Steiner, G. (Ed.). Mathematics as thinking education. a psychological exploration of the role of thinking in mathematical early education (pp. I-XV). Klett.
- Altieri, M., & Prediger, S. (2016). Unpacking procedural knowledge in mathematics exams for first-year engineering students. In R. Göller, R. Biehler, R. Hochmuth & H.-G. Rück (Eds.), Didactics of mathematics in higher education as a scientific discipline conference proceedings (p. 143–147). Kassel/Lüneburg: khdm.
- Atzert, R., John, R., Preisfeld, A., & Damerau, K. (2020). The influence of the criterial, social, and temporal reference norm on students’ experiment-related self-concept. Zeitschrift für Didaktik der Natur¬wissen¬schaften 26, 89–102. https://doi.org/10.1007/s40573-020-00114-x
- Bakhtin, M. (1975). Voprosy literatury i èstetiki [The dialogic imagination]. Austin: University of Texas Press.
- Bardini, C., Radford, L., & Sabena, C. (2005). Struggling with variables, parameters, and indeterminate objects or how to go insane in mathematics. The proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. https://hdl.handle.net/2318/67210
- Barwell, R. (2009). Multilingualism in mathematics classrooms – Global perspectives. Multilingual Matters.
- Barwell, R. (2016). A Bakhtinian perspective on language and content integration. In A. Ball & S. Freedman (Eds.), Bakhtinian perspectives on language, literacy, and learning (pp. 101–120). Cambridge: Cambridge University Press. https://doi.org/10.21832/9781783096145-008
- Barwell, R. (2020). Language background in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 441- 447). New York: Springer. https://doi.org/10.1007/978-3-030-15789-0_86
- Bauersfeld, H. (1980). Hidden dimensions in the so-called reality of a mathematics classroom. Educational Studies in Mathematics, 11, 23–41. https://doi.org/10.1007/BF00369158
- Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational Number Concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (91-125). New York: Academic Press.
- Bernstein, B. (1971). Theoretical studies towards a sociology of language. London: Routledge & Kegan Paul.
- Bonefeld, M., & Dickhäuser, O. (2018). (Biased) Grading of students’ performance: students’ names, performance level, and implicit attitudes. Frontiers in Psychology, 9 (481), 1-13. https://doi.org/10.3389/fpsyg.2018.00481
- Bourdieu, P. & Passeron, J.-C. (1977). Reproduction in education, society and culture. Sage.
- Bourdieu, P. (1991). Language and symbolic power [translated by G. Raymond & M. Adamson]. Oxford: Basil Blackwell.
- Brandt, B., & Schütte, M. (2010). Collective mathematical reasoning in classrooms with a multilingual body of pupils. In U. Gellert, E. Jablonka, & C. Morgan (Eds.), Proceedings of the 6th IMES Conference (pp. 111–114). Berlin: Freie Universität. https://www.ewi-psy.fu-berlin.de/en/v/mes6/documents/project_presentation/Brandt_Schuette_MES6.pdf
- Brandt, B., & Tiedemann, K. (2019). Mathematiklernen aus interpretativer Perspektive I. Waxmann.
- Büscher, C. (2018). Mathematical literacy on statistical measures. A design research study. Springer.
- Carrutthers, P. (2002). The cognitive functions of language. Behavioral and Brain Sciences, 25, 657-674. https://doi.org/10.1017/S0140525X02000122
- Clyne, M. (2008). The monolingual mindset as an impediment to the development of plurilingual potential in Australia. Sociolinguistic Studies, 2(3), 347–366. https://doi.org/10.1558/sols.v2i3.347
- Cobb, P., & Bauersfeld, H. (1995). The emergence of mathematical meaning. Interaction in classroom cultures. Routledge.
- Cummins, J. (1979). Cognitive/academic language proficiency, linguistic interdependence, the optimum age question and some other matters. Working Papers on Bilingualism, 19, 121- 129.
