Proof in School Mathematics: Insights from Psychological Research into Students' Ability for Deductive Reasoning

REFERENCES

• Acredolo , C. and Horobin , K. 1987 . Development of relational reasoning and avoidance of premature closure . Developmental Psychology , 23 : 13 – 21 .
   Web of Science ®Google Scholar
• Anderson , T. , Howe , C. and Tolmie , A. 1996 . “ Interaction and mental models of physics phenomena: Evidence from dialogues between learners ” . In Mental models in cognitive science , Edited by: Oakhill , J. and Garnham , A. 247 – 273 . Hove, , UK : Psychology Press .
   Google Scholar
• Balacheff , N. 1990 . Towards a problématique for research on mathematics teaching . Journal for Research in Mathematics Education , 21 : 258 – 272 .
   Web of Science ®Google Scholar
• Balacheff , N. 1991 . “ The benefits and limits of social interaction: The case of mathematical proof ” . In Mathematical knowledge: Its growth through teaching , Edited by: Bishop , A. J. 175 – 192 . Netherlands : Kluwer Academic Publishers .
   Google Scholar
• Balacheff, N. (2002/2004). The researcher epistemology: A deadlock for educational research on proof. In Fou Lai Lin (Ed.), Proceedings of International Conference on Mathematics: Understanding proving and proving to understand (pp. 23–44). Taipei: NSC and NTNU. Reprinted in Les cashiers du laboratoire Leibniz, no 109, August 2004. http://www-leibniz.imag.fr/NEWLEIBNIZ/LesCahiers/Cahier109/ResumCahier109.html (Accessed: 18 January 2006 ).
   Google Scholar
• Ball , D. L. and Bass , H. 2000 . “ Making believe: The collective construction of public mathematical knowledge in the elementary classroom ” . In Constructivism in education: Yearbook of the National Society for the Study of Education , Edited by: Philips , 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
• 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 : Higher Education Press .
   Google Scholar
• Bauer , M. I. and Johnson-Laird , P. N. 1993 . How diagrams can improve reasoning . Psychological Science , 4 : 372 – 378 .
   Web of Science ®Google Scholar
• Boero , P. , Garuti , R. and Mariotti , M. A. 1996 . “ Some dynamic mental processes underlying producing and proving conjectures ” . In Proceedings of the 20^th Conference of the International Group for the Psychology of Mathematics Education Edited by: Puig , L. and Gutiérrez , A. Vol. 2 , 121 – 128 . Valencia, , Spain
   Google Scholar
• Braine , M. D. S. and Rumain , B. 1983 . “ Logical reasoning ” . In Handbook of child psychology, Vol. 3: Cognitive development , Edited by: Flavell , J. and Markman , E. 263 – 340 . New York : Wiley .
   Google Scholar
• Byrne , J. P. and Overton , W. F. 1988 . Reasoning about logical connectives: A developmental analysis . Journal of Experimental Child Psychology , 46 : 194 – 218 .
   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
• Case , R. 1984 . “ The process of stage transition: A neo-Piagetian view ” . In Mechanisms of cognitive development , Edited by: Sternberg , R. J. 19 – 44 . New York : Freeman .
   Google Scholar
• Case , R. 1985 . “ A developmentally based approach to the problem of instructional design ” . In Thinking and learning skill: Research and open question , Edited by: Chipman , S. F. , Segal , J. W. and Glaser , R. Vol. 2 , 545 – 562 . Hillsdale, NJ : Laurence Erlbaum Associates .
   Google Scholar
• Chazan , D. 1993 . High school geometry students' justification for their views of empirical evidence and mathematical proof . Educational Studies in Mathematics , 24 : 359 – 387 .
   Google Scholar
• Cheng , P. W. and Holyoak , K. J. 1985 . Pragmatic reasoning schemas . Cognitive Psychology , 17 : 391 – 416 .
   PubMed Web of Science ®Google Scholar
• Cheng , P. W. and Holyoak , K. J. 1989 . On the natural selection of reasoning theories . Cognition , 33 : 285 – 313 .
   PubMed Web of Science ®Google Scholar
• Cheng , P. W. , Holyoak , K. J. , Nisbett , R. E. and Oliver , L. M. 1986 . Pragmatic versus syntactic approaches to training deductive reasoning . Cognitive Psychology , 18 : 293 – 328 .
   PubMed Web of Science ®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
• Cosmides , L. 1989 . The logic of social exchange: Has natural selection shaped how humans reason? Studies with the Wason selection task . Cognition , 31 : 187 – 276 .
