The effect of using Transparent Pseudo-Proofs in Linear Algebra

References

• Atkinson , R.K. , Derry , S.J. , Renkl , A. and Wortham , D. 2000 . Learning from examples: Instructional principles from the worked examples research . Review of Educational Research , 70 ( 2 ) : 181 – 214 .
   Web of Science ®Google Scholar
• Avital , S.M. , and S.J. Shettleworth . 1968 . Objectives for mathematics learning, some ideas for the teacher . OISE – The Ontario Institute for Studies in Education Bulletin 3 .
   Google Scholar
• Balacheff , N. 1987 . Processus de preuves et situations de validation . Educational Studies in Mathematics , 18 ( 2 ) : 147 – 76 .
   Google Scholar
• Carlson , D. 1993 . Teaching linear algebra: Must the fog always roll in? . College Mathematics Journal , 24 ( 1 ) : 29 – 40 .
   Google Scholar
• Carlson , D. , Johnson , C.R. , Lay , D.C. , Porter , A.D. , Watkins , A. and Watkins , W. 1993 . The linear algebra curriculum study group recommendations for the first course in linear algebra . College Mathematics Journal , 24 ( 1 ) : 41 – 46 .
   Google Scholar
• Cooper , M. and Sweller , J. 1987 . Effects of schema acquisition and rule automation of mathematical problem-solving transfer . Journal of Educational Psychology , 79 ( 4 ) : 347 – 62 .
   Web of Science ®Google Scholar
• Davis , R.B. 1988 . The interplay of algebra, geometry and logic . Journal of Mathematical Behavior , 7 : 9 – 28 .
   Google Scholar
• De Villiers , M. 1990 . The role and function of proof in mathematics . Pythagoras , 24 : 17 – 24 .
   Google Scholar
• Dubinsky , E. 2000 . Mathematical literacy and abstraction in the 21st century . School Science and Mathematics , 100 : 289 – 97 .
   Google Scholar
• Easdown , D. 2006 . Teaching mathematics: The gulf between semantics (meaning) and syntax (form) . In Proceedings of the 3rd International Conference on the Teaching of Mathematics at the Undergraduate Level . Istanbul : Turkish Mathematical Society .
   Google Scholar
• Easdown , D. 2007 . The role of proof in mathematics teaching and the plateau principle . http://www.maths.usyd.edu.au/u/pubs/publist/preprints/2007/easdown-15.pdf .
   Google Scholar
• Edwards , L. 1998 . Odds and evens: Mathematical reasoning and informal proof among high school students . Journal of Mathematical Behavior , 17 ( 4 ) : 489 – 504 .
   Google Scholar
• Hanna , G. and Barbeau , E. 2008 . Proofs as bearers of mathematical knowledge . ZDM – The International Journal on Mathematics Education , 40 ( 3 ) : 345 – 53 .
   Google Scholar
• Hanna , G. , M. de Villiers , F. Arzarello , T. Dreyfus , V. Durand-Gurrier , N. Jahnke , F. Lin , A. Selden , D. Tall , and O. Yvdokimov . 2009 . Proof and proving in mathematics education: Discussion document . In Proceedings of the ICMI Study 19 conference: Proof and Proving in Mathematics Education, ed. F.L. Lin, F. Hsieh, Hanna, M.G., and de Villiers, xix–xxx. Taipei, Taiwan: National Taiwan Normal University .
   Google Scholar
• Harel , G. 1989 . Learning and teaching linear algebra: Difficulties and an alternative approach to visualizing concepts and processes . Focus on Learning Problems in Mathematics , 11 : 139 – 48 .
   Google Scholar
• Harrison , A.G. and Treagust , D.F. 1998 . Modeling in science lessons: Are there better ways to learn with models? . School Science and Mathematics , 98 ( 8 ) : 420 – 9 .
   Google Scholar
• Hazzan , O. 1999 . Reducing abstraction level when learning abstract algebra concepts . Educational Studies in Mathematics , 40 ( 1 ) : 71 – 90 .
   Google Scholar
• Joyce , D.E. 2008 . Euclid's Elements Book ix , Proposition 20 . http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html .
   Google Scholar
• Kieran , C. and Sfard , A. 1999 . Seeing through symbols: The case of equivalent expressions . Focus on Learning Problems in Mathematics , 21 ( 1 ) : 1 – 17 .
   Google Scholar
• Leron , U. 1985 . A direct approach to indirect proofs . Educational Studies in Mathematics , 16 : 321 – 25 .
   Google Scholar
• Leron , U. , and O. Zaslavsky . 2009 . Generic proving: Reflections on scope and method . In Proceedings of the ICMI Study 19 conference: Proof and proving in mathematics education , F.L. Lin , F.J. Hsieh , G. Hanna , and M. de Villiers , Vol. 2, 53–8. Taipei, Taiwan: National Taiwan Normal University .
   Google Scholar
• Lin , F.L. , Hsieh , F.J. , Hanna , G. and de Villiers , M. 2009 . Proceedings of the ICMI Study 19 conference: Proof and proving in mathematics education , National Taiwan Normal University, Taipei, Taiwan .
   Google Scholar
• Malek , A. , and N. Movshovitz-Hadar . 2009 . The art of constructing a transparent p-proof . In Proceedings of the ICMI Study 19 conference: Proof and proving in mathematics education , F.L. Lin , F.J. Hsieh , G. Hanna , and M. , de Villiers , Vol. 2 , 70 – 5 . Taipei, , Taiwan : National Taiwan Normal University .
