Monitoring and Guiding Pupils' Problem Solving
DOI:
https://doi.org/10.15291/magistra.1493Ključne riječi:
problem solving, generalisation, guiding, primary school, secondary school, prospective teachers, teacher’s roleSažetak
This paper presents a discussion of the problem-solving approaches of primary and secondary school pupils in relation to the following issues: developing strategies, communicating, and receiving guidance. Guiding is the role of the teacher who should be sensitive enough to support pupils’ thinking, when necessary, but not direct it. A group of pupils (35 pupils between 10 and 19 years old) were given a geometrical problem that required them to define the number of parts created when a single plane was divided by straight lines. Each pupil tackled the problem individually, while prospective teachers from the Faculty of Education observed and guided them. After analysing the prospective teachers’ research reports on guiding pupils through the problem we came to the following conclusions: all the pupils needed guiding in order to make progress in problem solving towards general rule, most of the pupils need to learn about heuristics more systematically, prospective teachers got better inside view on thinking process for problem solving of different age groups of pupils. From the success at problem solving point of view we observed the following: until presented with a problem that required a geometrical approach, the differences among the age groups in terms of successful problem solving were not that noteworthy, the difference among age groups was observed in examples of more complex problem solving where a shift towards an arithmetical approach was needed.Reference
Ayalon, M., & Even, R. (2008) Deductive reasoning: in the eye of the beholder. Educational Studies in Mathematics, 69, 235–247.
Bennett, N. (1976). Teaching styles and pupil progress. London: Open Books.
Brandes, D., & Ginnis, P. (2001). A guide to student-centred learning. Cheltenham: Nelson Thornes.
Cañadas, M. C., & Castro, E. (2007). A proposal of categorisation for analysing inductive reasoning. PNA, 1(2), 67-78.
Ciosek, M. (2012). Generalization in the process of defining a concept and exploring it by students. In B. Maj-Tatsis & K. Tatsis (Eds.) Generalization in mathematics at all educational levels (pp. 38-56). Rzeszow: University of Rzeszow.
Dorfler,W. (1991). Forms and means of of generalization in mathematics, in A. Bishop (ed.), Mathematical knowledge: its growth through teaching, Mahwah, NJ Erlbaum,pp. 63-85
Haapasalo, L. (2003). The conflict between conceptual and procedural knowledge: should we need to understand in order to be able to do, or vice versa? In L. Haapasalo & K. Sormunen (Eds.), Towards meaningful mathematics and science education. Proceedings on the XIX Symposium of the Finnish Mathematics and Science Education Research Association (pp. 1-20). University of Joensuu, Finland: Bulletins of the Faculty of Education (no. 86).
Hiebert, J. (1986). Conceptual and procedural knowledge: the case of mathematics. Hillsdale: Erlbaum.
Gelman, R. & Meck, E. (1986). The notion of principle: The case of counting. In J. Hiebert (ed.) Conceptual and procedural knowledge: the case of mathematics (pp 29-57). Hillsdale: Erlbaum.
Gray, E. & Tall, D. (2001). Relationships between embodied objects and symbolic procepts: an explanatory theory of success and failure in mathematics. In M. van den Heuvel-Panhuizen (Ed.) Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education. Vol 3. (pp. 65-72) Utrecht University: Freudenthal Institute.
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.
Kilpatrick, J. (1985). A retrospective account of the past 25 years of research on teaching mathematical problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 1-5). Hillsdale, NJ: Erlbaum.
Kruetskii, V.A. (1976). The psychology of mathematical abilities in school children, University of Chicago press, Chicago.
Kuzle, A., & Conradi, C. (2016). Collaborative action research on the learning of heuristics: The case of informative figure, table and solution graph as heuristic auxiliary tools. In A. Kuzle, B. Rott, & T. Hodnik Čadež (Eds.), Problem solving in the mathematics classroom: perspectives and practices from different countries (pp. 211-225). Münster: WTM.
Manfreda Kolar, V., Mastnak, A., & Hodnik Čadež, T. (2012). Primary teacher students' competences in inductive reasoning. In T. Bergqvist (Ed.), Problem solving in mathematics education: Proceedings from the 13th ProMath conference: Learning problem solving and learning through problem solving (pp. 54-68). Umeå: Umeå Univeristy, Faculty of sciences and technology.
Maj-Tastis, B. & Tastis, K. (Eds.) (2012). Generalization in mathematics at all educational levels. Rzeszow: University of Rzeszow.
Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 173–203). Rotterdam: Sense.
Mason, J., Burton, L., & Stacey, K. (1982/2010). Thinking Mathematically. Dorchester: Pearson Education Limited. Second Edition.
Pólya, G. (1945). How to Solve It. Princeton, NJ: University Press.
Pólya, G. (1967). La découverte des mathématiques. París: DUNOD.
Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM Mathematics Education, 40, 83-96.
Radford, L. (2001). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42, 237–268.
Reid, D. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 33(1), 5-29.
Rivera, F. D., Rossi Becker, J. (2007) Abduction-induction (generalisation) process of elementary majors on figural patterns in algebra. Journal of Mathematical Behaviour 26, 140-155.
Rossi Becker, J., & Rivera, F. (2006). Sixth Graders’ figural and numerical strategies for generalizing patterns in algebra. In S. Alatore, J.L. Cortina, M. Saiz, & A. Mendez (Eds.), Proceedings of the 28th annual meeting of the North American chapter of the international group for the Psychology of Mathematics Education ( Vol. 2, pp. 95-101). Merida: Universisad Pedagogica nacional.
Rott, B. (2013). Process Regulation in the Problem-Solving Processes of Fifth Graders c e p s Journal | Vol.3 | No4 | Year 2013, 25 – 39
Schoenfeld, A. H. (1985). Mathematical Problem Solving. Orlando, Florida: Academic Press, Inc.
Swafford, J. O., Langrall, C. W. (2000). Grade 6 students preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education 31(1), 89-112.
Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics 14, 44 -55.
Skemp, R. (1979). Intelligence, learning and action. Chichester: Wiley.
Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics, with particular reference to limit and continuity. Educational Studies in Mathematics 12, 151-169.
Vinner, S. (2012). Generalizations in everyday thought processes and in mathematical contexts. In B. Maj-Tatsis and K. Tatsis (Eds.), Generalization in mathematics at all educational levels (pp.38-56). University of Rzeszow, Rzeszow.
Vygotski, L. N. (1986). Thought and language. Cambridge; London: MIT press
Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 227–236). Reston, VA: National Council of Teachers of Mathematics.
