[1] E. Bingolbali, Multiple solutions to problems in mathematics teaching: do teachers really value them?, Austral. J. Teach. Educ. 36 (2011), 18-31.
[2] A. Bogomolny, Ptolemy’s Theorem, Interactive Mathematics Miscellany and Puzzles. http://www.cut-the-knot.org/proofs/ptolemy.shtml (Accessed Sep. 2012).
[3] H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, Mathematical Association of America, 1967, pp. 29-30.
[4] T. Dreyfus and T. Eisenberg, On the aesthetics of mathematical thought, Learn. Math. 6 (1986), 2-10.
[5] F. A. Ersoz, Proof in different mathematical domains, ICME Study 1 (2009), 160-165.
[6] G. Hanna, Some pedagogical aspects of proof, Interchange 21 (1990), 6-13.
[7] G. Hanna, The ongoing value of proof, L. Puig and A. Gutiérrez, eds., Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education, Valencia, Spain, Vol. 2, 1996, pp. 21-34.
[8] G. Harel and L. Sowder, Toward comprehensive perspectives on the learning and teaching of proof, F. K. Lester, ed., Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics, Information Age Publishing Inc., Greenwich, CT, 2007, pp. 805-842.
[9] L. Healy and C. Hoyles, Proof conception in algebra, J. Res. Math. Educ. 31 (2000), 396-428.
[10] R. Hersh, Proving is convincing and explaining, Educ. Stud Math. 24 (1993), 389-399.
[11] R. Honsberger, Mathematical Morsels, Mathematical Association of America, 1978.
[12] R. Hoshino, The other side of inequalities, Part Four, Mathematical Mayhem 7(4) (1995).
[13] P. A. House and A. F. Coxford, Connecting Mathematics Across the Curriculum, Yearbook, NCTM, Reston, VA, 1995.
[14] F.-J. Hsieh, F.-T. Lee and T.-Y. Wang, How much proofs are students able to learn in mathematics class from their teachers, F.-L. Lin, F.-J. Hsieh, G. Hanna and M. de Villiers, eds., Proceedings of the ICMI Study 19 Conference: Proof and Proving in Mathematics Education, The Department of Mathematics, National Taiwan Normal University Taipei, Taiwan, Vol. 1, 2009, pp. 208-213.
[15] W.-Y. Hwang, N.-S. Chen, J.-J. Dung and Y.-L. Yang, Multiple representation skills and creativity effects on mathematical problem solving using a multimedia whiteboard system, Educational Technology & Society 10 (2007), 191-212.
[16] E. J. Knuth, Proof as a tool for learning mathematics, Math. Teach. 95 (2002), 486-490.
[17] E. J. Knuth, J. Choppin, M. Slaughter and J. Sutherland, Mapping the conceptual terrain of middle school students’ competencies in justifying and proving, D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Weigel, R. L. Bryant and K. Nooney, eds., Proceedings of the 24th Annual Meeting of the North America Chapter of the International Group for the Psychology of Mathematics Education, Athens, GA: Clearinghouse for Science, Mathematics, and Environmental Education, Vol. 4, 2002, pp. 1693-1670.
[18] B. Koichu and A. Berman, When do gifted high school students use geometry to solve geometry problems?, J. Secondary Gifted Education 16 (2005), 168-179.
[19] V. A. Krutetskii, The psychology of mathematical abilities in school children, J. Kilpatrick and I. Wirszuy, eds., (translated from Russian by J. Teller), The University of Chicago Press, Chicago, 1976.
[20] R. Leikin, Multiple proof tasks: teacher practice and teacher education, ICME Study 19(2) (2009), 31-36.
[21] R. Leikin and H. Lev, Multiple solution tasks as a magnifying glass for observation of mathematical creativity, J. H. Wo, H. C. Lew, K. S. Park and D. Y. Seo, eds., Proceedings of the 31st International Conference for the Psychology of Mathematics Education, The Korea Society of Educational Studies in Mathematics, Korea, Vol. 3, 2007, pp. 161-168.
[22] R. Leikin and A. Levav-Waynberg, Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks, Educ. Stud. Math. 66 (2007), 348-371.