- diSessa, A. A. (2007). An interactional analysis of clinical interviewing. Cognition and Instruction, 25(4), 523-565. https://doi.org/10.1080/07370000701632413
- Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1-2), 103-131. https://doi.org/10.1007/s10649-006-0400-z
- El-Mafaalani, A. (2012). BildungsaufsteigerInnen aus benachteiligten Milieus: Habitustransformationen und soziale mobilität bei einheimischen und türkeistämmigen. Springer.
- Erath, K., Prediger, S., Quasthoff, U., & Heller, V. (2018). Discourse competence as important part of academic language proficiency in mathematics classrooms: the case of explaining to learn and learning to explain. Educational Studies in Mathematics, 99, 161–179. https://doi.org/10.1007/s10649-018-9830-7
- Erickson, F. (2005). Definition and analysis of data from videotape: Some research procedures and their rationales. In J. L. Green et al. (Eds)., Handbook of complementary methods in education research. New York: Routledge.
- Esmonde, I. (2009). Explanations in mathematics classrooms: a discourse analysis. Canadian Journal of Science, Mathematics and Technology Education, 9(2), 86–99. https://doi.org/10.1080/14926150902942072
- Fausey, C. M., & Boroditsky, L. (2011). Who dunnit? Cross-linguistic differences in eye-witness memory. Psychon Bull Rev., 18(1), 150-157. https://doi.org/10.3758/s13423-010-0021-5
- Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. Dordrecht: Reidel.
- Foucault, M. (1977). History of systems of thought. In D. F. Bouchard (Ed.), Language, counter-memory, practice: Selected essays and interviews by Michel Foucault (pp. 119–204). Ithaca: Cornell University Press.
- Ginsburg, H. P. (1997). Entering the child’s mind. New York: Columbia University.
- Ginsburg, H. P. (2009). The challenge of formative assessment in mathematics education: children’s minds, teachers’ minds. Human Development, 52(2), 109–128. https://doi.org/10.1159/000202729
- Glade, M., & Prediger, S. (2017). Students’ individual schematization pathways - empirical reconstructions for the case of part-of- part determination for fractions. Educational Studies in Mathematics, 94(2), 185-203. https://doi.org/10.1007/s10649-016-9716-5
- Gogolin, I. (1994). Der monolinguale habitus der multilingualen schule. Waxmann.
- Gogus, A. (2012). Constructivist learning. In: Seel, N.M. (Ed.) Encyclopedia of the Sciences of Learning (pp. 783-786). Boston, MA: Springer. https://doi.org/10.1007/978-1-4419-1428-6_142
- Halliday, M. A. K. (1978). Language as social semiotic: The social interpretation of language and meaning. London: Edward Arnold.
- Hammond, J., & Gibbons, P. (2005). Putting scaffolding to work: The contribution of scaffold- ing in articulating ESL education. Prospect, 20(1), 6-30. https://search.informit.org/doi/10.3316/aeipt.143258
- Jungwirth, H. (2003). Interpretative forschung in der mathematikdidaktik – ein überblick für irrgäste, teilzieher und standvögel. Zentralblatt für Didaktik der Mathematik, 35(5), 189- 201. https://doi.org/10.1007/BF02655743
- Jussim, L., & Harber K.-D. (2005). Teacher expectations and self-fulfilling prophecies: knowns and unknowns, resolved and unresolved controversies. Personality and Social Psychology Review, 9(2), 131-155. https://doi.org/10.1207/s15327957pspr0902_3
- Klose, R., & Schreiber, C. (2018). TellMEE – Telling mathematics in elementary education. In Benz, C., Steinweg, A., Gasteiger, H., Schöner, P., Vollmuth, H., & Zöllner, J. (Eds.), Mathematics education in the early years (159-177). Cham: Springer International.
- Koch, P., & Oesterreicher, W. (1985). Language of immediacy - language of distance: orality and literacy from the perspective of language theory and linguistic history [translated by authors]. Romanistisches Jahrbuch 36(1), 15-43. http://doi.org/10.15496/publikation-20415
- Kollhoff, S. (2021). Analyse von transferprozessen in der entwicklung des bruchzahlbegriffs theoretische rahmung und empirische untersuchung. Wiesbaden: Springer.
