Reflection and Metacognition in Mathematics Education – Tools for the Improvement of Teaching Quality

On the basis of a category system that classifies metacognitive activities, the first part of this paper shows to what extent reflection can be understood as one of several metacognitive activities. It is then demonstrated that it proved to be useful to consider different nuances of reflection. Illustrated by examples taken from math classes on grammar school level, the second part of the essay shows what assignments look like that cause pupils to reflect, and how pupils face up to the demands to reflect on different matters in mathematics education. ZDM-Classification: D43, E44, E53, H33 1. Reflection, understood as a metacognitive activity The demand to cover central ideas concerning mathematics education in a challenging and thorough way, and to stimulate pupils’ thinking processes about mathematical matters, has been present in math didactics literature throughout the last two decades. Thereby, the term “reflection” is frequently used. Kilpatrick (1986, p. 8) describes how the connotation of this term, originally used to depict physical and geometric phenomena, changed and now serves as a metaphor for a variety of cognitive processes. Sjuts (1999a, p. 40) specifies “reflection” as “comparing and scrutinising cogitation, thinking, and examination, directed to the matter at hand, which is characterised through differentiation, detachment, and deepening.” One can find other descriptions like “to engage in soulsearching”, “to pass in revue”, as well as “to relate things”. Thus, “reflection” is used to describe a particular kind of high-level cognitive thinking process. The main lecture of Kilpatrick at ICME5 (Kilpatrick 1986) is considered trendsetting in the international discussion about making reflection a central part of mathematics education. Also, in his studies on a reform of mathematics education on late high school and undergraduate level in the USA, Dubinsky (1991a, b) emphasises the usefulness of reflection for an understanding of mathematics. By saying „ ... that we somehow move into another dimension when we reflect on what we have done.“ Kilpatrick (1986, p. 9) indicates that reflection is done from a superordinate point of view and that activities on the object level are viewed from a meta-perspective. Since the 1970s, the term “metacognition” has been established in cognitive psychology for this kind of cognitive activities (cp. Boekaerts 1996). The prefix “meta” suggests that internal processes are central to this concept. Wang, Haertel & Walberg (1993, p. 272f) emphasise the relevance of metacognition for learning achievements in general. In their metaanalysis of empirical studies on the success of school learning, they observe that metacognition is in an excellent rank regarding the influence on learning achievements. Schoenfeld (1992) and De Corte (1995) report on the importance of metacognitive activities to improve mathematical thinking and learning processes. Konrad (2005, p. 23) and Sjuts (1999a, p. 40-44) differentiate the terms “cognition” and “metacognition” from one another. However, Flawell (1979) shows on the one hand that “reflection” is not used consistently in everyday speech, and on the other hand his examples also indicate that both concepts are not clearly differentiable in everyday situations. Sjuts (1999b) characterised different metacognitive activities and documented their relevance to explain pupils’ achievements. By means of transcript passages and pupils’ solutions, the mechanism of action will be demonstrated, with a particular focus on the question, which metacognitive activities bear on which cognitive processes. 2. Category system to classify metacognitive activities 2.1. Formation of a category system From 2001 to 2004 a project supported by the “Deutsche Forschungsgemeinschaft” (German Research Foundation), subtitled “Analysis of Analyses ZDM 2006 Vol. 38 (4) 351 educational situations practicing reflection and metacognition in secondary school mathematics education” was conducted at the “Institut für kognitive Mathematik” (IKM) (Institute for Cognitive Mathematics) of the University of Osnabrück. The title already indicates the connection between metacognition and reflection. In the present project, the construct “metacognition” was decomposed under mathematics didactical aspects. Single activities were identified and used to build an extensive category system for metacognitive activities that were observed in teacher-pupil interactions in mathematics education. This system was then applied to some transcript passages. The analysis emanated from algebra lessons. In the process of the category formation, however, it became apparent that the observed and identified activities are also describable in a more general way. Consequently, the category system can more generally be applied so that it can now subsume metamathematical activities as well (cp. Cohors-Fresenborg & Kaune 2005a). This at first astonishing connection between metamathematics and metacognition can therefore be explained. In the context of an education according to the “Osnabrücker Curriculum” (Curriculum of Osnabrueck, Cohors-Fresenborg, 2001), an incentive could be to transform cognition, through metamathematical approaches, into a metacognition on mathematical prodedures. The importance of metacognition for the comprehension of mathematics is revealed by another observation: In mathematical science, the thinking about the nature of mathematical conceptions and the typical procedures that are used when practising mathematics (calculating, proving, abstracting, reifying) lead to the classical main components of metamathematics (mathe1 Supported by the German Research Foundation under reference Co 96/5-1. In German-speaking countries, this project is the first and so far only project that intensely investigates the role of metacognition in mathematics education. All the examples mentioned in the present study were analysed within this project. 2 Examples are the scene of the lesson on the barrel rule at the end of this paper as well as the episodes “Proving is nothing else but calculating” (CohorsFresenborg & Kaune 2005b) and “Do we need a 4 binomial formula?” in Kaune (2001). matical logic): computability theory, formal logic, axiomatic set theory. 2.2. Application of the category system to the analysis of scenes of lessons A scene of a lesson on “equation-solving” is chosen to illustrate to what extend the processes “reflection” and “metacognition” interact. It also shows which criteria are suitable to differentiate reflection from other metacognitive activities. Transcripts and pupils’ solutions that are analysed in the following are part of the mathematics educational databank MUMAS. Initial setting of the scene: The pupil Michaela stands in front of the blackboard which contains the equation ( ) ( ) x x x + = − ⋅ − − ⋅ 1 2 11 3 2 5 6 , 0 that is supposed to be solved. Michaela moderates the procedure; she calls up pupils, who are allowed to dictate one term rewriting only and she writes as they dictate. It is part of the classroom-culture that the pupil standing at the blackboard is not supposed to control the inputs given by the class concerning their completeness and correctness. Also, he or she is not allowed to change the contents of what is dictated.