Abstract
Formal multi-step mathematical argumentations are a typical case of a highly specified subject-specific genre in the technical language with challenging demands in the academic language of schooling. The design research project presented here has the dual aim of (1) specifying the structural and language demands of formal mathematical argumentations and (2) designing a teaching-learning arrangement that uses structural scaffolding to foster students to successively meet these demands. These dual aims are pursued by an in-depth empirical analysis of students’ processes on the micro-level. For this purpose, 15 design experiments were conducted with 10 pairs of students in three design experiment cycles. The first two cycles served to develop the structural scaffolding and the third cycle served to investigate the initiated learning processes and the language demands on the lexical and syntactical level. The qualitative in-depth analysis of the teaching-learning processes in the design experiments shows how students can successively learn to conduct multi-step argumentations when supported by structural scaffolding. Expressing the argumentations in an adequate way, however, is an additional challenge. The empirical analysis reveals deep insights into the complex interplay of conditional, causal, and consecutive phrases that are necessary to combine premises, arguments, and conclusions in a logical sound way. The design research study has theoretical consequences for conceptualizing subject-specific discursive demands as well as practical implications as one design outcome is a prototype of the learning arrangement to foster students in a mathematics- and language-integrated way.
Acknowledgements
We thank Helmut Johannes Vollmer, John Mason, and the anonymous reviewers for their valuable comments, which helped to clarify our ideas.
References
Bailey, Alison L. (ed.). 2007. The language demands of school: putting academic English to thetest. New Haven: Yale University Press.Search in Google Scholar
Bailey, Alison L., Francis A. Butler, Robin Stevens & Carol Lord. 2007. Further specifying the language demands of school. In Alison L. Bailey (ed.), The language demands of school. Putting academic English to the test, 103–156. New Haven: Yale University Press.Search in Google Scholar
Chazan, Daniel. 1993. High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational studies in mathematics 24(4). 359–387.10.1007/BF01273371Search in Google Scholar
Cho, Kyoo-Lak & David H. Jonassen. 2002. The effects of argumentation scaffolds on argumentation and problem solving. Educational Technology Research and Development 50(3). 5–22.10.1007/BF02505022Search in Google Scholar
Clarkson, Philip C. 2004. Researching the language for explanations in mathematics teaching and learning. Paper presented at the Australian Association of Research in Education, Melbourne. https://www.academia.edu/31128112 [accessed 1 May 2017]Search in Google Scholar
Council of Chief State School Officers. 2012. Framework for English Language Proficiency Development Standards Corresponding to the Common Core State Standards and the Next Generation Science Standards. Washington, DC: CCSSOSearch in Google Scholar
Douek, Nadia. 1999. Some remarks about argumentation and mathematical proof and their educational implications. In Inge Schwank (ed.), European Research in Mathematics Education, vol. 1, 125–139. Osnabrück: Forschungsinstitut für Mathematikdidaktik.Search in Google Scholar
Driver, Rosalind, Paul Newton & Jonathan Osborne. 2000. Establishing the norms of scientific argumentation in classrooms. Science Education 84. 287–312.10.1002/(SICI)1098-237X(200005)84:3<287::AID-SCE1>3.0.CO;2-ASearch in Google Scholar
Durand-Guerrier, Viviane, Paolo Boero, Nadia Douek, Susanna S. Epp & Denis Tanguay. 2011. Examining the Role of Logic in Teaching Proof. In Gila Hanna & Michael de Villiers (eds.), Proof and Proving in Mathematics Education, vol. 15, 369–389. Dordrecht: Springer Netherlands. (1 July, 2016).10.1007/978-94-007-2129-6_16Search in Google Scholar