- Kollhoff, S. (2022). Analysing processes of transfer in learning basic fraction concepts: a didactical approach to transfer. In C. Fernández, S. Llinares, Á. Gutiérrez, N. Planas, & PME (Eds.), Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education (pp. 51-58). Alicante. http://hdl.handle.net/10045/126618
- Krummheuer, G., & N. Naujok (1999). Grundlagen und beispiele interpretativer unterrichtsforschung. Opladen, Leske + Budrich
- Krummheuer, G., & Brandt, B. (2001). Paraphrase und traduktion: Partizipationstheoretische elemente einer interaktionstheorie des mathematiklernens in der grundschule. Weinheim, Basel: Beltz.
- Krummheuer, G. (2002). The comparative analysis in interpretive classroom research in mathematics education. In J. Novotná (Ed.), Proceedings of the 2nd Conference of the European Society for Research in Mathematics Education (pp. 339–346). Mariánské Lázně: Czech Republic: Charles University, Faculty of Education and ERME. https://doi.org/10.1007/BF02655650
- Kunter, M., Baumert, J., Blum, W., Klusmann, U., Krauss, S., & Neubrand M. (2013). Cognitive activation in the mathematics classroom and professional competence of teachers. Boston: Springer.
- Kuzu, T., & Prediger, S. (2017). Two languages – separate conceptualizations? Multilingual students’ processes of combining conceptualizations of the part-whole concept. In B. Kaur, W. K. Ho, T. L. Toh, & B. H. Choy (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education (pp. 121-128). PME.
- Kuzu, T., & Nührenbörger, M. (2021). The conceptual understanding of mental calculation strategies: Meaning-making in the case of the ‘Auxiliary task’. In J. Novotná, H. Moraová (Eds.), Proceedings of the International Symposium on Elementary Maths Teaching (pp. 270-280). Charles University, Faculty of Education. https://pub.uni-bielefeld.de/record/2967196
- Kuzu, T. (2019). Multilingual conceptual development: A study of learning processes with respect to the part-whole concept among German-Turkish learners. Wiesbaden: Springer.
- Kuzu, T. (2022). Pre-Algebraic aspects in arithmetic strategies – The generalization and conceptual understanding of the ‘Auxiliary task’. Eurasia Journal of Mathematics, Science and Technology Education, 18 (12), 1-17. https://doi.org/10.29333/ejmste/12656
- Kuzu, T. (2023a). Multilingual meaning making – An explorative study of German-Turkish 6th graders’ translanguaging processes regarding the part-whole-concept. Journal for Didactics of Mathematics, 1-25. https://doi.org/10.1007/s13138-023-00219-z
- Kuzu, T. (2023b). Mental calculation strategies as a ‘missing link’ between arithmetic and algebra – Insights into the 'Auxiliary Task' and its role in the ‘Cognitive Gap’. Turkish Journal of Mathematics Education, 4 (1), 1-23. https://tujme.org/index.php/tujme/article/view/69
- Kuzu, T. (2023c). Interactional forces in multilingual discourses – A teachers’ perspective on learners’ agency. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel & M. Tabach (Eds.). Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3 (pp. 235-242). University of Haifa, Israel: PME. https://www.igpme.org/publications/current-proceedings/
- Lupyan, G. (2017). How reliable is perception? Philosophical Topics, 45(1), 81-106. http://dx.doi.org/10.5840/philtopics20174515
- Maier, H., & Schweiger, F. (1999). Mathematik und sprache: zum verstehen und verwenden von fachsprache im mathematikunterricht. Wien: öbv & hpt.
