Identification of Crucial Steps and Skills in High-Achievers’ Solving Complex Mathematical Problem within Mathematical Contest

Keywords: Assessment, Competencies, Inquiry Based Learning, Mathematics Education, Open-ended Problems

Abstract

The aspects of inquiry based learning (IBL) are vigorously and frequently in the focus of recent studies. With the use of inquiry in mathematics in the daily school practice, some further questions are arising there: What kind of problems can be useful for an analysis of students’ competencies in the field of IBL and how to assess the performed level of competencies? In this paper, the Mathematics B-day contest assignment is introduced as a mean to assess the students’ performance in mathematical inquiry skills. Some new rubrics with didactical variables were designed as a tool for assessing students’ competencies. The statistical implicative analysis was used to investigate 29 solutions of Mathematics B-day 2017: Arrow clocks. We identified the key subtasks solutions directly related to the level of the IBL competencies performed in the final mathematical investigation. The subtask which required actually high level of algebraic thinking influenced the level of the final mathematical investigation the most. 

Author Biographies

Kristína Bulková, Constantine the Philosopher University in Nitra, Slovakia

Department of Mahtematics

graduate student

Soňa Čeretková, Constantine the Philosopher University in Nitra, Slovakia

Department of Mathematics

Assoc. prof.

References

Anderson, L. W., Krathwohl, D. R. (eds.) (2001) A Taxonomy for Learning, Teaching and Assessing: A revision of Bloom´s Taxonomy of Educational Objectives: Complete edition. New Yourk : Longman.

Arrow clocks (2017) Mathematics B-day 2017 assignment [Online] Available: http://www.fisme.science.uu.nl/wisbdag/opdrachten/assignment2017.pdf [2 Apr 2018].

Blaško, M. (2013) Kvalita v systéme modernej výučby. [The quality in system of modern education] Košice: Technical University in Košice.

Bloom, B. S., Engelhart, M. D., Furst, E. J., Hill, W. H. and Krathwohl, D. R. (1956) Taxonomy of educational objectives. The Classification of Educational Goals. Michigan: Edward Bros.

Brookhart, S. M. (2013) How to Create and Use Rubrics for Formative Assessment and Grading. Alexandria, VA: Association For Supervision & Curriculum Development.

Bruder, R., Prescott, A. (2013) ‘Research evidence on the benefits of IBL’, ZDM Mathematics Education, vol. 45, pp. 811-822. https://doi.org/10.1007/s11858-013-0542-2

Bulková, K., Čeretková, S. (2017a) ‘Rubrics as assessment tool of mathematical open-ended problems’, APLIMAT 2017: Proceedings from 16th Conference on Applied Mathematics, Bratislava, Bratislava: STU, pp. 235-245.

Bulková, K., Čeretková, S. (2017b) ‘Písomný prejav žiakov v riešeniach otvorených matematických problémov’, Študentská vedecká konferencia 2017: zborník recenzovaných príspevkov z konferencie, Nitra: UKF, pp. 458 – 463.

Bulková, K., Čeretková, S. (2017c) ‘Creativity as assessed attribute in mathematical open ended problém solving’, EDULEARN17: Proceeding from 9th International Conference on Education and New Learning Technologies, Barcelona: IATED, pp. 583 – 590. https://doi.org/10.21125/edulearn.2017

Couturier, R. (2008) ‘CHIC: Cohesive Hierarchical Implicative Classification ’, in Gras, R., Suzuki, E., Guillet, F. and Spagnolo, F. (ed.) Statistical Implicative Analysis, vol. 127, pp. 41-53. https://doi.org/10.1007/978-3-540-78983-3_2

Dorier, J. L. (2012) Context analysis for the implementation of IBL: International synthesis report, [Online] Available: http://www.scientix.eu/resources/details?resourceId=4039 [15 Jan 2015]

Dow, G. T. and Mayer, R. E. (2004) ‘Teaching students to solve insight problems: Evidence for domain specificity in creativity training’, Creativity Research Journal, vol. 16, no. 4, pp. 389-398. https://doi.org/10.1080/10400410409534550

Ďuriš, V. and Lengyelfalusy, T. (2019) Notes on Discrete Mathematics, Karlsruhe: Ste-Con,

Engeln, K., Euler, M., and Maass, K. (2013) ‘Inquiry-based learning in mathematics and science: A comparative baseline study of teachers’ beliefs and practices across 12 European countries ’, ZDM, vol. 45, pp. 823-836. https://doi.org/10.1007/s11858-013-0507-5

Gonda, D. and Tirpakova, A. (2018) ‘A new teaching method aimed at eliminating the causes of students'unsuccessful algorithmic problem solving with parameter’. Problems of education in the 21st century, vol. 76, no. 4, pp. 499-519.

Gras, R., Almouloud, S. A., Bailleul, M., Lahrer, A., Polo, M., Ratsimba-Rajohn, H., Totohasina, A. (1996) L'implication statistique: nouvelle méthode exploratoire de données, applications à la didactique. [Statistical implicative analysis: new exploratory method of data, applications to didactics: La Pensée sauvage]. La Pensée sauvage.

