# SECONDARY SCHOOL STUDENTS’ UNDERSTANDING OF MATHEMATICAL INDUCTION: STRUCTURAL CHARACTERISTICS AND THE PROCESS OF PROOF CONSTRUCTION

- 567 Downloads
- 6 Citations

## Abstract

In this study, we investigate the meaning students attribute to the structure of mathematical induction (MI) and the process of proof construction using mathematical induction in the context of a geometric recursion problem. Two hundred and thirteen 17-year-old students of an upper secondary school in Greece participated in the study. Students’ responses in 3 written tasks and the interviews with 18 of them are analyzed. Though MI is treated operationally in school, the students, when challenged, started to recognize the structural characteristics of MI. In the case of proof construction, we identified 2 types of transition from argumentation to proof, interwoven in the structure of the geometrical pattern. In the first type, MI was applied to the algebraic statement that derived from the direct translation of the geometrical situation. In the second type, MI was embedded functionally in the geometrical structure of the pattern.

## Key words

geometrical pattern mathematical induction proof structure of natural numbers## Preview

Unable to display preview. Download preview PDF.

## References

- Apostol, T. M. (1976).
*Introduction to analytic number theory*. New York: Springer.Google Scholar - Avital, S. & Hansen, R. T. (1976). Mathematical induction in the classroom.
*Educational Studies in Mathematics, 7,*399–411.Google Scholar - Avital, S. & Libeskind, S. (1978). Mathematical induction in the classroom: Didactical and mathematical issues.
*Educational Studies in Mathematics, 9*, 429–438.CrossRefGoogle Scholar - Bliss, J., Monk, M. & Ogborn, J. (1983).
*Qualitative data analysis for educational research*. London: Croom Helm.Google Scholar - Boero, P., Garuti, R. & Mariotti, M. A. (1996). Some dynamic mental processes underlying producing and proving conjectures. In A. Gutierrez & L. Puig (Eds.),
*Proceedings of the 2*0th*Conference of International Group for the Psychology of Mathematics Education,*vol. 2 (pp. 121–128). Valencia, Spain.Google Scholar - Burgess, R. G. (1985).
*Issues in educational research: Qualitative methods*. London: Falmer Press.Google Scholar - Cusi, A. & Malara, N.A. (2008). Improving awareness about the meaning of the principle of mathematical induction. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano & A. Sepulveda (Eds.),
*Proceedings of PME32 and PME-NAXXX,*vol. 2 (pp. 393–398). Morelia, Mexico.Google Scholar - Dubinsky, E. (1986). Teaching mathematical induction II.
*The Journal of Mathematical Behavior, 8*, 285–304.Google Scholar - Dubinsky, E. & Lewin, P. (1986). Reflective abstraction and mathematics education: The genetic decomposition of induction and compactness.
*The Journal of Mathematical Behavior, 5*, 55–92.Google Scholar - Ernest, P. (1984). Mathematical induction: A pedagogical discussion.
*Educational Studies in Mathematics, 15*, 173–189.CrossRefGoogle Scholar - Fischbein, E. & Engel, H. (1989). Psychological difficulties in understanding the principle of mathematical induction. In G. Vergnaud, J. Rogalski & M. Artigue (Eds.),
*Proceedings of the 13th Annual Conference of the International Group for the Psychology of Mathematics Education, vol. I*(pp. 276–282). Paris, France.Google Scholar - Hanna, G. & Barbeau, E. (2008). Proofs as bearers of mathematical knowledge.
*ZDM Mathematics Education, 40*, 345–353.CrossRefGoogle Scholar - Harel, G. (2002). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campell & R. Zaskis (Eds.),
*Learning and teaching number theory: Research in cognition and instruction*(pp. 185–212). New Jersey: Ablex Publishing.Google Scholar - Harel, G. & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput & E. Dubinsky (Eds.),
*Research in collegiate mathematics education, III*(pp. 234–283). Providence: American Mathematical Society.Google Scholar - Harel, G. & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester (Ed.),
