Exploring Logical Reasoning and Mathematical Proof in Grade 6 Elementary School Students

REFERENCES

• Balacheff , N. 1988 . “ Aspects of proof in pupils’ practice of school mathematics ” . In Mathematics, teachers and children , Edited by: Pimm , D. 216 – 235 . London , England : Hodder and Stoughton .
   Google Scholar
• Balacheff , N. 1991 . “ Benefits and limits of social interaction: The case of teaching mathematical proof ” . In Mathematical knowledge: Its growth through teaching , Edited by: Bishop , A. , Mellin-Olsen , S. and Van Dormolen , J. 175 – 192 . Dordrecht , , The Netherlands : Kluwer Academic Publisher .
   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
• Ball , D. L. , Hoyles , C. , Jahnke , H. N. and Movshovitz-Hadar , N. 2002 . “ The teaching of proof ” . In Proceedings of the International Congress of Mathematicians , Edited by: Tatsien , L. I. Vol. III , 907 – 920 . Beijing , , China : Higher Education Press .
   Google Scholar
• Boero , P. 1989 . “ Mathematical literacy for all: Experiences and problems ” . In Proceedings of the 13th PME International Conference Edited by: Vergnaud , G. , Rogalski , J. and Artigue , M. Vol. I , 62 – 76 . Paris , , France
   Google Scholar
• Brousseau , G. 2005 . “ The study of the didactical conditions of school learning in mathematics ” . In Activity and sign-grounding mathematics education , Edited by: Hoffmann , M. H. G. , Lenhard , J. and Seeger , F. 159 – 168 . New York, NY : Springer .
   Google Scholar
• California Department of Education . 2010 . K–12 California's common core content standards for mathematics Retrieved from http://www.scoe.net/castandards/agenda/2010/math_ccs_recommendations.pdf
   Google Scholar
• Dreyfus , T. 1999 . Why Johnny can't prove . Educational Studies in Mathematics , 38 : 85 – 109 .
   Google Scholar
• Fischbein , E. 1993 . The theory of figural concepts . Educational Studies in Mathematics , 24 ( 2 ) : 139 – 162 .
   Google Scholar
• Flores , A. 2002 . How do children know what they learn in mathematics is true? . Teaching Children Mathematics , 8 : 269 – 274 .
   Google Scholar
• Gawronski , J. D. 2005 . Mathematics assessment sampler: Grades 3–5 , Reston , VA : National Council of Teachers of Mathematics .
   Google Scholar
• Hanna , G. 2000 . Proof, explanation and exploration: An overview . Educational Studies in Mathematics , 44 ( 1 ) : 5 – 23 .
   Google Scholar
• Harel , G. and Sowder , L. 1998 . “ Students’ proof schemes: Results from exploratory studies ” . In Research on collegiate mathematics education , Edited by: Dubinsky , E. , Schoenfeld , A. H. and Kaput , J. J. Vol. 3 , 234 – 283 . Providence, RI : American Mathematical Society . Retrieved from http://math.ucsd.edu/~harel/Research/1Publications.htm
   Google Scholar
• Healy , L. and Hoyles , C. 1998a . Justifying and proving in school mathematics. Summary of the results from a survey of the proof conceptions of students in the UK Retrieved from http://www.lettredelapreuve.it/OldPreuve/Resumes/Healy/Healy98.html
   Google Scholar
• Healy , L. and Hoyles , C. 1998b . Justifying and proving in school mathematics (Tech. Rep.) , London , England : Institute of Education, University of London .
   Google Scholar
• Healy , L. and Hoyles , C. 2000 . A study of proof conceptions in algebra . Journal for Research in Mathematics Education , 31 ( 4 ) : 396 – 428 .
   Web of Science ®Google Scholar
• Hersh , R. 1993 . Proving is convincing and explaining . Educational Studies in Mathematics , 24 : 389 – 399 .
   Google Scholar
• Johnson , R. B. and Onwuegbuzie , A. J. 2004 . Mixed methods research: A research paradigm whose time has come . Educational Researcher , 33 ( 7 ) : 14 – 26 .