Kamp, A (2016) Engineering Education in the Rapidly Changing World. Rethinking the Vision for Higher Engineering Education. 2nd Edition. Delft: TU Delft.

Kenderov, P. S. (2006) ‘Competitions and Mathematical Education’, International Congres of Mathematicians: Proceedings, vol 3. https://doi.org/10.1007/978-3-319-62597-3

Kosko, K. W. and Wilkins, J. L. M. (2010) ‘Mathematical Communication and Its Relation to the Frequency of Manipulative Use’, nternational Electronic Journal of Mathematics Education, vol. 5, no. 2, pp. 79-90

Leikin, R. and Pitta-Pantazi, D. (2013) ‘Creativity and mathematics education: the state of the art’, ZDM Mathematics education, vol. 45. pp. 159-166. https://doi.org/10.1007/s11858-012-0459-1

Lock, R. (1990) ‘Open- ended, problem-solving investigations. What do we mean and how can we use them?’, School Science Review, vol. 71, no. 256, pp. 63-72.

Maaß, K. and Artigue, M. (2013) ‘Implementation of Inquiry-based learning in day-to-day teaching: A synthesis’, ZDM Mathematics Education, vol. 45, no. 6. pp 779-795. https://doi.org/10.1007/s11858-013-0528-0

Maaß, K. and Reitz-Koncebovski, K. (eds.) (2013) Inquiry-based learning in maths and science classes. What it is and how it works – examples – experiences, [Online], Freiburg: Pädagogische Hochschule Freiburg, Available: http://primas-project.eu/wpcontent/uploads/sites/323/2017/11/primas_final_publication.pdf [10 May 2019]

Medová, J., Bulková, K. and Čeretková, S. (2018) ‘Identification of crucial competencies in mathematical inquiry’, ERIE 2018: Proceedings of the 15th International Conference, Prague: Czech University of Life Sciences, pp. 203-212.

Moskal, B. M. (2000) ‘Scoring Rubrics: What, when and why?’, Practical Assessment, Research and Education, vol. 7, no. 3.

Nesher, P., Hershkovitz, S. and Novotna, J. (2003) ‘Situation model, Text Base and what else? Factors affecting Problem Solving ’, Educational Studies in Mathematics, vol. 52, pp. 151-176. https://doi.org/10.1023/A:1024028430965

Nohda, N. (2000) ‘Teaching by Open-Approach Method in Japanese Mathematics Classroom’, Proceedings of the Conference of the International Group for the Psychology of Mathematics Education. Japan. vol. 1, pp. 39-53.

Pantziara, M., Gagatsis, A. and Elia, I. (2009) ‘Using diagrams as tools for the solution of non-routine mathematical problems’, Educational Studies in Mathematics, vol 72, pp. 39-60. https://doi.org/10.1007/s10649-009-9181-5

R Core Team (2018) R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing.

Radford, L. (2008) ‘Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts’, ZDM Mathematics Education, vol. 40, pp. 83-96. https://doi.org/10.1007/s11858-007-0061-0

Russek, B. (1998) ‘Writing to Learn Mathematics’, Writing Across the Curriculum, vol. 9, pp. 36 - 45.

Samková, L. (2018) ‘Uplatnění otevřeného přístupu k matematice v přípravě kudoucích učitelů 1. stupně ZŠ - Empirická studie v kontextu badatelsky orientovaného kurzu [Employing an Open Approach go Mathematics in Pre-Service Primary School Teacher Training – An Empirical Study in the Context of an Inquiry-Based Course]’, Studia paedagogica, vol. 23, no. 3, pp. 49 - 67. https://doi.org/10.5817/SP2018-3-3

Schoenfeld, A. H. (1992) ‘Learning to think mathematically: problem solving, metacognition, and sense making in mathematics’ in Grous, D. (ed.) Handbook for Reasearch on Mathematics Teaching and Learning. New Yourk: Macmillan.

Stacey, K. (1992) ‘Mathematical problem solving in groups: Are two heads better than one?’, Journal of Mathematical behaviour, vol. 11, no. 3, pp. 261-275.

Sternberg, R. J. (ed.) (1998) Handbook of Creativity. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511807916

Utrecht University (2018) Mathematics B-day. [Online] Available: https://www.uu.nl/en/education/mathematics-b-day [1 Mar 2018]

Yackel, E. and Rasmussen, C. (2002) ‘Beliefs and norms in the mathematics classroom’ in Leder, G. C. (ed) Beliefs: A hidden variable in mathematics education?, pp. 313-330, Dordrecht: Kluwer Academic Publisher. https://doi.org/10.1007/0-306-47958-3_18

Zak, P. (2004) Kreativita a její rozvoj. [Creativity and its development]. Brno: Computer Press.

Published
2020-06-30
How to Cite
Bulková, K., Medová, J. and Čeretková, S. (2020) ’Identification of Crucial Steps and Skills in High-Achievers’ Solving Complex Mathematical Problem within Mathematical Contest’, Journal on Efficiency and Responsibility in Education and Science, vol. 13, no. 2, pp. 67-78 https://doi.org/10.7160/eriesj.2020.130202.
Section
Research Paper