- Mayring, P. (2015). Qualitative content analysis: theoretical background and procedures. In A. Bikner-Ahsbahs et al. (Eds.), Approaches to qualitative research in mathematics education, advances in mathematics education. Springer. https://doi.org/10.1007/978-94-017-9181-6_13
- Meissner, H. (1986). Cognitive conflicts in mathematics learning. European Journal of Psychology of Education, 1(2), 7-15. https://doi.org/10.1007/BF03172566
- Meyer, M. (2010). Abduction—A logical view for investigating and initiating processes of discovering mathematical coherences. Educational Studies in Mathematics, 74, 185–205. https://doi.org/10.1007/s10649-010-9233-x
- Meyer, M., & Tiedemann, K. (2017). Sprache im fach mathematik. Springer.
- Morek, M., & Heller, V. (2012). Academic language – Communicative, epistemic, social and interaktive aspects of its use. Journal for Applied Linguistics, 57(1), 67-101. http://doi.org/10.1515/zfal-2012-0011
- Norén, E. (2015). Agency and positioning in a multilingual mathematics classroom. Educational Studies in Mathematics, 89(2), 167-184. http://doi.org/10.1007/s10649-015-9603-5
- Nührenbörger, M. (2015). Mathematical argumentation processes of children between calculation and conversion. In J. Novotná, & H. Moraová (Eds.), Developing mathematical language and reasoning Proceedings of SEMT`15 (pp. 18-26). Prague: Charles University, Faculty of Education.
- Nührenbörger, M., & Steinbring, H. (2009). Forms of mathematical interaction in different social settings: examples from students’, teachers’ and teacher–students’ communication about mathematics. Journal of Mathematics Teacher Education, 12(2), 111–132. https://doi.org/10.1007/s10857-009-9100-9
- Nührenbörger, M., & Schwarzkopf, R. (2015). Processes of mathematical reasoning of equations in primary mathematics lessons. In N. Vondrová, & K. Krainer (Eds.), Proceedings of the 9th Congress of the European Society for Research in Mathematics Education (CERME 9) (pp. 316–323). Prague: ERME. https://www.hal.inserm.fr/CERME9-TWG02/hal-01281852v1
- Nührenbörger, M., Rösken-Winter, B., Ip Fung, C., Schwarzkopf, R., Wittmann, E. C., Akinwunmi, K., Lensing, F., & Schacht, F. (2016). Design Science and Its Importance in the German Mathematics Educational Discussion. (ICME-13 Topical Surveys). Springer.
- Oevermann, U., Allert, T., Konau E., & Krambeck, J. (1979). The methodology of a objective hermeneutics and its research-logical meaning in the social sciences. Science Review, 2(19), 25-31. https://doi.org/10.31435/rsglobal_sr/28022019/6366
- Ortmann, K., & Dipper, S. (2019). Variation between different discourse types: literate vs. Oral. In association for computational linguistics [Paper Presentation]. Proceedings of VarDial (pp. 64–79). Minneapolis: Minneapolis University. http://doi.org/10.18653/v1/W19-1407
- Ozçakır, B., Konca, A. S., & Arıkan, N. (2019). Children’s geometric understanding through digital activities: the case of basic geometric shapes. International Journal of Progressive Education, 15(3), 108-122. https://doi.org/10.29329/ijpe.2019.193.8
- Paavola, S. (2011). Lorenzo Magnani: Abductive cognition. The epistemological and eco-cognitive dimensions of hypothetical reasoning. Journal for General Philosophy of Science, 42(1), 201–205. https://doi.org/10.1007/s10838-011-9146-0
- Pant, H. A. (2020). Notengebung, leistungsprinzip und bildungsgerechtigkeit. In S. Beutel & H. A. Pant (Eds.), Lernen ohne Noten: Alternative konzepte der leistungsbeurteilung (pp. 22-58). Kohlhammer Verlag.
- Peirce, C. S. (1903). Harvard lectures on pragmatism: Lecture VII. MS [R] 315.
- Piaget, J. (1929/ 1976). The child’s conception of the world (J. Tomlinson & A. Tomlinson, transl). Littlefield, Adams & Co.
- Piaget, J. (1932/ 1965). The moral judgment of the child. The Free Press.
- Pinker, S. (1995). The language instinct: How the mind creates language. Harper Perennial.