[23] A. Levav-Waynberg and R. Leikin, Multiple solutions for a problem: a tool for evaluation of mathematical thinking in geometry, Proceedings of CERME 6, January 28th -February 1st 2009, Lyon, France, 2009, pp. 776-785.
[24] T. S. Martin, S. M. S. McCrone, M. L. W. Bower and J. Dindyal, The interplay of teacher and student actions in the teaching and learning of geometric proof, Educ. Stud. Math. 60 (2005), 95-124.
[25] E. Meitav, Matriculation Questionnaires and Their Solutions. A Challenging Problem Given by Dr. Peter Samovol, Eshel Hanasi High School, Israel, Shoresh Production, 2012, p. 46 (in Hebrew).
[26] National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, Author, Reston, VA, 2000.
[27] National Council of Teachers of Mathematics, Focus on High School Mathematics: Reasoning and Sense-making, Author, Reston, VA, 2009.
[28] G. Polya, How to Solve It: A New Aspect of Mathematical Method, Princeton University Press, Princeton, NJ, 1973.
[29] G. Polya, Mathematical Discovery: On Understanding Learning and Teaching Problem Solving, Wiley, New York, 1981.
[30] Y. Rav, Why do we prove theorems?, Philosophia Mathematica 7 (1999), 5-41.
[31] A. H. Schoenfeld, Mathematical Problem Solving, Academic Press, New York, 1985.
[32] A. H. Schoenfeld, When good teaching leads to bad results: the disasters of “well-taught” mathematics courses, Educ. Psychol. 23 (1988), 145-166.
[33] A. Shriki, Back to the Treasure Island, Math. Teach. 104 (2011), 658-664.
[34] E. A. Silver, Foresting creativity through instruction rich in mathematical problem solving and problem posing, ZDM 3 (1997), 75-80.
[35] E. Silver and W. Metzger, Aesthetic influences on expert problem solving, D. B. McLeod and V. M. Adams, eds., Affect and Mathematical Problem Solving, Springer-Verlag, New York, 1989, pp. 59-74.
[36] N. Sinclair, Mathematics and Beauty: Aesthetic Approaches to Teaching Children, Teachers College Press, New York, 2006.
[37] N. Sinclair, Aesthetic considerations in mathematics, J. Humanistic Mathematics 1 (2011), 2-32.
[38] R. Skemp, Relational understanding and instrumental understanding, Math. Teach. 77 (1976), 20-26.
[39] M. D. Steele and K. C. Rogers, Relationships between mathematical knowledge for teaching and teaching practice: the case of proof, J. Math. Teach. Educ. 15 (2012), 159-180.
[40] A. J. Stylianides, G. J. Stylianides and G. N. Philippou, University students’ conceptions of empirical proof and proof by counterexample, M. Tzekaki, ed., Proceedings of the 5th Panellenian Conference on “Didactics of Mathematics and Computers in Education”, Thessaloniki, Greece, Aristotle University of Thessaloniki, 2002, pp. 277-282.
[41] A. J. Stylianides, G. J. Stylianides and G. N. Philippou, Undergraduate students’ understanding of the contraposition equivalence rule in symbolic and verbal contexts, Educ. Stud. Math. 55 (2004), 133-162.
[42] A. J. Stylianides, G. J. Stylianides and G. N. Philippou, Prospective teachers’ understanding of proof: What if the truth set of an open sentence is broader than that covered by the proof?, H. L. Chick and J. L. Vincent, eds., Proceedings of the 29th Annual Meeting of the International Group for the Psychology of Mathematics Education. Melbourne, Australia, Vol. 4, 2005, pp. 241-248.
[43] D. Tall, Teachers as mentors to encourage both power and simplicity in active material learning, Plenary Lecture at the Third Annual Conference for Middle East Teachers of Science, Mathematics and Computing, Abu-Dhabi, 17-19, March, 2007.
[44] E. W. Weisstein, Euler’s Inequality, From Math World – A Wolfram Web Resource. http://mathworld.wolfram.com/EulersInequality.html, (Accessed Sep. 2012).
[45] E. Yackel and G. Hanna, Reasoning and proof, J. Kilpatrick, W. G. Martin and D. E. Schifter, eds., A Research Companion to Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA, 2003, pp. 227-236. |