   PubMed Web of Science ®Google Scholar
• Cosmides , L. and Tooby , J. 1989 . Evolutionary psychology and the generation of culture, Part II. Case study: A computational theory of social exchange . Ethnology and sociobiology , 10 : 51 – 97 .
   Web of Science ®Google Scholar
• English , L. D. 1998 . Children's reasoning in solving relational problems of deduction . Thinking and Reasoning , 4 : 249 – 281 .
   Google Scholar
• English , L. D. and Halford , G. S. 1995 . Mathematics education: Models and processes , Mahwah, NJ : Lawrence Erlbaum Associates .
   Google Scholar
• Ennis , R. H. 1976 . An alternative to Piaget's conceptualization of logical competence . Child Development , 47 : 903 – 919 .
   Web of Science ®Google Scholar
• Epp , S. S. 1987 . “ The logic of teaching calculus ” . In Toward a lean and lively calculus , Edited by: Douglas , R. G. 41 – 60 . Washington, D.C : Mathematical Association of America .
   Google Scholar
• Evans , J. St. B. T. 1982 . The psychology of deductive reasoning , London : Routledge and Kegan Paul .
   Google Scholar
• Evans , J. St. B. T. , Newstead , S. E. and Byrne , R. M. J. 1993 . Human reasoning: The psychology of deduction , Hove : Laurence Erlbaum Associates .
   Google Scholar
• Falmagne , R. J. 1980 . “ The development of logical competence: A psycholinguistic perspective ” . In Developmental models of thinking , Edited by: Kluwe , R. and Spada , H. 171 – 197 . New York : Academic Press .
   Google Scholar
• Fawcett , H. P. 1938 . The nature of proof (Yearbook of the National Council of Teachers of Mathematics) , New York : Bureau of Publications, Teachers College, Columbia University .
   Google Scholar
• Fischbein , E. 1982 . Intuition and proof . For the Learning of Mathematics , 3 : 9 – 18 . 24
   Google Scholar
• Foltz , C. , Overton , W. F. and Ricco , R. B. 1995 . Proof construction: Adolescent development from inductive to deductive problem-solving strategies . Journal of Experimental Child Psychology , 59 : 179 – 195 .
   PubMedGoogle Scholar
• Galbraith , P. L. 1981 . Aspects of proving: A critical investigation of process . Educational Studies in Mathematics , 12 : 1 – 29 .
   Google Scholar
• Galotti , K. M. , Komatsu , L. K. and Voelz , S. 1997 . Children's differential performance on deductive and inductive syllogisms . Developmental Psychology , 33 : 70 – 78 .
   PubMed Web of Science ®Google Scholar
• Girotto , V. , Blaye , A. and Farioli , F. 1989 . A reason to reason: Pragmatic basis of children's search for counter examples . European Bulletin of Cognitive Psychology , 9 : 297 – 321 .
   Google Scholar
• Girotto , V. , Gilly , M. , Blaye , A. and Light , P. 1989 . Children's performance in the selection task: Plausibility and familiarity . British Journal of Psychology , 80 : 79 – 95 .
   Web of Science ®Google Scholar
• Girotto , V. , Light , P. H. and Colbourn , C. J. 1988 . Pragmatic schemas and conditional reasoning in children . Quarterly Journal of Experimental Psychology , 40A : 469 – 482 .
   Web of Science ®Google Scholar
• Hanna , G. 2000 . Proof, explanation and exploration: An overview . Educational Studies in Mathematics , 44 : 5 – 23 .
   Google Scholar
• Hanna , G. and Jahnke , H. N. 1996 . “ Proof and proving ” . In International handbook of mathematics education , Edited by: Bishop , A. J. , Clements , K. , Keitel , C. , Kilpatrick , J. and Laborde , C. 877 – 908 . Dordrecht, , Netherlands : Kluwer Academic Publishers .
   Google Scholar
• Harel , G. and Sowder , L. 1998 . “ Students' proof schemes: Results from exploratory studies ” . In Research in collegiate mathematics education III , Edited by: Schoenfeld , A. H. , Kaput , J. and Dubinsky , E. 234 – 283 . Providence, RI : American Mathematical Society .
   Google Scholar
• Harel , G. and Sowder , L. 2007 . “ Toward comprehensive perspectives on the learning and teaching of proof ” . In Second handbook of research on mathematics teaching and learning , Edited by: Lester , F. K. 805 – 842 . Greenwich, CT : Information Age Publishing .
   Google Scholar
• Hawkins , J. , Pea , R. D. , Glick , J. and Scribner , S. 1984 . “Merds that laugh don't like mushrooms”: Evidence for deductive reasoning by preschoolers . Developmental Psychology , 20 : 584 – 594 .
