Students' competences in proving and argumentation

Proof plays an important role in mathematics and also in mathematics class. Moreover proof is an important topic of the mathematics curriculum and an essential aspect of mathematical competence. However, proof is one of the difficult issues for the students to learn. Recent studies have revealed wide gaps in students' understanding of proofs (e.g. Senk, 1985; Martin & Harel, 1989; Harel & Sowder, 1998).

Samples & research questions

In this research, we collected data of 659 German 7th grade students (8th grade: 528) in 27 classes and 189 Korean 7th grade students (8th grade: 182) in 5 classes with respect to their competencies in argumentation and proofs about geometry and questionnaire on mathematical beliefs. In addition, 22 German teachers and 58 Korean teachers were presented the same questionnaire on mathematical beliefs.

The quantitative research concerns the questions, which competencies students have with respect to proving, and how they perform proofs. Our aims are to analyse the factors (e.g. basic knowledge, methodological knowledge, etc) which could influence the geometrical competence and to identify aspects of geometrical competence of lower secondary students in Korea and Germany. To supplement quantitative research, individual interviews were taken place with the aim of investigating the following questions:

1. What are student's proof practices with 6 geometry items?
2. What is student's methodological knowledge with 4 different arguments for one same question?
3. How do students appreciate the proof in their mathematics learning?

The Quantitative results

As the first result, German students perform significantly better on items asking for basic competence. On the other hand, Korean students perform significantly better on argumentation and proofs items. However, it has also shown students in both countries have difficulties in proving and justifying.



As the second result, with respect to an achievement test on basic competence and competence in argumentation and proofs about geometry three competence levels were identified. This model could be verified empirically. In this table, the numbers mean the average of students responding correctly to all items in the corresponding levels. This model has three competence levels as follows:

Competence Level I: Simply use of rules and elementary reason:
In the first competence level, the application of concepts and rules and elementary reason (i.e., procedural knowledge) are located in the foreground. Partly simple arithmetic operations must be accomplished and elementary basic knowledge should be used. Formal mathematical reasoning is not required at this level.

Competence Level II: proving and justifying (one-step):
Certain aspects of the declarative knowledge are belonging to the second competence level. The students should find a correct reason of the given geometrical problems and make a grade of them appropriately. A flexible application of the concepts and factual knowledge within the range of middle level of geometry is suppositional.

Competence Level III: proving and justifying (several steps):
The third competence level is characterised by original and partially creative problem solving and arguing and justifying. In this level, abilities are expected regarding an adequate notation of arguments and reasons to a proof step as well as meaningful chains of several arguments.

Table 1: Percentage of correct solution for 7th grade students

References:

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
Martin, W.G. & Harel, G. (1989) Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41-51
Senk, S.L (1985) How Well Do Students Write Geometry Proofs. Math Teach. v. 78(6) p. 448-56.

Heinze, A. & Kwak, J. (2001). Mathematical understanding of grade 8 students, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, Utrecht, Netherlands, 2001Vol. 1. p. 398

Heinze, A. and Kwak, J. (2002). Informal Prerequisites for Informal Proofs. Zentralblatt für Didaktik der Mathematik (ZDM) 34 (1), pp. 9 - 16.

Kwak, J. & Reiss, K (2002) Different view of Korean and German teachers beliefs about mathematics, Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, Norwich, UK, 2002 Vol. 1.pp.287

Kwak, J., Reiss, K. & Thomas, J. (2002). Performance of German grade 7 students in proving and reasoning (in Korean). Journal of the Korea Society of Mathematical Education, Series E: Communications of Mathematical Education, 13, 265-274

Kwak, J (2003) Students' Beliefs about Proving, In M. Toepell (ed.), Beiträge zum Mathematikunterricht 2003. Hildesheim: Franzbecker, pp. 377-380

Poster presentations

Heinze, A., Kwak, J., & Köhler, B (2001). Mathematical understanding in secondary school. Poster at the annual conference of the Gesellschaft für Didaktik der Mathematik, Ludwigsburg, March 2001.

Heinze, A. & Kwak, J. (2001). Mathematical understanding of grade 8 students, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, July 2001, Utrecht, Netherlands.

Presentations

Kwak, J (2003) Beliefs von Schülerinnen und Schülern zum Beweisen, presentation at the 27th annual conference of the Gesellschaft für Didaktik der Mathematik, March 2003, Dortmund.

Kwak, J., Reiss, K. & Thomas, J. (2002). Performance of German grade 7 students in proving and reasoning (in Korean). Korea Society of Mathematical Education, February, 2002

Kwak, J. & Reiss, K (2002) Different view of Korean and German teachers beliefs about mathematics, Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, July, 2002 Norwich, UK

Kwak, J. (2001). Beweisen und Begründen in der Sekundarstufe I: Eine komparative Studie - Korea und Deutschland- Vortrag im Doktorandenkolloquium der Gesellschaft für Didaktik der Mathematik, September 2001, Benediktbeuern.

Participation in conferences