An Oral Proof in a Geometry Class: How Linguistic Tools Can Help Map the Content of a Proof

REFERENCES

• Aaron , W. R. 2011 . The position of the geometry student in instruction: A study from three perspectives , Ann Arbor , MI : The University of Michigan . Unpublished doctoral dissertation
   Google Scholar
• Ball , D. L. and Bass , H. 2000 . “ Making believe: The collective construction of public mathematical knowledge in the elementary classroom ” . In Yearbook of the national society for the study of education, constructivism in education , Edited by: Phillips , D. 193 – 224 . Chicago , IL : University of Chicago Press .
   Google Scholar
• Ball , D. L. and Bass , H. 2003 . “ Making mathematics reasonable in school ” . In A research companion to principles and standards for school mathematics , Edited by: Kilpatrick , J. , Martin , W. G. and Schifter , D. 27 – 44 . Reston , VA : National Council of Teachers of Mathematics .
   Google Scholar
• Boyd , C. , Burrill , G. , Cummins , J. , Kanold , T. and Malloy , C. 1998 . Geometry , Westerville , OH : Glencoe .
   Google Scholar
• Cazden , C. B. 2001 . Classroom discourse , 2nd ed. , Portsmouth , NH : Heinemann .
   Google Scholar
• Chieu , V. M. , Aaron , W. and Herbst , P. Impact of critical events in an animated classroom story on teacher learners' online comments . Paper presented at the 2013 Annual Meeting of the American Educational Research Association, San Francisco, CA , Retrieved from Deep Blue at the University of Michigan: http://hdl.handle.net/2027.42/97551
   Google Scholar
• Christie , F. 2002 . Classroom discourse analysis: A functional perspective , London , , UK : Continuum .
   Google Scholar
• Cole , R. , Becker , N. , Towns , M. , Sweeney , G. , Wawro , M. and Rasmussen , C. 2012 . Adapting a methodology from mathematics education research to chemistry education research: Documenting collective activity . International Journal of Science and Mathematics Education , 10 : 193 – 211 .
   Web of Science ®Google Scholar
• Dimmel , J. and Herbst , P. Analyzing the diagrammatic register in geometry textbooks: Toward a semiotic architecture . Proceedings of the 34th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Kalamazoo, MI , Retrieved from Deep Blue at The University of Michigan: http://hdl.handle.net/2027.42/91288
   Google Scholar
• Eggins , S. and Slade , D. 1997 . Analysing casual conversation , London , , UK : Equinox .
   Google Scholar
• Forman , E. A. , Larreamendy-Joerns , J. , Stein , M. K. and Brown , C. 1998 . You're going to want to find out which and prove it”: Collective argumentation in mathematics classrooms . Learning and Instruction , 8 ( 6 ) : 527 – 548 .
   Web of Science ®Google Scholar
• González , G. 2009 . Mathematical tasks and the collective memory: How do teachers manage students' prior knowledge when teaching geometry with problems? , Ann Arbor , MI : The University of Michigan . Unpublished doctoral dissertation
   Google Scholar
• González , G. and DeJarnette , A. 2012 . Agency in a geometry review lesson: A linguistic view on teacher and student division of labor . Linguistics and Education , 23 ( 2 ) : 182 – 199 .
   Google Scholar
• González , G. and Herbst , P. 2006 . Competing arguments for the geometry course: Why were American high school students supposed to study geometry in the twentieth century? . International Journal for the History of Mathematics Education , 1 ( 1 ) : 7 – 33 .
   Google Scholar
• Halliday , M. A. K. 1975 . “ Some aspects of sociolinguistics. ” . In Interactions between linguistics and mathematical education: Final report of the symposium sponsored by UNESCO, CEDO and ICMI, Nairobi, Kenya, September 1–11, 1974 (UENSCO Report, No. ED-74/CONF.808 , Edited by: Jacobsen , E. 64 – 73 . Paris , , France : UNESCO .
   Google Scholar
• Halliday , M. A. K. 1978 . Language as social semiotic , London , , UK : Edward Arnold .
   Google Scholar
• Halliday , M. A. K. and Hasan , R. 1976 . Cohesion in English , London , , UK : Longman .
   Google Scholar
• Halliday , M. A. K. and Matthiessen , C. 1999 . Constructing experience through meaning: A language based approach to cognition , London , , UK : Continuum .
   Google Scholar
• Halliday , M. A. K. and Matthiessen , C. 2004 . An introduction to functional grammar , 3rd ed. , London , UK : Hodder Arnold .
