Session Information
24 SES 07 A, Issues in Mathematics Teacher Education I
Paper Session
Contribution
Our study intends to acquire information on the ways in which prospective class and subject teachers who are pursuing university studies of mathematics grasp and make sense of the concept of a plane angle. In fact, an angle has been in the course of centuries a concept which even the mathematical science community has found hard to define and hard to approach from one single point of view (Matos 1990, Keiser 2004). The three following modes of definition have been the ones most frequently applied at different times when seeking a definition of an angle: an angle defined (1) as a rotation by which one of two intersecting straight lines is made to merge into the other, (2) as a region defined by two half lines starting from the same point or (3) as the common region defined by two intersecting half planes (Mitchelmore & White 2000, 209). According to these definition alternatives, an angle can be understood to be either a measurable quantity, or a geometric construction, or a plane region. Keiser’s studies (Keiser 2004 and Keiser et al. 2003), in particular, show that these different interpretations of the concept of an angle which have been applied throughout the history of mathematics reflect rather well the differences in the interpretations of individuals as regards this concept which have been discovered in didactic research.
It is possible to examine the way in which an individual grasps a mathematical concept either by observing how the individual uses the concept spontaneously in speech and action without actually being aware of the observation, or by planning a test in which the informant is asked to do something that reveals as much as possible of the way in which this individual interprets the meaning of the concept. In our earlier case study (Joutsenlahti & Silfverberg 2007) we used the former method of collecting research data from schoolchildren, whereas in the present study the latter method was used for the purpose of examining teacher students’ interpretations of an angle.
An analysis of the definitions given by the informants can be made in several ways. For instance, we can check (1) how correctly a definition defines the concept in comparison with the normative interpretation; (2) how adequately the form of a given definition meets the criteria set for a mathematical definition (Hershkowitz 1990; Leikin & Winicki-Landman 2000a and 2000b; de Villiers 1995); (3) how well a definition given by an informant corresponds with the concept form which this informant seems to have on the basis of the situations in which the concept was actually applied (Tall & Vinner 1981; Vinner & Dreyfus 1989; Vinner 1991); (4) what kind of linguistic form the informant used in providing a definition (Barnbrook 2002), etc. Our approach is the one which was applied by Barnbrook (2002) in his study of the structure of dictionary-type definitions.
Method
Expected Outcomes
References
Barnbrook, G. 2002. Defining Language. A local grammar of definition sentences. Philadelphia, PA, USA: John Benjamin’s Publishing Company. Hershkowitz, R., Bruckheimer, M. & Vinner, S. 1987. Activities with teachers based on cognitive research. In M.M. Lindquist & A.P. Schulte (eds.) Learning and teaching geometry, K-12: NCTM 1987 Yearbook. Reston VA: NCTM, 222-235. Houdement, C. & Kuzniak, A. 2003. Elementary geometry split into different geometrical paradigms. In M.A. Mariotti (eds.) Proceedings of CERME 3. Belaria, Italy. http://www.dm.unipi.it/~didattica/CERME3/proceedings/ Joutsenlahti, J. & Silfverberg, H. 2007. Case-analyysi kulman käsitteen tulkinnasta. [In Finnish] In J. Lavonen (ed.) Tutkimusperustainen opettajankoulutus ja kestävä kehitys. Ainedidaktinen symposium 3.2.2006.Helsingin yliopisto. Tutkimuksia 285, s. 341-348. Keiser, J.M. 2004. Struggles with developing the concept of angle: Comparing sixth-grade students’ discourse to the history of angle concept. Mathematical Thinking and Learning 6 (3), 285-306. Keiser, J.M., Klee, A. & Fitch, K. 2003. An assessment of students' understanding of Angle. Mathematics Teaching in the Middle School 9 (2) 116. Leikin, R. & Winicki-Landman, G. 2000. On equivalent and non-equivalent definitions: part 1. For the Learning of Mathematics 20 (1), 17-21. Leikin, R. & Winicki-Landman, G. 2000. On equivalent and non-equivalent definitions: part 2. For the Learning of Mathematics 20 (2), 24-29. Matos, J. 1990. The historical development of the concept of angle. The Mathematics Educator 1 (1), 4-11. Mitchelmore, M.C. & White, P. 2000. Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics 41, 209–238. Tall, D. & Vinner S. 1981. Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics 12 (2), 151-169. Vinner, S. 1991. The role of definitions in the teaching and learning of mathematics. In D. Tall (ed.) Advanced mathematical thinking. Mathematics Education Library. Dordrecht, Netherlands: Kluwer Academic Publishers, 65-81.
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