Session Information
24 SES 06 B, Teaching and Learning of Mathematics
Paper Session
Contribution
The actual question in upper-secondary schools and in universities is: what kind of mathematical proficiency and mathematical thinking new students have? If we recognize features of students’ mathematical thinking we could develop the curricula, learning materials and teaching methods more effective.
The concept ‘mathematical thinking’ is very often used in mathematics education literature but very seldom it is defined. In fact there are a lot of various descriptions about mathematical thinking in international literature of mathematics didactic (e.g. Gray & Tall 1994, Pirie & Kieren 1994, Sfard 1991, Sierpinska & Nnadozie 2001 ) . Sternberg (1996) has studied approaches to the concept ‘mathematical thinking’ and he has recognized at least five different points of view to the concept: anthropological, information process, mathematical, pedagogical and psychometric. It is possible to describe the kernel of these approaches also by such concepts as ‘culture’, ‘ knowledge’, ‘belief’, ‘problem solving’ and ‘ability’ (Joutsenlahti 2005, 65). We have chosen the approach of information process. This leads us to use the concept ‘knowledge’, which can be studied for example from point of view of psychological or philosophical epistemology. In the psychological epistemology ‘knowledge’ can be divided into conceptual, procedural and perhaps strategic knowledge (Hiebert & Lefevre 1986, Joutsenlahti 2005). The philosophical epistemology includes the concept ‘belief’, which is important with the concept ‘metacognition’ if we describe students thinking and learning processes. The role of students’ beliefs as part of one’s metacognition is crucial in the monitoring of thinking process. If student has understood knowledge then the knowledge is conceptual and it has a lot of connections to others in the student’s mind (cf. connectionism (Bereiter 2002)). Skills are related to procedural knowledge and problem solving requires the use of strategic knowledge.
Mathematical proficiency is defined by five components (Kilpatrick, Swafford & Findell 2001, 116): 1) conceptual understanding: comprehension of mathematical concepts, operations, and relations, 2) procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately, 3) strategic competence: ability to formulate, represent, and solve mathematical problems, 4) adaptive reasoning: capacity for logical thought, reflection, explanation, and justification and 5) productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile , coupled with a belief in diligence and one’s own efficacy. We have changed the last component ‘productive disposition’ to ‘view of mathematics’ (Pehkonen 1998), which is a broader concept and is better suited to the study.
The empirical part of the study examines upper-secondary school students' mathematical thinking in the longer course (advanced course) in mathematics from three perspectives: the perspective of society, the teacher's and the student's perspective. The main problem in the study is to describe features of the student's mathematical thinking. The sub problems consider what kinds of differences exist in the mathematical proficiency and in the view of mathematics between genders and between students who chose a compulsory test or an optional test in the matriculation examination.
Method
Expected Outcomes
References
Bereiter, C. 2002. Education and mind in the knowledge age. Mahwah (NJ): Erlbaum. Gray, E. & Tall, D. 1994. Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. Journal for Research in Mathematics Education 26 (2), 115–141. Hiebert, J. & Lefevre, P. 1986. Conceptual and procedural knowledge in mathematics: an introductory analysis. In J. Hiebert (Eds.) Conceptual and procedural knowledge: the case of mathematics. Hillsdale (NJ): Lawrence Erlbaum, 1–27. Joutsenlahti, J. 2005. Lukiolaisen tehtäväorientoituneen matemaattisen ajattelun piirteitä: 1990–luvun pitkän matematiikan opiskelijoiden matemaattisen osaamisen ja uskomusten ilmentämänä. [Characteristics of task-oriented mathematical thinking among students in upper-secondary school]. Publication of university of Tampere. Acta Universitatis Tamperensis 1061. Kilpatrick, J., Swafford, J. & Findell, B. (Eds.) 2001. Adding it up. Washington DC: National Academy Press. Pehkonen, E. 1998. On the concept ''mathematical belief''. In E. Pehkonen & G. Törner (Eds.) The state-of-art in mathematics-related belief research. Results of the MAVI activities. University of Helsinki. Department of Teacher Education. Research report 195, 37–72. Pirie, S. & Kieren, T. 1994. Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics 26 (2), 191–228. Sfard, A. 1991. On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics 22 (1), 1–36. Sierpinska, A. & Nnadozie, A. 2001. Methodological problems in analyzing data from a small scale study on theoretical thinking in high achieving linear algebra students. In van den Heuvel-Panhuizen (Eds.) Proceedings of the 25. conference of the international group for the psychology of mathematics education, Utrecht (The Netherlands). Volume 4, 177–184. Sternberg, R. 1996. What is mathematical thinking? In R. Sternberg & T. Ben-Zeev (Eds.) The nature of mathematical thinking. Mahwah (NJ): Erlbaum, 303–318.
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