Paradoxes as a window to infinity

References

• Allis , V. and Koetsier , T. 1995 . On some paradoxes of the infinite II . The British Journal for the Philosophy of Science , 46 : 235 – 247 .
   Web of Science ®Google Scholar
• Bolzano , B. 1950 . Paradoxes of the infinite . Trans. F. Prihonsky, with introd. by D. Steele . London : Routledge and Kegan Paul .
   Google Scholar
• Cantor , G. 1915 . Contributions to the founding of the theory of transfinite numbers . Trans. P. Jourdain, repr. 1955 . New York : Dover Publications .
   Google Scholar
• Cottrill , J. , Dubinsky , E. , Nichols , D. , Schwingendorf , K. , Thomas , K. and Vidakovic , D. 1996 . Understanding the limit concept: Beginning with a coordinated process scheme . Journal of Mathematical Behavior , 15 : 167 – 192 .
   Google Scholar
• Dreyfus , T. and Tsamir , P. 2004 . Ben's consolidation of knowledge structures about infinite sets . Journal of Mathematical Behavior , 23 : 271 – 300 .
   Google Scholar
• Dubinsky , E. and McDonald , M. 2001 . “ APOS: A constructivist theory of learning in undergraduate mathematics education research ” . In The teaching and learning of mathematics at university level: An ICMI study , Edited by: Holton , D. 273 – 80 . Dordrecht : Kluwer Academic Publishers .
   Google Scholar
• Dubinsky , E. , Weller , K. , McDonald , M.A. and Brown , A. 2005 . Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 1 . Educational Studies in Mathematics , 58 : 335 – 359 .
   Google Scholar
• Dubinsky , E. and Yiparaki , O. 2000 . “ On student understanding of AE and EA Quantification ” . In Research in collegiate mathematics education IV , Edited by: Dubinsky , E. , Schoenfeld , A.H. and Kaput , J. 239 – 89 . Providence, RI : American Mathematical Society .
   Google Scholar
• Ely , R. 2007 . Nonstandard models of arithmetic found in student conceptions of infinite processes . Paper presented at the 29th International Conference for the Psychology of Mathematics Education – North American Chapter, October 25–28, in Nevada, USA .
   Google Scholar
• Fischbein , E. 2001 . Tacit models of infinity . Educational Studies in Mathematics , 48 : 309 – 329 .
   Google Scholar
• Fischbein , E. , Tirosh , D. and Hess , P. 1979 . The intuition of infinity . Educational Studies in Mathematics , 10 : 3 – 40 .
   Google Scholar
• Fischbein , E. , Tirosh , D. and Melamed , U. 1981 . Is it possible to measure the intuitive acceptance of a mathematical statement? . Educational Studies in Mathematics , 12 : 491 – 512 .
   Google Scholar
• Hazzan , O. 1999 . Reducing abstraction level when learning abstract algebra concepts . Educational Studies in Mathematics , 40 : 71 – 90 .
   Google Scholar
• Mamolo , A. 2008 . Constrained by knowledge: The case of the infinite ping-pong balls . Paper presented at the 11th SIGMAA on RUME Conference, February 28–March 2, in San Diego, USA .
   Google Scholar
• Mamolo , A. , and R. Zazkis . 2007 . Confronting infinity via the ping-pong ball conundrum . Paper presented at the 29th International Conference for the Psychology of Mathematics Education – North American Chapter, October 25–28, in Nevada, USA .
   Google Scholar
• Movshovitz-Hadar , N. and Hadass , R. 1990 . Preservice education of math teachers using paradoxes . Educational Studies in Mathematics , 21 : 265 – 287 .
   Google Scholar
• National Council of Teachers of Mathematics 2000 . Principles and standards for school mathematics . Reston : NCTM .
   Google Scholar
• Quine , W.V.O. 1966 . Ways of paradox and other essays , Kingsport : Random House .
   Google Scholar
• Russell , B. 1903 . The principles of mathematics. Repr. 1996 . New York : W.W. Norton. Electronic version : http://fair-use.org/bertrand-russell/the-principles-of-mathematics/
   Google Scholar
• Tall , D.O. 1980 . The notion of infinite measuring numbers and its relevance in the intuition of infinity . Educational Studies in Mathematics , 11 : 271 – 284 .
   Google Scholar
• Tirosh , D. 1999 . Finite and infinite sets . International Journal of Mathematical Education in Science and Technology , 30 : 341 – 9 .
   Google Scholar
• Tirosh , D. and Tsamir , P. 1996 . The role of representations in students’ intuitive thinking about infinity . Journal of Mathematical Education in Science and Technology , 27 : 33 – 40 .
   Google Scholar
• Tsamir , P. 1999 . The transition from finite to infinite sets . Educational Studies in Mathematics , 38 ( 1–3 ) : 209 – 34 .
   Google Scholar
• Tsamir , P. and Tirosh , D. 1999 . Consistency and representations: The case of actual infinity . Journal for Research in Mathematics Education , 30 : 213 – 219 .
   Web of Science ®Google Scholar
• Van Bendegem , J.P. 1994 . Ross’ paradox is an impossible super-task . British Journal for the Philosophy of Science , 45 : 743 – 748 .
   Web of Science ®Google Scholar
• Williams , S. 1991 . Models of limits held by college calculus students . Journal for Research in Mathematics Education , 22 : 219 – 236 .
   Web of Science ®Google Scholar
• Zazkis , R. and Chernoff , E. 2008 . What makes a counterexample exemplary? . Educational Studies in Mathematics , 68 : 195 – 208 .
   Google Scholar
• Zazkis , R. and Leikin , R. 2007 . Generating examples: From pedagogical tool to a research tool . For the Learning of Mathematics , 27 : 15 – 21 .
   Google Scholar