This chapter was published in N. Bednarz et al. (eds.), Approaches to Algebra, Kluwer Academic Publishers, 1996, 107-111 CHAPTER 7 SOME REFLECTIONS ON TEACHING ALGEBRA THROUGH GENERALIZATION LUIS RADFORD This brief commentary chapter devoted to issues suggested by the Mason and Lee chapters raises a number of fundamental questions concerning generalization: the epistemological status of generalization and the nature and complexities of generalization as it is manifested in the didactic context of the algebra classroom. 1. THE EPISTEMOLOGICAL STATUS OF GENERALIZATION Mrs. Smith, sitting in her living room, hears the doorbell ring. She gets up to see who is at the door. No one is there, and she returns to the room. Mr. Smith pursues their conversation when once again the doorbell rings. Mrs. Smith gets up once again to answer, but no one is there. This scene repeats itself a third time. The fourth time the doorbell rings, Mrs. Smith exclaims to her husband: "Do not send me to open the door! You have seen that it is useless! Experience has shown us that when we hear the doorbell, it implies that no one is there!" You surely remember the above scene from Cantatrice chauve by Eugne Ionesco. What is so captivating about this scene is that it reflects in an impeccable way the dynamics of a procedure of generalization (in fact, Mrs. Smith remains faithful to the "observed facts") and surely the conclusion is absurd (for us). The fragile status of knowledge obtained through a generalization process brings us to the question of what constitutes a "good" or "bad" generalization. The answer to this question, which has puzzled mathematicians and philosophers for the last 25 centuries, has been given to us in various forms: normative logic, inductive or probabilistic logic, statistics, and so on. Surely generalization is not specific to mathematics: From a certain point of view, it is perhaps one of the deepest characteristics of the whole of scientific knowledge and even, perhaps, of daily non- scientific knowledge, as shown by the little extract of Ionesco's theater piece quoted above. From a mathematical teaching perspective that favors generalizing activities, it may be convenient to try to answer the question: Why, in the construction of his/her knowledge, does the cognizer make generalizations? The "why" should be understood, of course, in its deeper meaning, so that we may specify the epistemological role of the generalization as well as the nature of the relation between generalization and the resulting knowledge. From the same perspective, other interesting questions are: • What is the significance of generalization in mathematics? and more specifically: • What are the kinds and the characteristics of generalizations involved in algebra? • What are the algebraic concepts that we can reach through numeric generalizations? 107