Duval, Raymond. 1991. Structure du Raisonnement Deductif et Apprentissage de la Demonstration. Educational Studies in Mathematics 22(3). 233–261.10.1007/BF00368340Search in Google Scholar
Epp, Susanna S. 2003. The Role of Logic in Teaching Proof. The American Mathematical Monthly 110(10). 886–899. 10.1080/00029890.2003.11920029Search in Google Scholar
Gibbons, Pauline. 2002. Scaffolding language, scaffolding learning: teaching second language learners in the mainstream classroom. Portsmouth, NH: Heinemann.Search in Google Scholar
Gravemeijer, Koeno & Paul Cobb. 2006. Design research from the learning design perspective. In Jan J. H. Akker, Koeno Gravemeijer, Susan McKenney & Nienke Nieveen (eds.), Educational design research: The design, development and evaluation of programs, processes and products, 17–51. London: Routledge.Search in Google Scholar
Grice, H. Paul. 1975. Logic and conversation. In Peter Cole & Jerry L. Morgan (eds.), Speech acts (Syntacs and semantics 3), 41–58. New York: Academic Press.10.1163/9789004368811_003Search in Google Scholar
Halliday, Michael A. K. & Christian M. I. M. Matthiessen. 2004. An introduction to functional grammar. 3rd edn. London & New York: Oxford University Press.10.4324/9780203783771Search in Google Scholar
Hanna, Gila & Michael de Villiers (eds.). 2012. Proof and proving in mathematics education: the 19th ICMI study. (New ICMI study series 15). Dordrecht: Springer.10.1007/978-94-007-2129-6Search in Google Scholar
Harel, Guershon & Larry Sowder. 1998. Students’ Proof Schemes: Results from Exploratory Studies. CBMS Issues Mathematics Education 7. 234–283.10.1090/cbmath/007/07Search in Google Scholar
Hein, Kerstin & Susanne Prediger. 2017. Fostering and investigating students’ pathways to formal reasoning: A design research project on structural scaffolding for 9th graders. Paper presented at the 10th Conference of European Research in Mathematics Education, Dublin, 1–5 February.Search in Google Scholar
Jimenez-Aleixandre, M. Pilar, Anxela Bugallo Rodriguez & Richard A. Duschl. 2000. “Doing the lesson” or “doing science”: Argument in high school genetics. Science Education 84(6). 757–792.10.1002/1098-237X(200011)84:6<757::AID-SCE5>3.0.CO;2-FSearch in Google Scholar
Knipping, Christine & David Reid. 2015. Reconstructing Argumentation Structures: A Perspective on Proving Processes in Secondary Mathematics Classroom Interactions. In Angelika Bikner-Ahsbahs, Christine Knipping & Norma Presmeg (eds.), Approaches to Qualitative Research in Mathematics Education: Examples of Methodology and Methods, 75–101. Dordrecht: Springer.10.1007/978-94-017-9181-6_4Search in Google Scholar
Krummheuer, Götz. 1995. The Ethnography of Argumentation. In Paul Cobb & Heinrich Bauersfeld (eds.), The Emergence of Mathematical Meaning: Interaction in Classroom Cultures, 229–269. Hillsdale: Lawrence Erlbaum Associates.Search in Google Scholar
Lajoie, Susanne P. 2005. Extending the Scaffolding Metaphor. Instructional Science 33(5–6). 541–557.10.1007/s11251-005-1279-2Search in Google Scholar
Lampert, Magdalene & Paul Cobb. 2003. “Communication and learning in the mathematics classroom.” In Jeremy Kilpatrick & Deborah Shifter (eds.), Research Companion to the NCTM Standards, 237–249. Reston: National Council of Teachers of Mathematics.Search in Google Scholar
Miyazaki, Mikio, Taro Fujita & Keith Jones. 2015. Flow-chart proofs with open problems as scaffolds for learning about geometrical proofs. ZDM Mathematics Education 47(7). 1211–1224.10.1007/s11858-015-0712-5Search in Google Scholar
Miyazaki, Mikio, Taro Fujita & Keith Jones. 2017. Students’ understanding of the structure of deductive proof. Educational Studies in Mathematics 94(2). 223–239.10.1007/s10649-016-9720-9Search in Google Scholar
