Latest reasearch:Elaborate something with Infinity and then compare it with something Finite
It is Allah who has created seven heavens and of the earth, the like of them. [His] command descends among them so you may know that Allah is over all things competent and that Allah has encompassed all things in knowledge. (Talaq 12)
The heavens are mentioned first and the case of Earth is compared to them.
It suggests the order of creation and something else too.
1.The Universe is continuously expanding.The measurements/dimensions of space fall into infinity.
2.The diameter of Earth,its gravity and almost everything is measurable.Nothing falls into infinity.
Read the following research:
Research in mathematics education indicates that in the transition from given systems to wider ones prospective teachers tend to attribute all the properties that hold for the former also to the latter. In particular, it has been found that, in the context of Cantorian Set Theory, prospective teachers have been found to erroneously attribute properties of finite sets to infinite ones — using different methods to compare the number of elements in infinite sets. These methods which are acceptable for finite sets, lead to contradictions with infinite ones. This paper describes a course in Cantorian Set Theory that relates to prospective secondary mathematics teachers’ tendencies to overgeneralize from finite to infinite sets. The findings clearly indicate that when and were also more consistent in their use of a single method than those who studied the traditional, formally based Cantorian Set Theory course. the prospective teachers who took the course were more successful comparing the number of elements in infinite sets
The best way is to compare something with infinity to something finite.
Allah has started from the Heavens for us.
Almog, N.: 1988, Conceptual Adjustment in Progressing from Real to Complex Numbers: An Educational Approach, Unpublished thesis for the Master’s degree. Tel Aviv University, Tel Aviv, Israel, (in Hebrew).
Ball, D. L.: 1990, ‘Prospective elementary and secondary teachers’ understanding of division’, Journal for Research in Mathematics Education 21(2), 132–144.
Boolos, G.: 1964/1983, ‘The iterative concept of set’, in P. Benacerraf and H. Putman (eds.), Philosophy of Mathematics, Cambridge University Press, Cambridge, pp. 486–502.
Borasi, R.: 1985, ‘Errors in the enumeration of infinite sets’, Focus on Learning Problems in Mathematics 7, 77–88.
Davis, P. J. and Hersh, R.: 1980/1990, The Mathematical Experience, Penguin, London, pp. 136–140, 161–162.
Duval, R.: 1983, ‘L’obstacle du dedoublement des objects mathematiques’, Educational Studies in Mathematics 14, 385–414. View Article
Falk, R., Gassner, D., Ben Zoor, F. and Ben Simon, K.: 1986, ‘How do children cope with the infinity of numbers?’ Proceedings of the 10th Conference of the International Group for the Psychology of Mathematics Education, London, England, pp. 7–12.
Fischbein, E.: 1983, ‘The role of implicit models in solving elementary arithmetical problems’, Proceedings of the 7th Conference of the International Group for the Psychology of Mathematics Education, Rehovot, Israel, pp. 2–18.
Fischbein, E.: 1987, Intuition in Science and Mathematics, D. Reidel, Dordrecht, The Netherlands.
Fischbein, E.: 1993, ‘The interaction between the formal and the algorithmic and the intuitive components in a mathematical activity’, in R. Biehler, R. W. Scholz, R. Straser and B. Winkelmann (eds.), Didactic of Mathematics as a Scientific Discipline, Kluwer, Dordrecht, The Netherlands, pp. 231–345.
Fischbein, E., Jehiam, R. and Cohen, D.: 1995, ‘The concept of irrational numbers in high-school students and prospective teachers’, Educational Studies in Mathematics 29(1), 29–44. View Article
Fischbein, E. and Tirosh, D.: 1996, Mathematics and Reality, unpublished manuscript, Tel Aviv University, Tel Aviv, Israel (in Hebrew).
Fischbein, E., Tirosh, D. and Hess, P.: 1979, ‘The intuition of infinity’, Educational Studies in Mathematics 10, 3–40. View Article
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. View Article
Fraenkel, A. A.: 1953/1961, Abstract Set Theory, North-Holland, Amsterdam.