*The second handbook of research on mathematics teaching and learning*(pp. 805–842). Washington: NCTM.Google Scholar - Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra.
*Journal for Research in Mathematics Education, 31*, 396–428.CrossRefGoogle Scholar - Lin, F.L, Hsieh, F. J., Hanna, G. & de Villiers, M. (2009).
*Proceedings of the ICMI Study 19 Conference: Proof and Proving in Mathematics Education*, vol. 1, vol.2. National Taiwan Normal University, Taipei, Taiwan.Google Scholar - Mayring, P. (2000). Qualitative content analysis [28 paragraphs].
*Forum Qualitative Sozialforschung/Forum: Qualitative Social Research*,*1*(2), Art. 20. http://nbnresolving.de/urn:nbn:de:0114-fqs0002204. - Movshovitz-Hadar, N. (1993). The false coin problem, mathematical induction and knowledge fragility.
*The Journal of Mathematical Behavior, 12*, 253–268.Google Scholar - Nardi, E. & Iannone, P. (2003). The rough journey towards a consistent mathematical proof: The
*P*(*n*)→*P*(*n*+ 1) step in mathematical induction. In A. Gagatsis & S. Papastavridis (Eds.),*Proceedings of the*3rd*Mediterranean Conference on Mathematical Education*(pp. 621–628). Athens: Hellenic and Cyprus Mathematical Societies.Google Scholar - National Council of Teachers of Mathematics (2000).
*Principles and standards for school mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed?
*Educational Studies in Mathematics, 66*, 23–41.CrossRefGoogle Scholar - Pinto M. M. F. & Tall D. (1999). Student constructions of formal theory: Giving and extracting meaning. In Zaslavsky (Ed.),
*Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education, vol. 3*(pp. 281–288). Israel Institute of Technology, Haifa.Google Scholar - Radford, L. (2003). Gestures, speech, and the sprouting of sings: A semiotic-cultural approach to students’ types of generalization.
*Mathematical Thinking and Learning, 5*, 37–70.CrossRefGoogle Scholar - Ron, G. & Dreyfus, T. (2004). The use models in teaching proof by mathematical induction. In M. J. Hoines & A. B. Fuglestad (Eds.),
*Proceedings of the 2*8th*Conference of the International Group for the Psychology of Mathematics Education, vol. 4*(pp. 113–120). Bergen, Norway.Google Scholar - Sfard, A. (1991). On the dual nature of the mathematical conceptions: Reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics, 22*, 1–36.CrossRefGoogle Scholar - Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reifications—The case of function. In G. Harel & E. Dubinsky (Eds.),
*The concept of function: Aspects of epistemology and pedagogy*(pp. 59–84). Washington, DC: MAA Notes 25.Google Scholar - Stylianides, G. J. & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof.
*Journal for Research in Mathematics Education, 40*, 314–352.Google Scholar - Stylianides, G. J., Stylianides, A. J. & Philippou, G. N. (2007). Preservice teachers’ knowledge of proof by mathematical induction.
*Journal of Mathematics Teacher Education, 10*, 145–166.CrossRefGoogle Scholar - Tall, D., Thomas, M., Davis, G., Gray, E. & Simpson, A. (2000). What is the object of the encapsulation of a process?
*The Journal of Mathematical Behavior, 18*, 223–241.CrossRefGoogle Scholar - Vamvakoussi, X. & Vosniadou, S. (2007). How many numbers are there in a rational numbers interval? Constraints, synthetic models and the effect of the number line. In S. Vosniadou, A. Baltas & X. Vambakoussi (Eds.),
*Reframing the conceptual change approach in learning and instruction*(pp. 265–282). Oxford: Elsevier.Google Scholar - Weber, K. (2005). Problem-solving, proving and learning: The relationship between problem-solving and learning opportunities in the activity of proof construction.
*The Journal of Mathematical Behavior, 24*, 351–360.CrossRefGoogle Scholar - Wistedt, I. & Brattstrom, G. (2005). Understanding mathematical induction in a cooperative setting. In A. Chronaki & I. M. Christiansen (Eds.),
*Challenging perspectives on mathematics classroom communication*(pp. 173–203). Greenwich: Information Age Publishing.Google Scholar