- Planas, N., & Setati-Phakeng, M. (2014). On the process of gaining language as a resource in mathematics education. Mathematics Education, 46(6), 883-893. https://doi.org/10.1007/s11858-014-0610-2
- Planas, N. (2018). Language as resource: a key notion for understanding the complexity of mathematics learning. Educational Studies in Mathematics, 98, 215–229. https://doi.org/10.1007/s10649-018-9810-y
- Planas, N. (2021). Challenges and opportunities from translingual research on multilingual mathematics classrooms. In A. A. Essien & A. Msimanga (Eds.), Multilingual education yearbook 2021 (pp. 1–19). Cham: Springer. https://doi.org/10.1007/978-3-030-72009-4_1
- Pöhler, B., & Prediger, S. (2015). Intertwining lexical and conceptual learning trajectories – A design research study on dual macro- scaffolding towards percentages. Eurasia Journal of Mathematics, 11(6), 1697-1722. https://doi.org/10.12973/eurasia.2015.1497a
- Pöhler, B., Prediger, S., & Neugebauer, P. (2017). Content- and language integrated learning: A field experiment for percentages. In Berinderjeet Kaur, Wen Kin Ho, Tin Lam Toh, & Ban Heng Choy (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (p. 73-80). PME.
- Prediger, S. (2006). Continuities and discontinuities for fractions. A proposal for analyzing in different levels. In Novotna, J. et al. (Eds.), Proceedings of the 30th PME (pp. 377-384). Prague.
- Prediger, S. (2019). Theorizing in design research: Methodological reflections on developing and connecting theory elements for language-responsive mathematics classrooms. Avances de Investigación en Educación Matemática, 15, 5-27. https://doi.org/10.35763/aiem.v0i15.265
- Prediger, S., & Şahin-Gür, D. (2020). Eleventh graders’ increasingly elaborate language use for disentangling amount and change: A case study on the epistemic role of syntactic language complexity. Journal for Didactics of Mathematics, 41, 43–79. https://doi.org/10.1007/s13138-019-00155-x
- Prediger, S., Clarkson, P., & Bose, A. (2016). Purposefully relating multilingual registers: Building theory and teaching strategies for bilingual learners based on an integration of three traditions. In R. Barwell, P. Clarkson, A. Halai, M. Kazima, J.N. Moschkovich, N. Planas, M. Phakeng, P. Valero, & M. Villavicencio-Ubillús (Eds.), Mathematics education and language diversity. ICMI 21 (pp. 193-215). Springer. https://doi.org/10.1007/978-3-319-14511-2_11
- Prediger, S., Gravemeijer, K., & Confrey, J. (2015). Design research with a focus on learning processes – an overview on achievements and challenges. ZDM Mathematics Education, 47(6), 877-891. https://doi.org/10.1007/s11858-015-0722-3
- Prediger, S., Şahin-Gür, D., & Zindel, C. (2019). What language demands count in subject-matter classrooms? A study on mathematics teachers’ language-related orientations and diagnostic categories for students’ explanations. RISTAL, 2(1), 102–117. https://doi.org/10.23770/rt1827
- Prediger, S, Kuzu, T., Schüler-Meyer, A., & Wagner, J. (2019). One mind, two languages, separate conceptualisations? A case study on students’ bilingual modes for dealing with language-related conceptualisations of fractions. Research in Mathematics Education, 21(2), 188-207. https://doi.org/10.1080/14794802.2019.1602561
- Radford, L. (2018). The emergence of symbolic algebraic thinking in primary school. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12- year-olds (pp. 3-25). Springer. https://doi.org/10.1007/978-3-319-68351-5_1
- Redder, A. (2008). Functional pragmatics. In G. Antos & E. Ventola (Ed.), Handbook of interpersonal communication (pp. 133-178). Berlin, New York: De Gruyter Mouton. https://doi.org/10.1515/9783110211399.1.133
- Riccomini, P. J., Smith, G. W., Hughes, E. M., & Fries, K. M. (2015). The language of mathematics: the importance of teaching and learning mathematical vocabulary. Reading and Writing Quarterly, 31(3), 235-252. https://doi.org/10.1080/10573569.2015.1030995
- Rosenthal, G. (2018). Interpretative social research. an introduction. University Göttingen.