   Web of Science ®Google Scholar
• Healy , L. and Hoyles , C. 2000 . Proof conceptions in algebra . Journal for Research in Mathematics Education , 31 : 396 – 428 .
   Web of Science ®Google Scholar
• Herbst , P. G. 2002a . Engaging students' in proving: A double bind on the teacher . Journal for Research in Mathematics Education , 33 : 176 – 203 .
   Web of Science ®Google Scholar
• Herbst , P. G. 2002b . Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century . Educational Studies in Mathematics , 49 : 283 – 312 .
   Google Scholar
• Hitch , G. J. and Halliday , M. S. 1983 . Working memory in children . Philosophical Transactions of the Royal Society of London, Series B , 302 : 325 – 340 .
   Web of Science ®Google Scholar
• Hoyles , C. and Küchemann , D. 2002 . Students' understanding of logical implication . Educational Studies in Mathematics , 51 : 193 – 223 .
   Google Scholar
• Inhelder , B. and Piaget , J. 1964 . The early growth of logic in the child: Classification and seriation , New York : Harper and Raw .
   Google Scholar
• Johnson-Laird , P. N. 1983 . Mental models: Towards a cognitive science of language, inference, and consciousness , Cambridge : Cambridge University Press .
   Google Scholar
• Johnson-Laird , P. N. 1990 . “ The development of reasoning ability ” . In Causes of development , Edited by: Butterworth , G. and Bryant , P. 85 – 110 . Hillsdale, NJ : Laurence Erlbaum Associates .
   Google Scholar
• Johnson-Laird , P. N. and Bara , B. G. 1984 . Syllogistic inference . Cognition , 16 : 1 – 61 .
   PubMed Web of Science ®Google Scholar
• Johnson-Laird , P. N. and Byrne , R. M. J. 1991 . Deduction , Hillsdale, NJ : Laurence Erlbaum Associates .
   Google Scholar
• Johnson-Laird , P. N. and Steedman , M. J. 1978 . The psychology of syllogisms . Cognitive Psychology , 10 : 64 – 99 .
   Web of Science ®Google Scholar
• Johnson-Laird , P. N. and Wason , P. C. 1970 . Insight into a logical relation . Quarterly Journal of Experimental Psychology , 22 : 49 – 61 .
   Web of Science ®Google Scholar
• Johnson-Laird , P. N. , Oakhill , J. V. and Bull , D. 1986 . Children's syllogistic reasoning . Quarterly Journal of Experimental Psychology , 38A : 35 – 58 .
   Web of Science ®Google Scholar
• Jones , K. 2000 . Providing a foundation for deductive reasoning: Students' interpretations when using dynamic geometry software and their evolving mathematical explanations . Educational Studies in Mathematics , 44 : 55 – 85 .
   Google Scholar
• Klaczynski , P. A. and Narasimham , G. 1998 . Representations as mediations of adolescent deductive reasoning . Developmental Psychology , 34 : 865 – 881 .
   PubMed Web of Science ®Google Scholar
• Knuth , E. J. , Choppin , J. , Slaughter , M. and Sutherland , J. . Mapping the conceptual terrain of middle school students' competencies in justifying and proving . 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. G. , Bryant , R. L. and Nooney , K. Vol. 4 , pp. 1693 – 1670 . Athens, GA : Clearinghouse for Science, Mathematics, and Environmental Education .
   Google Scholar
• Lakatos , I. 1976 . Proofs and refutations: The logic of mathematical discovery , Cambridge, , UK : University Press .
   Google Scholar
• Lampert , M. 1990 . When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching . American Educational Research Journal , 27 : 29 – 63 .
   Web of Science ®Google Scholar
• Lampert , M. 1992 . “ Practices and problems in teaching authentic mathematics ” . In Effective and responsible teaching: The new synthesis , Edited by: Oser , F. K. , Dick , A. and Patry , J. 295 – 314 . San Francisco, CA : Jossey-Bass Publishers .
   Google Scholar
• Light , P. , Blaye , A. , Gilly , M. and Girotto , V. 1989 . Pragmatic schemas and logical reasoning in 6- to 8-year-old children . Cognitive Development , 4 : 49 – 64 .
   Google Scholar
• Maher , C. A. and Martino , A. M. 1996 . The development of the idea of a mathematical proof: A 5-year case study . Journal for Research in Mathematics Education , 27 : 194 – 214 .
