### Change in the Level of Justification in Problem Solving Over Time

#### Abstract

*Students’ actions in a mathematical context contain some sort of proving in the forms of justification, explanation, verification, etc. Each form has levels in terms of the quality that adds to mathematical understanding of students. In this study, 58 teacher candidates’ problem solving processes were analyzed across time in terms of the level of justification. The key findings from this study were as the use of external justifications decreased over time, that of internal increased especially for the first five weeks and the use of schematic justifications was fluctuated. The findings suggest that through teacher feedback and structured writing with prompted questions, students can develop an awareness of rigorous mathematical justifications over time.*

#### Keywords

#### Full Text:

PDF#### References

Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D.Pimm (Ed.), Mathematics, Teachers and Children (pp. 216-235). London: Hodder and Stoughton.

Ellis, A. B. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38 (3), 194-229.

Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany, NY: State of University of New York Press.

Francisco, J. M. & Maher, C. A. (2005). Conditions for promoting reasoning in problem solving: Insights from a longitudinal study. Journal of Mathematical Behavior, 24, 361-372.

Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23.

Hanna, G. & Barbeau, E. (2010). Proofs as bearers of mathematical knowledge. In G. Hanna, H. N. Jahnke and H. Pulte (Eds.), Explanation and Proof in Mathematics: Philosophical and Educational Perspectives (pp. 85-100). New York, NY: Springer.

Harel, G. & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics (pp. 805-842). Reston, VA: National Council of Teachers of Mathematics.

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, RI: American Mathematical Society.

Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31 (4), 396-428.

Kenyon, R. W. (1989). Writing is problem solving. In P. Connolly & T. Vilardi (Eds.), Writing to learn mathematics and science (pp. 73-87). New York and London: Teachers College.

Maher, C. A. & Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27 (2), 194-214.

Mariotti, M. A. (2000). Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics, 44, 25-53.

Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. Addison-Wesley, London.

Mayer R.E (1982). The psychology of mathematical problem solving. In F.K. Lester and J. Garofalo (Eds), Mathematical problem solving: Issues in research (pp. 1-13).Philadelphia: The Franklin Institute Pres.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

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.

Reid, D. A. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 33 (1), 5-29.

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.

Sandefur, J., Mason, J., Stylianides, G. J., & Watson, A. (2013). Generating and using examples in the proving process. Educational Studies in Mathematics, 83 (3), 323-340. doi: 10.1007/s10649-012-9459-x.

Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. Journal of Mathematical Behavior 31, 447-462.

Stylianides, A. J. (2006). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65, 1-20.

Stylianides, A. J. (2009). Towards a more comprehensive “knowledge package” for teaching proof. In J. H. Meyer & A. van Biljon (Eds.), Proceedings of the 15th Annual Congress of the Association for Mathematics Education of South Africa (AMESA) (Vol. 1, pp. 242-263). University of the Free State, Bloemfontein, South Africa.

Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks, Mathematical Thinking and Learning, 11 (4), 258-288.

Stylianides, G. J., Stylianides, A. J., & Shilling-Traina, L. N. (2013). Prospective teachers’ challenges in teaching reasoning-and-proving. International Journal of Science and Mathematics Education, 11, 1463-1490.

Varghese, T. (2011). Considerations concerning Balacheff’s 1988 taxonomy of mathematical proofs. Eurasia Journal of Mathematics, Science & Technology Education, 7 (3), 181-192.

Weber, K. (2005). Problem-solving, proving, and learning: The relationship between problem-solving processes and learning opportunities in the activity of proof construction. Journal of Mathematical Behavior 24 (3-4), 351–360.

Weber, K. (2010). Mathematics majors' perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12 (4), 306-336.

Yopp, D., & Ely, R. (2016). When does an argument use a generic example? Educational Studies in Mathematics, 91 (1), 37-53. doi: 10.1007/s10649-015-9633-z

Zaslavsky, O. & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36 (4), 317-346.

DOI: https://doi.org/10.24106/kefdergi.3050

### Refbacks

- There are currently no refbacks.

indexed by TR-DİZİN, DOAJ, SOBİAD

Kastamonu Eğitim'de yayınlanan tüm içerik ve makaleler "Creative Commons Alıntı 4.0 Uluslararası Lisansı" ile lisanslanmıştır.