   Google Scholar
• Herbel-Eisenmann , B. and Otten , S. 2011 . Mapping mathematics in classroom discourse . Journal for Research in Mathematics Education , 42 ( 5 ) : 451 – 485 .
   Web of Science ®Google Scholar
• Herbel-Eisenmann , B. and Wagner , D. 2010 . Appraising lexical bundles in mathematics classroom discourse: Obligation and choice . Educational Studies in Mathematics , 75 : 43 – 63 .
   Web of Science ®Google Scholar
• Herbel-Eisenmann , B. , Wagner , D. and Cortes , V. 2010 . Lexical bundle analysis in mathematics classroom discourse: The significance of stance . Educational Studies in Mathematics , 75 : 23 – 42 .
   Web of Science ®Google Scholar
• Herbst , P. 2002a . Engaging students in proving: A double bind on the teacher . Journal for Research in Mathematics Education , 33 ( 3 ) : 176 – 203 .
   Web of Science ®Google Scholar
• Herbst , P. 2002b . Establishing a custom of proving in American school geometry: Evolution of the two-column proof in early twentieth century . Educational Studies in Mathematics , 49 : 283 – 312 .
   Google Scholar
• Herbst , P. and Chazan , D. 2009 . Methodologies for the study of instruction in mathematics classrooms . Recherches en Didactique des Mathématiques , 29 ( 1 ) : 11 – 32 .
   Google Scholar
• Herbst , P. , Chen , C. , Weiss , M. , González , G. , Nachlieli , T. , Hamlin , M. and Brach , C. 2009 . “ Doing proofs” in geometry classrooms ” . In Teaching and learning proof across the grades: A K–16 perspective , Edited by: Blanton , M. , Stylianou , D. and Knuth , E. 250 – 263 . New York , NY : Routledge . with
   Google Scholar
• Herbst , P. , González , G. , Hsu , H. Y. , Chen , C. , Weiss , M. and Hamlin , M. 2010 . Instructional situations and students' opportunities to reason in the high school geometry class , Unpublished manuscript. Retrieved from Deep Blue at the University of Michigan . http://hdl.handle.net/2027.42/78372 with
   Google Scholar
• Herbst , P. , Nachlieli , T. and Chazan , D. 2011 . Studying the practical rationality of mathematics teaching: What goes into “installing” a theorem in geometry? . Cognition and Instruction , 29 ( 2 ) : 218 – 255 .
   Web of Science ®Google Scholar
• Hollebrands , K. , Conner , A. and Smith , R. C. 2010 . The nature of arguments provided by college geometry students with access to technology while solving problems . Journal for Research in Mathematics Education , 41 ( 4 ) : 324 – 350 .
   Web of Science ®Google Scholar
• Hsu , H-Y. 2010 . The study of Taiwanese students' experiences with geometric calculation with number (GCN) and their performance on GCN and geometric proof (GP) , Ann Arbor , MI : The University of Michigan . Unpublished doctoral dissertation
   Google Scholar
• Inglis , M. , Mejia-Ramos , J. P. and Simpson , A. 2007 . Modelling mathematical argumentation: The importance of qualification . Educational Studies in Mathematics , 66 : 3 – 21 .
   Google Scholar
• Kosko , K. and Herbst , P. 2012 . A deeper look at how teachers say what they say: A quantitative modality analysis of teacher-to-teacher talk . Teaching and Teacher Education , 28 : 589 – 598 .
   Web of Science ®Google Scholar
• Krummheuer , G. 1995 . “ The ethnography of argumentation ” . In The emergence of mathematical meaning: Interaction in classroom cultures , Edited by: Cobb , P. and Bauersfeld , H. 229 – 269 . Hillsdale , NJ : Lawrence Erlbaum .
   Google Scholar
• Lemke , J. 1990 . Talking science: Language, learning, and values , Westport , CT : Ablex .
   Google Scholar
• Lemke , J. L. 2003 . “ Mathematics in the middle: Measure, picture, gesture, sign, and word ” . In Educational perspectives on mathematics as semiosis: From thinking to interpreting to knowing , Edited by: Anderson , M. , Sáenz-Ludlow , A. , Zellweger , S. and Cifarelli , V. V. 215 – 234 . Brooklyn, NY : Legas .
   Google Scholar
• Leron , U. 1983 . Structuring mathematical proofs . The American Mathematical Monthly , 90 ( 3 ) : 174 – 185 .
   Web of Science ®Google Scholar
• Livingston , E. 1999 . Cultures of proving . Social Studies of Science , 29 ( 6 ) : 867 – 888 .