   Google Scholar
• Knuth , E. , Slaughter , M. , Choppin , J. and Sutherland , J. 2002 . “ Mapping the conceptual terrain of middle school students’ competencies in justifying and proving ” . In Proceedings of the 24th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education , Edited by: Mewborn , D. S. , Sztajn , P. , White , D. Y. , Weigel , H. L. , Bryant , R. L. and Nooney , K. Vol. 4 , 1693 – 1700 . Athens, GA : Clearinghouse for Science, Mathematics, and Environmental Education .
   Google Scholar
• Kuchemann , D. and Hoyles , C. 2004 . Year 10 students’ proofs of a statement in number/algebra and their responses to related multiple choice items: Longitudinal and cross-sectional comparisons . Retrieved from http://www.mathsmed.co.uk/ioe-proof/dek-chBSRLM-IP24%2C1-2004.pdf
   Google Scholar
• Lester , F. K. 1975 . Developmental aspects of children's ability to understand mathematical proof . Journal for Research in Mathematics Education , 6 ( 1 ) : 14 – 25 .
   Google Scholar
• Maher , C. A. and Martino , A. M. 1996 . The development of the idea of mathematical proof: A 5-year case study . Journal for Research in Mathematics Education , 27 ( 2 ) : 194 – 214 .
   Web of Science ®Google Scholar
• Mariotti , M. A. 2000 . Introduction to proof: The mediation of a dynamic software environment . Educational Studies in Mathematic , 44 ( 1–2 ) : 25 – 53 .
   Google Scholar
• Mariotti , M. A. 2006 . “ Proof and proving in mathematics education ” . In Handbook of research on the psychology of mathematics education: Past, present and future Edited by: Guiterez , A. and Boero , P. 173 – 204 . Rotterdam, the Netherlands: Sense Publishers.
   Google Scholar
• National Council of Teachers of Mathematics . 2000 . Principles and standards for school mathematics , Reston , VA : Author . Retrieved from http://standardstrial.nctm.org/document/prepost/cover.htm
   Google Scholar
• Ontario Ministry of Education . 2005 . The Ontario curriculum, Grades 1–8: Mathematics Retrieved from http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf
   Google Scholar
• Reid , D. and Zack , V. 2009 . “ Aspects of teaching proving in upper elementary school ” . In Teaching and learning proof across the grades , Edited by: Stylianou , D. , Blanton , M. and Knuth , E. 133 – 146 . New York , NY : Routledge .
   Google Scholar
• Schoenfeld , A. H. 1994 . A discourse on methods . Journal for Research in Mathematics Education , 25 ( 6 ) : 697 – 710 .
   Web of Science ®Google Scholar
• Segal , J. 2000 . Learning about mathematical proof: Conviction and validity . Journal of Mathematical Behavior , 18 ( 2 ) : 191 – 210 .
   Google Scholar
• Stylianides , A. J. 2007a . Introducing young children to the role of assumptions in proving . Mathematical Thinking and Learning , 9 ( 4 ) : 361 – 385 .
   Google Scholar
• Stylianides , A. J. 2007b . The notion of proof in the context of elementary school mathematics . Educational Studies in Mathematics , 65 : 1 – 20 .
   Google Scholar
• Stylianides , A. J. 2007c . Proof and proving in school mathematics . Journal for Research in Mathematics Education , 38 ( 3 ) : 289 – 321 .
   Web of Science ®Google Scholar
• Stylianides , A. J. and Ball , D. L. 2008 . Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving . Journal of Mathematics Teacher Education , 11 ( 4 ) : 307 – 332 .
   Google Scholar
• Stylianides , G. J. and Stylianides , A. J. 2008 . Proof in school mathematics: Insights from psychological research into students’ ability for deductive reasoning . Mathematics, Thinking and Learning , 10 ( 2 ) : 103 – 133 .
   Google Scholar
• Zack , V. You have to prove us wrong: Proof at the elementary level . Proceedings of the 21st International Conference for the Psychology of Mathematics Education , Edited by: Pehkonen , E. Vol. 4 , pp. 291 – 298 . Lahti, Finland: University of Helsinki.
   Google Scholar