   Google Scholar
• Mason , J. and Pimm , D. 1984 . Generic examples: Seeing the general in the particular . Educational Studies in Mathematics , 15 : 277 – 89 .
   Google Scholar
• Maurer , S.B. 1984 . Two meanings of algorithmic mathematics . Mathematics Teacher , 11, no. 6 : 430 – 35 .
   Google Scholar
• McLoughlin , M.P.M.M. 2008 . Inquiry based learning: A modified Moore method approach to encourage student research . Paper presented at the 11th Annual Legacy of R. L. Moore Conference, Austin, Texas .
   Google Scholar
• Meel , D.E. and T.A. Hern . 2005 . Tool building: Web-based linear algebra modules . Journal of Online Mathematics and its Applications . http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=560 .
   Google Scholar
• Moore , R.C. 1994 . Making transition to formal proof . Educational Studies in Mathematics , 27 : 249 – 66 .
   Google Scholar
• Movshovitz-Hadar , N. 1988a . Stimulating presentations of theorems followed by responsive proofs . For the Learning of Mathematics , 8 ( 2 ) : 12 – 30 .
   Google Scholar
• Movshovitz-Hadar , N. 1988b . School mathematics theorems – an endless source of surprise . For the Learning of Mathematics , 8 ( 3 ) : 34 – 40 .
   Google Scholar
• Movshovitz-Hadar , N. 1993 . The false coin problem, mathematical induction and knowledge fragility . Journal of Mathematical Behavior , 12 ( 3 ) : 253 – 268 .
   Google Scholar
• Movshovitz-Hadar , N. ( 2011, in press ). Bridging between mathematics and education courses: Strategy games as generators of problem solving and proving tasks . In Constructing knowledge for teaching secondary mathematics. Mathematics teacher education Vol. 6 , O. Zaslavsky and P. Sullivan , 117 – 40 . Berlin : Springer . doi: DOI: 10.1007/978-0-387-09812-8_8 .
   Google Scholar
• Movshovitz-Hadar , N. and Malek , A. 1998 . “ Transparent Pseudo-Proofs – a bridge to formal proofs ” . In Proceedings of the International Conference on the Teaching of Mathematics, 221–3. , New York : John Wiley & Sons .
   Google Scholar
• Mwangi , W. and Sweller , J. 1998 . Learning to solve compare word problems: The effect of example format and generating self-explanations . Cognition and Instruction , 16 ( 2 ) : 173 – 99 .
   Web of Science ®Google Scholar
• Retzer , K.A. 1996 . A geometry proof making model ; In Proceedings of ICME 8 – The 8th International Congress on Mathematics Education , ed. C. Alsina, J. M. Alvarez, M. Niss, A. Perez, L. Rico, A. Sfard , 60 – 85 . Seville , Spain .
   Google Scholar
• Rowland , T. 1998 . Conviction, explanation and generic examples . In Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education , A. Olivier , and K. Newstead , Vol. 4 , 65 – 72 . Stellenbosch, , South Africa : University of Stellenbosch .
   Google Scholar
• Rowland , T. 2001 . Generic proofs: setting a good example . Mathematics Teaching , 177 : 40 – 3 .
   Google Scholar
• Rowland , T. , 2002 . Generic proofs in number theory . In Learning and teaching number theory: Research in cognition and instruction , S. Campbell and R. Zazkis , 157 – 84 . Westport, CT : Ablex Publishing .
   Google Scholar
• Sierpinska , A. 1997 . Formats of interaction and model readers . For the Learning of Mathematics , 17 ( 2 ) : 3 – 12 .
   Google Scholar
• Selden , A. and Selden , J. 1995 . Unpacking the logic of mathematics statements . Educational Studies in Mathematics , 29 ( 2 ) : 123 – 51 .
   Google Scholar
• Sfard , A. and Linchevski , L. 1994 . The gains and the pitfalls of reification – the case of algebra . Educational Studies in Mathematics , 26 : 191 – 228 .
   Google Scholar
• Skemp , R. 1976 . Relational understanding and instrumental understanding . Mathematics Teaching , 77 : 20 – 6 .
   Google Scholar
• Steinbring , H. 1989 . Routine and meaning in the mathematics classroom. For the Learning of Mathematics 9 . no. , 1 : 24 – 33 .
   Google Scholar
• Tall , D. 1979 . Cognitive aspects of proof, with special reference to the irrationality of √2 . In Proceedings of the Third International Conference for the Psychology of Mathematics Education , Warwick: 203–5 .
   Google Scholar
• Tall , D. and Vinner , S. 1981 . Concept image and concept definition in mathematics with particular reference to limits and continuity . Educational Studies in Mathematics , 24 ( 2 ) : 151 – 69 .
   Google Scholar
• Thompson , D.R. 1996 . Learning and teaching indirect proof . Mathematics Teacher , 89 : 474 – 82 .
   Google Scholar
• Uhlig , F. 2002 . The role of proof in comprehending and teaching elementary linear algebra . Educational Studies in Mathematics , 50 ( 3 ) : 335 – 46 .
   Google Scholar
• Weber , K. and Alcock , L. 2004 . Semantic and syntactic proof production . Educational Studies in Mathematics , 56 ( 2 ) : 209 – 34 .
   Google Scholar