   Web of Science ®Google Scholar
• Mariotti , M. A. , Bussi , M. G. B. , Boero , P. , Ferri , F. and Garutti , R. . Approaching geometry theorems in contexts: From history and epistemology to cognition . Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education . Edited by: Penhoken , E. pp. 180 – 195 . Lahti, , Finland
   Google Scholar
• Mariotti , M. A. and Fischbein , E. 1997 . Defining in classroom activities . Educational Studies in Mathematics , 34 : 219 – 248 .
   Google Scholar
• Markovits , H. , Schleifer , M. and Fortier , L. 1989 . Development of elementary deductive reasoning in young children . Developmental Psychology , 25 : 787 – 793 .
   Web of Science ®Google Scholar
• Marrades , R. and Gutiérrez , A. 2000 . Proofs produced by secondary school students learning geometry in a dynamic computer environment . Educational Studies in Mathematics , 44 : 87 – 125 .
   Google Scholar
• Martin , W. G. and Harel , G. 1989 . Proof frames of preservice elementary teachers . Journal for Research in Mathematics Education , 10 : 41 – 51 .
   Google Scholar
• Mason , J. , Burton , L. and Stacey , K. 1982 . Thinking mathematically , London : Addison-Wesley .
   Google Scholar
• Moore , R. C. 1994 . Making the transition to formal proof . Educational Studies in Mathematics , 27 : 249 – 266 .
   Google Scholar
• Moshman , D. 1998 . “ Cognitive development beyond childhood ” . In Handbook of child psychology, Vol. 2: Cognition, perception, and language , 5th , Edited by: Damon , W. , Kuhn , D. and Siegler , R. S. 947 – 978 . New York : Wiley .
   Google Scholar
• NCTM . 2000 . Principles and standards for school mathematics , Reston, VA : National Council of Teachers of Mathematics .
   Google Scholar
• O'Brien , T. C. 1972 . Logical thinking in adolescents . Educational Studies in Mathematics , 4 : 401 – 428 .
   Google Scholar
• O'Brien , T. C. 1975 . Deformation and the four card problem . Educational Studies in Mathematics , 6 : 23 – 39 .
   Google Scholar
• Osherson , D. N. 1974 . Logical abilities in children. Logical inference: Underlying operations , Vol. 2 , Hillsdale, N.J : Erlbaum .
   Google Scholar
• Osherson , D. N. 1975 . Logical abilities in children. Reasoning in adolescence: Deductive inference , Vol. 3 , Hillsdale, N.J : Erlbaum .
   Google Scholar
• Osherson , D. N. 1976 . Logical abilities in children: Reasoning and concepts , Vol. 4 , Hillsdale, N.J : Erlbaum .
   Google Scholar
• Overton , W. F. , Byrne , J. P. and O'Brien , D. P. 1985 . Developmental and individual differences in conditional reasoning: The role of contradiction training and cognitive style . Developmental Psychology , 21 : 692 – 702 .
   Web of Science ®Google Scholar
• Overton , W. F. , Ward , S. L. , Black , J. , Noveck , I. A. and O'Brien , D. P. 1987 . Form and content in the development of deductive reasoning . Developmental Psychology , 23 : 22 – 30 .
   Web of Science ®Google Scholar
• Piaget , J. 1928 . Judgment and reasoning in the child , New York : Harcourt, Brace, and Co .
   Google Scholar
• Piaget , J. 1987 . The role of necessity in cognitive development , Minneapolis : University of Minnesota Press .
   Google Scholar
• Polya , G. 1954 . Induction and analogy in mathematics , Princeton, NJ : Princeton University Press .
   Google Scholar
• Porteous , K. 1990 . What do children really believe? . Educational Studies in Mathematics , 21 : 589 – 598 .
   Google Scholar
• Reid, D. A. (2001). Proof, proofs, proving and probing: Research related to proof. Short Oral presentation. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. I, p. 360). Utrecht, Netherlands. http://ace.acadiau.ca/∼dreid/publications/proof/proof.htm (ace.acadiau.ca/~dreid/publications/proof/proof.htm) (Accessed: 18 January 2006 ).
   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
• Reid, D. A. (2005). The meaning of proof in mathematics education. http://cerme4.crm.es/Papers%20definitius/4/Reid.pdf (Accessed: 18 January 2006 ).
   Google Scholar
• Richards , C. A. and Sanderson , J. A. 1999 . The role of imagination in facilitating deductive reasoning in 2-, 3- and 4-year-olds . Cognition , 72 : B1 – B9 .
   PubMed Web of Science ®Google Scholar
• Rips , L. J. 1994 . The psychology of proof: Deductive reasoning in human thinking , Cambridge, MA : MIT Press .
