Using patterns generically to see structure

A common approach to generalization activities in school algebra is to set them in the context of figurative patterns (e.g., a row of touching squares formed of matchsticks, with a relationship between the numbers of squares and matchsticks; or a row of touching tables surrounded by chairs, with a relationship between the numbers of tables and chairs). This approach allows students to see and explain numerical relationships in terms of the figurative pattern's structure. However, patterns are all too often presented, in textbooks and by teachers and researchers, in the form of a systematic sequence of elements, with the result that students generate a systematic set of ordered pairs for which they try to induce an empirical relationship, divorced from the structure of the pattern that produced them. In this paper, I argue that one reason that students appear to have difficulty in seeing structure is that they are not sufficiently initiated into this way of thinking. Further, I argue that the search for structure can often be done by focusing on an individual element of a pattern and treating it as a prototype, that is, generically. I begin by looking at some of the influences that might have fostered an empirical approach and point to manifestations of this approach in a recent special issue of ZDM – The International Journal on Mathematics Education. I then discuss how well various generalization tasks are suited to a generic approach. Last, I report on a short intervention where students were encouraged to use a generic approach, with positive results.