Can a proof and a counterexample coexist? Students' conceptions about the relationship between proof and refutation

References

• Balacheff , N. 1988 . A study of students' proving processes at the junior high school level In Proceedings of the Second UCSMP International Conference on Mathematics Education I. Wirszup R. Streit , 284 97 . Reston, VA : National Council of Teachers of Mathematics .
   Google Scholar
• Ball , D.L. , H. Bass 2003 . Making mathematics reasonable in school In A research companion to principles and standards for school mathematics J. Kilpatrick , W.G. Martin , D. Schifter , 27 44 . Reston, VA : National Council of Teachers of Mathematics .
   Google Scholar
• Carpenter , T.P. , Franke , M.L. and Levi , L. 2003 . Thinking mathematically: Integrating arithmetic & algebra in elementary school , Portsmouth, NH : Heinemann .
   Google Scholar
• Coe , R. and Ruthven , K. 1994 . Proof practices and constructs of advanced mathematics students . British Educational Research Journal , 20 : 41 – 53 .
   Google Scholar
• Denzin , N. 1988 . The research act , New York : McGraw-Hill .
   Google Scholar
• Fischbein , E. 1982 . Intuition and proof . For the Learning of Mathematics 3 , 2 : 9 18 , 24
   Google Scholar
• Hanna , G. , H.N. Jahnke 1996 . Proof and proving International handbook of mathematics education A.J. Bishop , K. Clements , C. Keitel , J. Kilpatrick , C. Laborde , 877 908 . Dordrecht, , The Netherlands : Kluwer Academic Publishers .
   Google Scholar
• Harel , G. , L. Sowder 2007 . Toward comprehensive perspectives on the learning and teaching of proof Second handbook of research on mathematics teaching and learning F.K. Lester , 805 42 . Greenwich, CT : Information Age Publishing .
   Google Scholar
• Healy , L. and Hoyles , C. 2000 . A study of proof conceptions in algebra . Journal for Research in Mathematics Education , 31 : 396 – 428 .
   Web of Science ®Google Scholar
• Küchemann , D. , C. Hoyles 2001–03 Longitudinal Proof Project (Technical reports for Year 8–10 surveys) . London : Institute of Education . Retrieved 3 May, 2009, from http://www.mathsmed.co.uk/ioe-proof/techreps.html .
   Google Scholar
• Lakatos , I. 1976 . Proofs and refutations: The logic of mathematical discovery , Cambridge : Cambridge University Press .
   Google Scholar
• Lampert , M. 1992 . Practices and problems in teaching authentic mathematics Effective and responsible teaching: The new synthesis F.K. Oser , A. Dick , J. Patry , 295 314 . San Francisco, CA : Jossey-Bass Publishers .
   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 : 194 – 214 .
   Web of Science ®Google Scholar
• Marshall , C. and Rossman , G.B. 1999 . Designing qualitative research , 3rd.ed. , Thousand Oaks, CA : Sage .
   Google Scholar
• Mason , J. Burton , L. Stacey , K. 1982 . Thinking mathematically . London : Addison-Wesley .
   Google Scholar
• Merton , R. and Kendall , P. 1946 . The focused interview . American Journal of Sociology , 51 : 541 – 557 .
   Web of Science ®Google Scholar
• Reading , C. and Reid , J. 2007 . Reasoning about variation: Student voice . International Electronic Journal of Mathematics Education , 2 ( 3 ) : 110 – 127 .
   Google Scholar
• Reid , D.A. 2002 . Conjectures and refutations in grade 5 mathematics . Journal for Research in Mathematics Education , 33 : 5 – 29 .
   Web of Science ®Google Scholar
• Robson , C. 1993 . Real world research , Oxford : Blackwell .
   Google Scholar
• Schoenfeld , A.H. 1983 . Problem solving in the mathematics curriculum: A report, recommendations, and an annotated bibliography . Washington, DC : Mathematical Association of America .
   Google Scholar
• Schoenfeld , A.H. 2006 . Design experiments Handbook of complementary methods in education research J.L. Green , G. Camilli , P.B. Elmore A. Skukauskaite E. Grace 193 205 . Washington, DC : American Educational Research Association .
   Google Scholar
• Stylianides , A.J. 2007 . Proof and proving in school mathematics . Journal for Research in Mathematics Education , 38 : 289 – 321 .
   Web of Science ®Google Scholar
• Stylianides , A.J. 2009 . Breaking the equation “empirical argument = proof.” Mathematics Teaching 213 : 9 14 .
   Google Scholar
• Stylianides , G.J. and Stylianides , A.J. 2008 . Proof in school mathematics: Insights from psychological research into students's ability for deductive reasoning . Mathematical Thinking and Learning , 10 : 103 – 133 .
   Google Scholar
• Stylianides , G.J. Stylianides A.J. 2009 Facilitating the transition from empirical arguments to proof Journal for Research in Mathematics Education 40 314 52 .
   Web of Science ®Google Scholar
• Tuckman , B. 1972 . Conducting educational research , New York : Harcourt Brace Jovanovich .
   Google Scholar
• Zack , V. 1997 . “You have to prove us wrong”: Proof at the elementary school level In Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education E. Pehkonen , 4 : 291 98 . Lahti : University of Helsinki .
   Google Scholar