   Google Scholar
• Schoenfeld , A. H. 1983 . Problem solving in the mathematics curriculum: A report, recommendations, and an annotated bibliography . The Mathematical Association of America , MAA Notes, No. 1
   Google Scholar
• Schoenfeld , A. H. 1991 . “ On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics ” . In Informal reasoning and education , Edited by: Voss , J. F. , Perkins , D. N. and Segal , J. W. 311 – 343 . Hillsdale, NJ : Lawrence Erlbaum Associates .
   Google Scholar
• Schoenfeld , A. H. 1994 . What do we know about mathematics curricula? . Journal of Mathematical Behavior , 13 : 55 – 80 .
   Google Scholar
• Selden , A. and Selden , J. 2003 . Validations of proofs written as texts: Can undergraduates tell whether an argument proves a theorem? . Journal for Research in Mathematics Education , 34 : 4 – 36 .
   Web of Science ®Google Scholar
• Shibli , J. 1932 . Recent developments in the teaching of geometry , State College, PA : Author .
   Google Scholar
• Simon , M. A. and Blume , G. W. 1996 . Justification in mathematics classrooms: A study of prospective elementary school students . Journal of Mathematical Behavior , 15 : 3 – 31 .
   Google Scholar
• Stylianides , A. J. 2007a . Introducing young children to the role of assumptions in proving . Mathematical Thinking and Learning , 9 : 361 – 385 .
   Google Scholar
• Stylianides , A. J. 2007b . Proof and proving in school mathematics . Journal for Research in Mathematics Education , 38 : 289 – 321 .
   Web of Science ®Google Scholar
• Stylianides , A. J. , Stylianides , G. J. and Philippou , G. N. 2004 . Undergraduate students' understanding of the contraposition equivalence rule in symbolic and verbal contexts. . Educational Studies in Mathematics , 55 : 133 – 162 .
   Google Scholar
• Stylianides , G. J. 2007c . Investigating the guidance offered to teachers in curriculum materials: The case of proof in mathematics . International Journal of Science and Mathematics Education , 6 : 191 – 215 .
   Google Scholar
• Stylianides , G. J. , Stylianides , A. J. and Philippou , G. N. 2007 . Preservice teachers' knowledge of proof by mathematical induction . Journal of Mathematics Teacher Education , 10 : 145 – 166 .
   Google Scholar
• Sweller , J. , van Merrienboer , J. J. G. and Paas , F. W. C. 1998 . Cognitive architecture and instructional design . Educational Psychology Review , 10 : 251 – 296 .
   Web of Science ®Google Scholar
• Usiskin , Z. 1987 . “ Resolving the continuing dilemmas in school geometry ” . In Learning and Teaching Geometry, K–12: 1987 Yearbook , Edited by: Lindquist , M. M. and Shulte , A. P. 17 – 31 . Reston, VA : National Council of Teachers of Mathematics .
   Google Scholar
• Vinner , S. 1991 . “ The role of definitions in the teaching and learning of mathematics ” . In Advanced mathematical thinking , Edited by: Tall , D. 65 – 81 . Dordrecht, , Netherlands : Kluwer Academic Publishers .
   Google Scholar
• Ward , S. L. and Overton , W. F. 1990 . Semantic familiarity, relevance, and the development of deductive reasoning . Developmental Psychology , 26 : 488 – 493 .
   Google Scholar
• Wason , P. C. 1966 . “ Reasoning ” . In New horizons in psychology , Edited by: Foss , B. M. Vol. I , 135 – 151 . Harmondsworh : Penguin .
   Google Scholar
• Weber , K. 2001 . Student difficulty in constructing proof: the need for strategic knowledge . Educational Studies in Mathematics , 48 : 101 – 119 .
   Google Scholar
• Yackel , E. and Cobb , P. 1996 . Sociomathematical norms, argumentation, and autonomy in mathematics . Journal for Research in Mathematics Education , 27 : 458 – 477 .
   Web of Science ®Google Scholar
• Yackel , E. and Hanna , G. 2003 . “ Reasoning and proof ” . In A research companion to Principles and Standards for School Mathematics , Edited by: Kilpatrick , J. , Martin , W. G. and Schifter , D. 22 – 44 . Reston, VA : National Council of Teachers of Mathematics .
   Google Scholar
• Zack , V. . “You have to prove us wrong”: Proof at the elementary school level . Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education . Edited by: Pehkonen , E. Vol. 4 , pp. 291 – 298 . Lahti, , Finland : University of Helsinki .
   Google Scholar
• Zaslavsky , O. and Shir , K. 2005 . Students' conceptions of a mathematical definition . Journal for Research in Mathematics Education , 36 : 317 – 346 .
   Web of Science ®Google Scholar