- Robins, R. H., & Crystal, D. (2021). Language. Encyclopedia Britannica.
- Ruwisch, S. (2015). Wie die Zahlen im Kopf wirksam werden. Merkmale tragfähiger Zahlvorstellungen. Grundschule Mathematik, 44 (1), 4-5.
- Scherer, P. (2019). The potential of substantial learning environments for inclusive mathematics – student teachers’ explorations with special needs students [Paper Presentation]. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht. https://hal.science/hal-02431500/
- Schüler-Meyer, A., Prediger, S., Kuzu, T., Wessel, L., & Redder, A. (2019). Is formal language proficiency in the home language required to profit from a bilingual teaching intervention in mathematics? A mixed methods study on fostering multilingual students’ conceptual understanding. International Journal of Science and Mathematics Education, 17(2), 317-339. https://doi.org/10.1007/s10763-017-9857-8
- Schütte M., Friesen R.A., & Jung J. (2019). Interactional analysis: A method for analysing mathematical learning processes in interactions. In G. Kaise & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education (pp. 101-129). Springer. https://doi.org/10.1007/978-3-030-15636-7_5
- Schwarzkopf, R., Nührenbörger, M., & Mayer, C. (2018). Algebraic understanding of equalities in primary classes. In C. Kieran (Ed.), Early algebra (pp. 195–212). Springer. https://doi.org/10.1007/978-3-319-68351-5_8
- Selter, C., Prediger, S., Nührenbörger, M., & Hußmann, S. (2012). Taking away and determining the difference—a longitudinal perspective on two models of subtraction and the inverse relation to addition. Educational Studies in Mathematics, 79(3), 389–408. https://doi.org/10.1007/s10649-011-9305-6
- Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36. https://doi.org/10.1007/BF00302715
- Snow, C. E., & Uccelli, P. (2009). The challenge of academic language. In D. R. Olson & N. Torrance (Eds.), The Cambridge handbook of literacy (pp. 112-133). Cambridge University Press. http://dx.doi.org/10.1017/CBO9780511609664.008
- Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404-411. https://doi.org/10.2307/30034943
- Steinbring, H. (2005). The construction of new mathematical knowledge in classroom interaction. An epistemological perspective. Springer.
- Steinbring, H. (2006). What makes a sign a mathematical sign? An epistemological perspective on mathematical interaction. Educational Studies in Mathematics, 61(1/2), 133-162. https://doi.org/10.1007/s10649-006-5892-z
- Steiner, G. (1973). Mathematics as thinking education. A psychological exploration of the role of thinking in mathematical early education. Klett.
- Steinweg A. S., Akinwunmi K., & Lenz D. (2018). Making implicit algebraic thinking explicit: exploiting national characteristics of german approaches. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 283-307). Springer. https://doi.org/10.1007/978-3-319-68351-5_12
- Stephan, M. (2014). Sociomathematical norms in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 563-566). Springer. https://doi.org/10.1007/978-94-007-4978-8_143
- Straehler-Pohl, H., & Gellert, U. (2013). Towards a Bernsteinian language of description for mathematics classroom discourse. British Journal of Sociology of Education, 34(3), 313-332. https://doi.org/10.1080/01425692.2012.714250
- Swain, M. (2006). Languaging, agency and collaboration in advanced second language proficiency. In H. Byrnes (Ed.), Advanced language learning (pp. 95-108). Continuum.
- Thierry, G., Athanasopoulos, P., Wiggett, A., Dering, B., & Kulper, J.-R. (2009). Unconscious effects of language-specific terminology on preattentive color perception. Proceedings of the National Academy of Sciences of the United States of America, 106(11), 4567-4570. https://doi.org/10.1073/pnas.0811155106
- Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50(1), 29–47. https://doi.org/10.1023/A:1020572803437
- Tiedemann, K. (2012). Mathematik in der familie: Zur familialen unterstützung früher mathematischer lernprozesse in vorlese- und spielsituationen. Waxmann.
