Conditions for proving by mathematical induction to be explanatory
Tài liệu tham khảo
Bell, 1976, A study of pupil’s proof-explanations in mathematical situations, Educational Studies in Mathematics, 7, 23, 10.1007/BF00144356
Boero, 1996, Challenging the traditional school approach to theorems: a hypothesis about the cognitive unity of theorems, vol. 2, 113
Boero, 1996, Some dynamic mental processes underlying producing and proving conjectures, vol. 2, 121
Brown, 2008, Exploring epistemological obstacles to the development of mathematics induction, Proceedings of the 11th conference for research on undergraduate mathematics education
Cariani, 2011
Davis, 2009, Learning proof by mathematical induction, Proceedings of the 12th conference on research in undergraduate mathematics education
de Villiers, 1999, The role and function of proof, 3
Dubinsky, 1986, Reflective abstraction and mathematics education: the generic decomposition of induction and compactness, Journal of Mathematical Behavior, 5, 55
Dubinsky, 1986, Teaching mathematical induction I, Journal of Mathematical Behavior, 5, 305
Dubinsky, 1990, Teaching mathematical induction II, Journal of Mathematical Behavior, 8, 285
Garuti, 1998, Cognitive unity of theorems and difficulty of proof, vol. 2, 345
Hanna, 1990, Some pedagogical aspects of proof, Interchange, 21, 6, 10.1007/BF01809605
Hanna, 2000, Proof: explanation and exploration: an overview, Educational Studies in Mathematics, 44, 5, 10.1023/A:1012737223465
Harel, 1998, Students’ proof schemes: results from exploratory studies, 234
Harel, 2007, Toward comprehensive perspectives on the learning and teaching of proof, 805
Harel, 2001, The development of mathematical induction as a proof scheme: a model for DNR-based instruction, 185
Hersh, 1993, Proving is convincing and explaining, Educational Studies in Mathematics, 24, 389, 10.1007/BF01273372
Kitcher, 1989, Explanatory unification and the causal structure of the world, 410
Knuth, 2002, Secondary school mathematics teachers’ conceptions of proof, Journal for Research in Mathematics Education, 33, 379, 10.2307/4149959
Lange, 2009, Why proofs by mathematical induction are generally not explanatory, Analysis, 69, 203, 10.1093/analys/anp002
Maher, 1996, The development of the idea of mathematical proof: a 5-year case study, Journal for Research in Mathematics Education, 27, 194, 10.2307/749600
Maher, 2007, The development of proof making by students, 197
Mariotti, 2006, Proof and proving in mathematics education, 173
Movshovitz-Hadar, 1993, The false coin problem, mathematical induction and knowledge fragility, Journal of Mathematical Behavior, 12, 253
Raman, 2009, Is that a proof? Using video to teach and learn how to prove at the university level, vol. 2, 154
Raman, 2003, Key ideas: what are they and how can they help us understand how people view proof?, Educational Studies in Mathematics, 52, 319, 10.1023/A:1024360204239
Reid, 2002, Conjectures and refutations in grade 5 mathematics, Journal for Research in Mathematics Education, 33, 5, 10.2307/749867
Resnik, 1987, Explanation, independence, and realism in mathematics, British Journal for the Philosophy of Science, 38, 141, 10.1093/bjps/38.2.141
Ron, 2004, The use of models in teaching proof by mathematical induction, vol. 4, 113
Sandefur, 2013, Generating and using examples in the proving process, Educational Studies in Mathematics, 83, 323, 10.1007/s10649-012-9459-x
Smith, 2006, A sense-making approach to proof: strategies of students in traditional and problem-based number theory courses, Journal of Mathematical Behavior, 25, 73, 10.1016/j.jmathb.2005.11.005
Steiner, 1978, Mathematical explanation, Philosophical Studies, 34, 135, 10.1007/BF00354494
Stylianides, 2009, Facilitating the transition from empirical arguments to proof, Journal for Research in Mathematics Education, 40, 314, 10.5951/jresematheduc.40.3.0314
Stylianides, 2007, Preservice teachers’ knowledge of proof by mathematical induction, Journal of Mathematics Teacher Education, 10, 145, 10.1007/s10857-007-9034-z
Stylianides, 2016, Research on the teaching and learning of proof: taking stock and moving forward
Stylianides, 2007, Proof and proving in school mathematics, Journal for Research in Mathematics Education, 38, 289
Stylianides, 2009, Reasoning-and-proving in school mathematics textbooks, Mathematical Thinking and Learning, 11, 258, 10.1080/10986060903253954
Weber, 2010, Proofs that develop insight, For the Learning of Mathematics, 30, 32