Moschkovich, Judit N. 2015. Academic literacy in mathematics for English Learners. The Journal of Mathematical Behavior 40. 43–62. 10.1016/j.jmathb.2015.01.005Search in Google Scholar
Nunes, Terezinha, Analúcia Dias Schliemann & David William Carraher. 1993. Street mathematics and school mathematics. (Learning in Doing). New York: Cambridge University Press.Search in Google Scholar
Pimm, David. 1987. Speaking mathematically: communication in mathematics classrooms. London: Routledge.Search in Google Scholar
Prediger, Susanne & Nadine Krägeloh. 2016. “X-Arbitrary Means Any Number, but You Do Not Know Which One.” In Anjum Halai & Philip Clarkson (eds.), Teaching and Learning Mathematics in Multilingual Classrooms: Issues for Policy, Practice and Teacher Education, 89–108. Rotterdam: Sense Publishers. http://dx.doi.org/10.1007/978-94-6300-229-5_7. (accessed 24 May 2017)10.1007/978-94-6300-229-5_7Search in Google Scholar
Prediger, Susanne & Carina Zindel. 2017, in press. School academic language demands for understanding functional relationships – A design research project on the role of language in reading and learning. In press for Eurasia Journal of Mathematics Science and Technology Education.10.12973/eurasia.2017.00804aSearch in Google Scholar
Prediger, Susanne & Larissa Zwetzschler. 2013. Topic-specific design research with a focus on learning processes: The case of understanding algebraic equivalence in grade 8. In Tjeerd Plomp & Nienke Nieveen (eds.), Educational design research – Part B: Illustrative cases, 407–424. Enschede: SLO.Search in Google Scholar
Rapanta, Chrysi, Merce Garcia-Mila & Sandra Gilabert. 2013. What is meant by argumentative competence? An integrative review of methods of analysis and assessment in education. Review of Educational Research 83(4). 483–520.10.3102/0034654313487606Search in Google Scholar
Schleppegrell, Mary. 2004. The language of schooling: a functional linguistics perspective. Mahwah, N.J: Lawrence Erlbaum Associates.Search in Google Scholar
Selden, John & Annie Selden. 1995. Unpacking the logic of mathematical statements. Educational Studies in Mathematics 29(2). 123–151.10.1007/BF01274210Search in Google Scholar
Simpson, Adrian. 2015. The anatomy of a mathematical proof: Implications for analyses with Toulmin’s scheme. Educational Studies in Mathematics 90(1). 1–17.10.1007/s10649-015-9616-0Search in Google Scholar
Snow, Catherine E. & Paola Uccelli. 2009. The Challenge of Academic Language. In David R. Olson & Nancy Torrance (eds.), The Cambridge Handbook of Literacy, 112–133. Cambridge: Cambridge University Press. http://ebooks.cambridge.org/ref/id/CBO9780511609664A016 (accessed 2 December 2016).10.1017/CBO9780511609664.008Search in Google Scholar
Thürmann, Eike, Helmut Vollmer & Irene Pieper. 2010. Language(s) of schooling: Focusing on vulnerable learners. Paper presented at Policy Forum The right of learners to quality and equity in education – The role of linguistic and intercultural competences, Geneva, 2–4 November. http://www.coe.int/t/dg4/linguistic/source/source2010_forumgeneva/2-VulnerLLearnersThurm_EN.doc (accessed 2 December 2016).Search in Google Scholar
Toulmin, Stephen E. 1958. The Uses of Arguments. Cambridge: Cambridge University Press.10.1017/CBO9780511840005Search in Google Scholar
Ufer, Stefan, Aiso Heinze, Sebastian Kuntze & Franziska Rudolph-Albert. 2009. Beweisen und Begründen im Mathematikunterricht. Journal für Mathematik-Didaktik 30(1). 30–54.10.1007/BF03339072Search in Google Scholar
Villiers, Michael de. 1990. The role and function of proof in mathematics. Pythagoras 24. 17–24.Search in Google Scholar
Wood, David, Jerome S. Bruner & Gail Ross. 1976. The role of tutoring in problem solving. Journal of child psychology and psychiatry 17(2). 89–100.10.1111/j.1469-7610.1976.tb00381.xSearch in Google Scholar
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