- Tubach, D., & Nührenbörger, M. (2016). Mathematical understanding in transition from Kindergarten to Primary School: Play as a bridge between two education institutions. In T. Meaney, O. Helenius, M. L. Johansson, T. Lange, & A. Wernberg (Eds.), Mathematics Education in the Early Years (pp.81-98). Springer. https://doi.org/10.1007/978-3-319-23935-4_5
- Tunç-Pekkan, Z. (2015). An analysis of elementary school children’s fractional knowledge depicted with circle, rectange, and number line representations. Educational Studies in Mathematics, 89(3), 419-441. https://doi.org/10.1007/s10649-015-9606-2
- Umansky, I., & Dumont, H. (2019). English learner labeling: How English learner status shapes teacher perceptions of student skills & the moderating role of bilingual instructional settings. American Educational Research Journal, 58(5), 993–1031. https://doi.org/10.3102/0002831221997571
- Uribe, Á., & Prediger, S. (2021). Students’ multilingual repertoires-in-use for meaning-making: Contrasting case studies in three multilingual constellations. The Journal of Mathematical Behaviour, 62, 1-33. https://doi.org/10.1016/j.jmathb.2020.100820
- Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83-94. https://doi.org/10.1159/000202727
- Vygotsky, L. (1934/ 1986). Thought and language. The MIT Press.
- Vygotsky, L. (1978). Mind in society: The development of higher psychological processes. Harvard University Press.
- Vygotsky, L. (2004). Imagination and creativity in childhood. Journal of Russian and East European Psychology, 42(1), 7-97. https://doi.org/10.1080/10610405.2004.11059210
- Waxer, M., & Morton, J. B. (2012). Cognitive conflict and learning. In N. M. Seel (Ed.), Encyclopedia of the sciences of learning (pp. 585–587). Springer. https://doi.org/10.1007/978-1-4419-1428-6_280
- Wessel, L. (2020). Vocabulary in learning processes towards conceptual understanding of equivalent fractions—specifying students’ language demands on the basis of lexical trace analyses. Mathematics Education Research Journal 32, 653-681. https://doi.org/10.1007/s13394-019-00284-z
- Whorf, B. L. (1940). Science and linguistics. Technology Review, 42(6), 229-248.
- Wilson, T. P. (1981). Theorien der Interaktion und Modelle Soziologischer Erklärung. In: Arbeitsgruppe Bielefelder Soziologen (eds.): Alltagswissen, Interaktion und gesellschaftliche Wirklichkeit (p. 54- 79). Opladen: Westdeutscher Verlag. https://doi.org/10.1007/978-3-663-14511-0_3
- Winawer, J., Witthoft, N., Frank, M. C., Wu, L., Wade, A. R., & Boroditsky, L. (2007). Russian blues reveal effects of language on color discrimination. Proceedings of the National Academy of Sciences of the United States of America, 104(19), 7780-7785. https://doi.org/10.1073/pnas.0701644104
- Wittgenstein, L. (1953). Philosophical investigations [translated by A.E.M. Anscombe]. Oxford: Basil Blackwell.
- Wittmann, E. C. (1985). Objects-operations-effects: The operative principle in mathematics education. Teaching Mathematics 11, 7–11.
- Wittmann, E. C. (1995). Mathematics education as a ‘design science’. Educational Studies in Mathematics 29, 355-374. https://doi.org/10.1007/BF01273911
- Wittmann, E. C. (2021). Connecting mathematics and mathematics education. Collected papers on mathematics education as a design science. Springer.
- Zahner, W. & Moschkovich, J. (2011). Bilingual students using two languges during peer mathemtaics discussions: ¿qué significa? Estudiantes bilingües usando dos idiomas en sus discusiones matemáticas: what does it mean? In K. Téllez, J. N. Moschkovich, & M. Civil (Eds.), Latinos/as and Mathematics Education: Research on Learning and Teaching in Classrooms and Communities (pp. 37-63). Information Age.