   Web of Science ®Google Scholar
• Martin , J. R. and Rose , D. 2003 . Working with discourse: Meaning beyond the clause , London , , UK : Continuum .
   Google Scholar
• Martin , J. R. and Rose , D. 2007 . Working with discourse: Meaning beyond the clause , 2nd ed. , London , UK : Continuum .
   Google Scholar
• Martin , J. R. and Rose , D. 2008 . Genre relations: Mapping culture , London , , UK : Equinox .
   Google Scholar
• Martin , T. 2007 . Mathematics teaching today: improving practice, improving student learning , Reston , VA : NCTM .
   Google Scholar
• Martin , T. , Soucy , S. , Wallace , M. and Dindyal , J. 2005 . The interplay of teacher and student actions in the teaching and learning of geometric proof . Educational Studies in Mathematics , 60 : 95 – 124 .
   Google Scholar
• McClain , K. 2009 . “ When is an argument just an argument? ” . In Teaching and learning mathematical proof across the grades , Edited by: Stylianou , D. , Knuth , E. and Blanton , M. 222 – 234 . New York, NY : Routledge .
   Google Scholar
• Mehan , H. 1979 . Learning lessons , Cambridge , MA : Harvard University Press .
   Google Scholar
• Mesa , V. and Chang , P. 2010 . The language of engagement in two highly interactive undergraduate mathematics classrooms . Linguistics and Education , 21 : 83 – 100 .
   Google Scholar
• Moore-Russo , D. , Conner , A. and Rugg , K. I. 2011 . Can slope be negative in 3-space? Studying concept image of slope through collective definition construction . Educational Studies in Mathematics , 76 : 3 – 21 .
   Web of Science ®Google Scholar
• Morgan , C. 1996 . The language of mathematics”: Towards a critical analysis of mathematics texts . For the Learning of Mathematics , 16 ( 3 ) : 2 – 10 .
   Google Scholar
• Morgan , C. 2006 . What does social semiotics have to offer to mathematics education research? . Educational Studies in Mathematics , 61 ( 1/2 ) : 219 – 245 .
   Google Scholar
• Nachlieli , T. , Herbst , P. and González , G. 2009 . Seeing a colleague encourage a student to make an assumption while proving: What teachers put to play in casting an episode of geometry instruction . Journal for Research in Mathematics Education , 40 ( 4 ) : 427 – 459 .
   Web of Science ®Google Scholar
• Nardi , E. , Biza , I. and Zachariades , T. 2012 . Warrant” revisited: Integrating mathematics teachers' pedagogical and epistemological considerations into Toulmin's model for argumentation . Educational Studies in Mathematics , 79 : 157 – 173 .
   Web of Science ®Google Scholar
• National Council of Teachers of Mathematics . 2000 . Principles and standards for school mathematics , Reston , VA : NCTM .
   Google Scholar
• O'Connor , M. C. 1998 . “ Language socialization in the mathematics classroom: Discourse practices and mathematical thinking ” . In Talking mathematics in school: Studies of teaching and learning , Edited by: Lampert , M. and Blunk , M. 17 – 55 . Cambridge , UK : Cambridge University Press .
   Google Scholar
• O'Halloran , K. 1998 . Classroom discourse in mathematics: A multisemiotic analysis . Linguistics and Education , 10 ( 3 ) : 359 – 388 .
   Google Scholar
• O'Halloran , K. 1999 . Towards a systemic functional analysis of multisemiotic mathematics texts . Semiotica , 124 : 1 – 2 . 1 – 29 .
   Web of Science ®Google Scholar
• O'Halloran , K. 2005 . Mathematical discourse: Language, symbolism and visual images , London , , UK : Continuum .
   Google Scholar
• Pedemonte , B. 2007 . How can the relationship between argumentation and proof be analysed? . Educational Studies in Mathematics , 66 : 23 – 41 .
   Google Scholar
• Pedemonte , B. and Reid , D. 2011 . The role of abduction in proving process . Educational Studies in Mathematics , 76 : 281 – 303 .
   Web of Science ®Google Scholar
• Pimm , D. 1987 . Speaking mathematically: Communication in mathematics classrooms. , London , , UK : Routledge & Kegan Paul .
   Google Scholar
• Pontecorvo , C. and Girardet , H. 1993 . Arguing and reasoning in understanding historical topics . Cognition and Instruction , 11 ( 3 & 4 ) : 365 – 395 .
