Finnish mathematics teacher students’ informal and formal arguing skills in the case of derivative

Antti Viholainen

doi:10.7146/nomad.v13i2.148119

Authors

Antti Viholainen

DOI:

https://doi.org/10.7146/nomad.v13i2.148119

Abstract

In this study, formal and informal reasoning skills of 146 Finnish subject-teacher students in mathematics are investigated. The students participated in a test in which they were asked to argue two claims concerning derivative both informally and formally. The results show that the success in the formal tasks and the success in the informal tasks were dependent. However, there were several students who did well in the formal tasks despite succeeding poorly in the informal tasks. The success both in the formal tasks and in the informal tasks was dependent also on the amount of passed studies in mathematics and on the success in these studies. Moreover, these factors could have a stronger effect on the formal than on the informal reasoning skills.

References

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52 , 215-241. https://doi.org/10.1023/A:1024312321077

Dreyfus, T. (1994). Imagery and reasoning in mathematics and mathematics education. In D.F. Robitaille, D.H. Wheeler & C. Kieran (Eds.), Selected Lectures from the 7th International Congress on Mathematical Education (pp. 107-122). Quebec: Les Presses d l'Universit ́e Laval.

Dreyfus, T. (2000). Some views on proofs by teachers and mathematicians. In A. Gagatsis (Ed.), Proceedings of the Second Mediterranean Conference on Mathematics Education, Vol. I, Nikosia, Cyprus (pp.11-25). The University of Cyprus.

Goldin, G. A. (2003). Developing complex understandings: on the relation of mathematics education research to mathematics. Educational Studies in Mathematics, 54 , 171-202. https://doi.org/10.1023/B:EDUC.0000006180.20493.3c

Hanna, G. (1991). Mathematical proof. In Tall, D. (Ed.), Advanced mathematical thinking (pp. 54-61). Dordrecht: Kluwer. https://doi.org/10.1007/0-306-47203-1_4

Harel, G. & Sowder, L. (1998). Students' proof schemes: results from exploratory studies. In A. H. Schoenfeld, J. Kaput & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234-283). Providence, RI: American Mathematical Society. https://doi.org/10.1090/cbmath/007/07

Hähkiöniemi, M. (2006a). Perceiving the derivative: the case of Susanna. Nordic Studies in Mathematics Education, 11 (1), 51-73.

Hähkiöniemi, M. (2006b). The role of representations in learning the derivative (Report 104). Department of mathematics and statistics, University of Jyväskylä.

Parameswaran, R. (2007). On understanding the notion of limits and in-finitesimal quantities. International journal of Science and mathematics Education, 5 , 193-216. https://doi.org/10.1007/s10763-006-9050-y

Raman, M. (2002). Proof and justification in collegiate calculus (Doctoral dissertation). Berkeley: University of California.

Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52, 319-325. https://doi.org/10.1023/A:1024360204239

Rodd, M.M. (2000). On mathematical warrants: proof does not always warrant, and a warrant may be other than a proof. Mathematical Thinking and Learning, 2 (3), 221-244. https://doi.org/10.1207/S15327833MTL0203_4

Shulman, L. S. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15 (2), 4-14. https://doi.org/10.2307/1175860

Shulman, L. S. (1987). Knowledge and teaching: foundations of the new reform. Harvard Educational Review, 57 (1), 1-22. https://doi.org/10.17763/haer.57.1.j463w79r56455411

Stylianou, D.A. (2002). On the interaction of visualization and analysis: the negotiation of a visual representation in expert problem solving. Journal of Mathematical Behavior, 21 , 303-317. https://doi.org/10.1016/S0732-3123(02)00131-1

Toulmin, S. E. (2003). The uses of argument. New York: Cambridge University Press. https://doi.org/10.1017/CBO9780511840005

Weber, K. & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56 , 209-234. https://doi.org/10.1023/B:EDUC.0000040410.57253.a1