Forms of Mathematical Knowledge pp 209-234 | Cite as

# The Transition from Comparison of Finite to the Comparison of Infinite Sets: Teaching Prospective Teachers

- 1 Citations
- 243 Downloads

## Abstract

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 comparing the number of elements in infinite sets the prospective teachers who took the course were more successful and were also more consistent in their use of a single method than those who studied the traditional, formally-based Cantorian Set Theory course.

## Keywords

Prospective Teacher Educational Study Mathematical System Actual Infinity Valid Judgment## Preview

Unable to display preview. Download preview PDF.

## References

- 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*Conceptual Adjustment in Progressing from Real to Complex Numbers: An Educational Approach*, Unpublished thesis for the Master’s degree.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).Google Scholar - Ball, D. L.: 1990, ‘Prospective elementary and secondary teachers’ understanding of division’,
*Journal for Research in Mathematics Education*21 (2), 132–144.CrossRefGoogle Scholar - 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.Google Scholar - Borasi, R.: 1985, ‘Errors in the enumeration of infinite sets’,
*Focus on Learning Problems in Mathematics*7, 77–88.Google Scholar - Davis, P. J. and Hersh, R.: 1980/1990,
*The Mathematical Experience*, Penguin, London, pp. 136–140, 161–162.Google Scholar - Duval, R.: 1983, ‘L’obstacle du dedoublement des objects mathematiques’,
*Educational Studies in Mathematics*14, 385–414.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - Fischbein, E.: 1987,
*Intuition in Science and Mathematics*, D. Reidel, Dordrecht, The Netherlands.Google Scholar - Fischbein, E.: 1993, ‘The interaction between the formal and the algorithmic and the intuitive components in a mathematical activity’, in R. Biehler, R. W.Google Scholar
- Scholz, R. Straser and B. Winkelmann (eds.),
*Didactic of Mathematics as a Scientific Discipline*,Kluwer, Dordrecht, The Netherlands, pp. 231–345.Google Scholar - 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.CrossRefGoogle Scholar - Fischbein, E. and Tirosh, D.: 1996,
*Mathematics and Reality*, unpublished manuscript, Tel Aviv University, Tel Aviv, Israel (in Hebrew).Google Scholar - Fischbein, E., Tirosh, D. and Hess, P.: 1979, ‘The intuition of infinity’,
*Educational Studies in Mathematics*10, 3–40.CrossRefGoogle 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.CrossRefGoogle Scholar - Fraenkel, A. A.: 1953/1961,
*Abstract Set Theory*,North-Holland, Amsterdam.Google Scholar - Fraenkel, A. A. and Bar-Hillel, Y.: 1958,
*Foundations of Set Theory*, North-Holland, Amsterdam.Google Scholar - Greer, B.: 1994, ‘Rational numbers’, in T. Husen and N. Postlethwaite (eds.),
*International Encyclopedia of Education*( Second ed. ), Pergamon, London.Google Scholar - 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.Google Scholar - Kitcher, P.: 1947/1984,
*The Nature of Mathematical Knowledge*,Oxford University Press, pp. 101–148.Google Scholar - 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.Google Scholar - 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.Google Scholar - Moloney, K. and Stacey, K.: 1996, ‘Understanding decimals’,
*Australian Mathematics Teacher*52 (1), 4–8.Google Scholar - Papert, S.: 1980,
*Mindstorms: Children*,*Computers and Powerful Ideas*, Harvester, England.Google Scholar - 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.Google Scholar - Smullyan, R. M.: 1971, ‘The continuum hypothesis’, in
*The mathematical Sciences*, The M.I.T. Press, Cambridge, pp. 252–260.Google Scholar - Streefland, L.: 1996, ‘Negative numbers: reflection of a learning researcher’,
*Journal of Mathematical Behavior*15 (1), 57–77.CrossRefGoogle Scholar - Tall, D.: 1980, ‘The notion of infinite measuring numbers and its relevance in the intuition of infinity’,
*Educational Studies in Mathematics*11, 271–284.CrossRefGoogle Scholar - Tall, D.: 1990, ‘Inconsistencies in the learning of calculus and analysis’, Focus on
*Learning Problems in Mathematics*12 (3 and 4), 49–64.Google Scholar - 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.CrossRefGoogle Scholar - Tirosh, D.: 1990, ‘Inconsistencies in students’ mathematical constructs’,
*Focus on Learning Problems in Mathematics*12, 111–129.Google Scholar - Tirosh, D.: 1991, ‘The role of students’ intuitions of infinity in teaching the cantonal theory’, in D. Tall (ed.),
*Advanced Mathematical Thinking*, Kluwer, Dordrecht, The Netherlands, pp. 199–214.Google Scholar - 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.CrossRefGoogle Scholar - Tsamir, P.: 1990,
*Students’ Inconsistent Ideas about Actual Infinity*, Unpublished thesis for the Master’s degree. Tel Aviv University, Tel Aviv, Israel (in Hebrew).Google Scholar - 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.Google Scholar - 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.Google Scholar - Tsamir, P. and Tirosh, D.: ‘Consistency representations: The case of actual infinity’,
*Journal for Research in Mathematics Education*,in press.Google Scholar - Vinner, S.: 1990, ‘Inconsistencies: Their causes and function in learning mathematics’,
*Focus on Learning Problems in Mathematics*12 (3 and 4), 85–98.Google Scholar - Wilson, P.: 1990, ‘Inconsistent ideas related to definition and examples’,
*Focus on Learning Problems in Mathematics*12 (3and4), 31–48.Google Scholar