Fraenkel, A. A. and Bar-Hillel, Y.: 1958, Foundations of Set Theory, North-Holland, Amsterdam.
Greer, B.: 1994, ‘Rational numbers’, in T. Husen and N. Postlethwaite (eds.), International Encyclopedia of Education (Second ed.), Pergamon, London.
Hart, K.: 1981, Children’s Understanding of Mathematics, 11–16, Murray, London.
Hefendehl, H. L.: 1991, ‘Negative numbers: Obstacles in their evolution from intuitive to intellectual constructs’, For the Learning of Mathematics 11(1), 26–32.
Kitcher, P.: 1947/1984, The Nature of Mathematical Knowledge, Oxford University Press, pp. 101–148.
Klein, R. and Tirosh, D.: 1997, ‘Teachers’ pedagogical content knowledge of multiplication and division of rational numbers’, Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, Lahti, Finland, 3, 144–151.
Martin, W. G. and Wheeler, M. M.: 1987, ‘Infinity concepts among preservice elementary school teachers’, Proceedings of the 11th Conference of the International Group for the Psychology of Mathematics Education, France, pp. 362–368.
Moloney, K. and Stacey, K.: 1996, ‘Understanding decimals’, Australian Mathematics Teacher 52(1), 4–8.
Papert, S.: 1980, Mindstorms: Children, Computers and Powerful Ideas, Harvester, England.
Putt, I. J.: 1995, ‘Preservice teachers ordering of decimal numbers: When more is smaller and less is larger!’ Focus on Learning Problems in Mathematics 17(3), 1–15.
Smullyan, R. M.: 1971, ‘The continuum hypothesis’, in The mathematical Sciences, The M.I.T. Press, Cambridge, pp. 252–260.
Streefland, L.: 1996, ‘Negative numbers: reflection of a learning researcher’, Journal of Mathematical Behavior 15(1), 57–77. View Article
Tall, D.: 1980, ‘The notion of infinite measuring numbers and its relevance in the intuition of infinity’, Educational Studies in Mathematics 11, 271–284. View Article
Tall, D.: 1990, ‘Inconsistencies in the learning of calculus and analysis’, Focus on Learning Problems in Mathematics 12(3&4), 49–64.
Tall, D. and Vinner, S.: 1981, ‘Concept image and concept definition in mathematics with particular reference to limit and continuity’, Educational Studies in Mathematics 12, 151–169. View Article
Tirosh, D.: 1990, ‘Inconsistencies in students’ mathematical constructs’, Focus on Learning Problems in Mathematics 12, 111–129.
Tirosh, D.: 1991, ‘The role of students’ intuitions of infinity in teaching the cantorial theory’, in D. Tall (ed.), Advanced Mathematical Thinking, Kluwer, Dordrecht, The Netherlands, pp. 199–214.
Tirosh, D. and Tsamir, P.: 1996, ‘The role of representations in students’ intuitive thinking about infinity’, International Journal of Mathematics Education in Science and Technology 27(1), 33–40.
Tsamir, P.: 1990, Students’ Inconsistent Ideas about Actual Infinity, Unpublished thesis for the Master’s degree. Tel Aviv University, Tel Aviv, Israel (in Hebrew).
Tsamir, P. and Tirosh, D.: 1992, ‘Students’ awareness of inconsistent ideas about actual infinity’, Proceedings of the 16th Conference of the International Group for the Psychology of Mathematics Education, Durham, USA, 3, 90–97.
Tsamir, P. and Tirosh, D.: 1994, ‘Comparing infinite sets: intuitions and representations’, Proceedings of the 18th Conference of the International Group for the Psychology of Mathematics Education Lisbon, Portugal, 4, 345–352.
Tsamir, P. and Tirosh, D.: ‘Consistency representations: The case of actual infinity’, Journal for Research in Mathematics Education, in press.
Vinner, S.: 1990, ‘Inconsistencies: Their causes and function in learning mathematics’, Focus on Learning Problems in Mathematics 12(3&4), 85–98.
Wilson, P.: 1990, ‘Inconsistent ideas related to definition and examples’, Focus on Learning Problems in Mathematics 12(3&4), 31–48.