Cognitive models for the concept of angle

ion was found to be not a departure from "real world" characterizations of angles into more intangible realms of knowledge, but the development of a categorization that made use of a multitude of cognitive models, a cluster of models, that competed with each other for primacy at solvins a siven task. The idea of the trading floor of a "stock exchange" (Kilpatrick, 1985, p. 13) of models, which is itself a metaphor, seems an apt description of the kind of reasoning used by some of the more advanced students. This characterization of the category of angles also refutes the idea that its structure could be completely accounted for by describing the relevant social environment and its influence upon students' concept of angle. Social interactions in the classrooms guided by the teachers played a fundamental part in shaping the students' categorization of ansles. But social influences alone cannot explain why an angle was thought to be like a point or a corner, for example. They cannot explain the determinant role of image schemas structuring the category of anzles. This study barely scratched the surface of the role played by social interactions in shaping categories, and much research needs to be done in this area. This role is of special importance for research on improving teaching methodologies. Implications for the van Hiele Tlzeon The van Hiele theory was found to be useful in describing complexity in geometric thinking, yielding valuable insights in interpreting students' verbalizations and actions. Although the van Hiele theory was not the focus of this study, several changes in the theory were needed to account for the results. Several changes are discussed in chapter 4 and were incorporated into the interpretation of the theory used in this study. Two types of changes were addressed there: changes in the implicit cognitive theory (that there are external spontaneous structures ready to be perceived by the learner and that learning proceeds uniformly) and changes in the characterization of the levels (that movement through the levels is continuous and that a full understanding of the classification of quadrilaterals requires Level 4). This study shows that further modifications should be made, all of them consequences of assuming that cognif on is embodied. The description of the first two levels can be enriched by the findings of this study. Level 1 is characterized by basic-level categorizations based on rich images and motor actions. as van Hiele anticipated. But, contrary to van Hiele's ideas, there are no spontaneous structures of the material on which our cognition is based. Instead, image schemas grounded in our material and social life yield idiosyncratic metaphoric models that are construed by each individual. Level 2, as van Hiele proposed, is characterized by the appearance of propositional models built upon those of the previous level. These new models imply. however, the disappearance of neither the basic-level of categorization nor the metaphoric projections. Neither do they imply that the language of previous levels is not understood at higher levels. Contrary to van Hiele theory, previous levels are not lost; they remain an important asset in a student's cognitive repertoire. They play a crucial role in imagination. understood as the formation of similarity-creating metaphors. The relation between primary and complex metaphors and the van Hiele levels was not clear in the study. Although students with primary metaphors were at Level 1 and students with complex metaphors were at Level 2, there was no clear evidence for a sharp distinction. An alternative possibility would be that complex metaphors emerge as students progress through Level 1. This study did not address thinking above Level 2. Therefore, there is no evidence as to the kind of models that characterize Level 3 and above. Since from van Hiele's perspective Level 3 is characterized by a local logical ordering of properties, it is only possible to conjecture that a corzdilit metaphor (Johnson, 1987) for this ordering might come into play at this level. Table 6 summarizes this discussion. Table 6 Relation Between van Hiele Levels and Categorization van Hiele Manipulated Observed Categorization level object object 1 Figure Basic-level. schemas, metaphors Figure Property Propositional models 1 3 Property Ordering of properties Conduit metaphor? 4 Ordering of properties Axiomatic system ? 5 Axiomatic system Logic ? Reco~? rr? zendations forTeachi~zg The ways in which students conceive of the category of angles suggest some recommendations for teaching strategies. The present study showed that a new mathematical concept is lsarrled in the same way as other concepts are: by establishing a basic-level structure that relates the new concept metaphorically to previous models of other concepts. Abstraction is gradually built up by enriching the available models for thinking about an object and by separating the domains involved in the metaphor. Complex mathematical reasoning is supported by the existence of a rich background of models. These developments require a departure from practices in which the initial statement of definitions is an essential teaching mechanism. Mathematical categories are not established cognitively by verbalizing necessary and sufficient conditions. In other words, for most students stating a definition does not help their learning. Teachers need to pay special attention to prototype effects. There is no way to learn a category that is general enough to exclude prototype effects. Research shows the pervasiveness of prototype effects, as one of humanity's cognitive tools for relating to the environment. Mathematics is no different in this respect from other subjects, and i t is impossible. even if it were desirable, to remove these effects from the teaching and learning process. But then, given that the propositional structure of mathematics attempts to counter those very effects. what should teachers do? In this study, the teachers moved gradually from an initial basic-level category of angles to the more complex categorization described above. Pitfalls originating in prototype effects were occasionally pointed out by the teachers so as to build a more robust category. The centrality of certain subcategories (acute angles and right angles) was not denied, but occasionally noncentral exemplars of the category of angles were discussed. This stud), showed the role played by image-schematic models in promoting mathematical reasoning. One student's explorations of the similarities between angles and corners illustrates the sort of activities that should take place occasionally in mathematics classrooms. Comparing and contrasting mathematical entities informally may prove to be a powerful means of strengthening students' mathematical knowledge. Finally, teachers should be aware that several models are available on which to build specific mathematical topics. Appendix A, for example, shows several options for the case of angles. and this study showed connections between those types of angle, the cognitive models of angles of students, and very basic image schemas. Not all models, hon.ever. have the same educational value. Lakoff and Nuiiez (1997) discuss an e-xample of a metaphoric model for teaching negative numbers that proves to be inadequate for specific mathematical properties of these numbers. Although there is a good correspondence between the source domain and the target domain for adding and subtracting, the model fails in the case of multiplication. Reconzrnendations for Research The perspective on cognition outlined above raises a number of research questions in mathematics education. A first area concerns enlarging the experiential basis for understanding the role played by metaphors and basic-level categorization in school mathematics. There is no reason to believe that this role is confined to teaching and learning angles, and, although there is mounting evidence from the history of mathematics of the ways in which mathematics uses metaphors, much research still needs to be done at the school level on how metaphors might be introduced and used. Longitudinal studies investigating the development of metaphors at several grade levels or linking the formation of primary metaphors to the formation of advanced mathematical thinking are needed. In what concerns the concept of angle, three types of angle used in school mathematics that were identified in Appendix A, the continuous external rotation, the slope, and the rrigonoi? letric nrlgle, belong to more advanced mathematics curricula and were not found in this study. Research focusing on these types of angle may uncover relations that broaden the scope of the diagram in Figure 10 (p. 136). Although the corrti~zuous external rotation type of angle has, plausibly, roots in the angles turn model, the other two types of angle are apparently of a very distinct nature, and the metaphoric projections may prove to be very different. The slope type of angle may be related to schemas associated with actions of climbing (up or down), which are experientially very different from the four schemas identified in this study. There is historical support for this conjecture (hhtos, 1990), as Chinese or Esyptian mathematics, for example, did not use ansles as Greek mathematics did, bur [{. ere very proficient in calculating slopes. The trigonometric ungle may be a cluster of several models relating to experiences involving vibrations and very complex metaphoric mathematical models such as the number line. A second area refers to the consequences of the perspective for an epistemology of mathematics. Social influences on the development of mathematics have been pointed out for some time. The role of external social factors (Struik, 1942) and of internal social factors (Restivo, 1990; Wilder, 1981) in shaping mathematics has been acknowledged. The study of the influences of social fact