   Web of Science ®Google Scholar
• Rasmussen , C. and Stephan , M. 2008 . “ A methodology for documenting collective activity ” . In Handbook of design research methods in education , Edited by: Kelly , A. E. , Lesh , R. A. and Baek , J. Y. 195 – 215 . New York , NY : Routledge .
   Google Scholar
• Resnick , L. B. , Salmon , M. , Zeitz , C. M. , Wathen , S. H. and Holowchack , M. 1993 . Discourse and shared reasoning . Cognition and Instruction , 11 ( 3&4 ) : 347 – 364 .
   Web of Science ®Google Scholar
• Schiffrin , D. 1987 . Discourse markers , Cambridge , UK : Cambridge University Press .
   Google Scholar
• Schleppegrell , M. J. 2007 . The linguistic challenges of mathematics teaching and learning: A research review . Reading & Writing Quarterly , 23 : 139 – 159 .
   Google Scholar
• Schleppegrell , M. J. 2010 . “ Language in mathematics teaching and learning: A research review from a systemic functional linguistics perspective ” . In Research on language and mathematics education: Multiple perspectives, challenges, and new directions , Edited by: Moschkovich , J. 73 – 112 . Charlotte , NC : Information Age Publishing .
   Google Scholar
• Sekiguchi , Y. 1991 . An investigation on proofs and refutations in the mathematics classroom , Dissertation Abstracts International . 52, 03A
   Google Scholar
• Shreyar , S. , Zolkower , B. and Pérez , S. 2010 . Thinking aloud together: A teacher's semiotic mediation of a whole-class conversation about percents . Educational Studies in Mathematics , 73 : 21 – 53 .
   Web of Science ®Google Scholar
• Steele , M. D. 2005 . Comparing knowledge bases and reasoning structures in discussions of mathematics and pedagogy . Journal of Mathematics Teacher Education , 8 ( 4 ) : 291 – 328 .
   Google Scholar
• Stephan , M. and Rasmussen , C. 2002 . Classroom mathematical practices in differential equations . Journal of Mathematical Behavior , 21 : 459 – 490 .
   Google Scholar
• Strom , D. , Kemeny , V. , Lehrer , R. and Forman , E. 2001 . Visualizing the emergent structure of children's mathematical argument . Cognitive Science , 25 : 733 – 773 .
   Web of Science ®Google Scholar
• Toulmin , S. 1958 . The uses of argument , Cambridge, , UK : Cambridge University Press .
   Google Scholar
• Unsworth , L. 1998 . Sound” explanations in school science: A Functional Linguistic perspective on effective apprenticing texts . Linguistics and Education , 9 ( 2 ) : 199 – 226 .
   Google Scholar
• Unsworth , L. 2001 . Evaluating the language of different types of explanations in junior high school science texts . International Journal of Science Education , 23 ( 6 ) : 585 – 609 .
   Web of Science ®Google Scholar
• Veel , R. 1999 . “ Language, knowledge and authority in school mathematics ” . In Pedagogy and the shaping of consciousness: Linguistic and social processes , Edited by: Christie , F. 185 – 216 . London , , UK : Continuum .
   Google Scholar
• Weber , K. and Alcock , L. 2005 . Using warranted implications to understand and validate proofs . For the Learning of Mathematics , 25 ( 1 ) : 34 – 38 . 51
   Google Scholar
• Weiss , M. , Herbst , P. and Chen , C.-L. 2009 . Teachers' perspectives on “authentic mathematics” and the two-column proof form . Educational Studies in Mathematics , 70 : 275 – 293 .
   Web of Science ®Google Scholar
• Wood , T. 1999 . Creating a context for argument in mathematics class . Journal for Research in Mathematics Education , 30 : 171 – 191 .
   Web of Science ®Google Scholar
• Young , R. F. and Nguyen , N. T. 2002 . Modes of meaning in high school science . Applied Linguistics , 23 ( 3 ) : 348 – 372 .
   Web of Science ®Google Scholar
• Zack , V. and Graves , B. 2001 . Making mathematical meaning through dialogue: “Once you think of it, the Z minus three seems pretty weird . Educational Studies in Mathematics , 46 ( 1–3 ) : 229 – 271 .
   Google Scholar
• Zolkower , B. and de Freitas , E. 2012 . Mathematical meaning-making in whole class conversation: Functional grammatical analysis of a paradigmatic text . Language and Dialogue , 2 ( 1 ) : 60 – 79 .
   Google Scholar
• Zolkower , B. and Shreyar , S. 2007 . A teacher's mediation of a thinking-aloud discussion in a 6th grade mathematics classroom . Educational Studies in Mathematics , 65 : 177 – 202 .
   Google Scholar