Search

The article discusses the role of abduction and induction in patterning activity. It explores the concept of abduction, beginning with a general discussion of Charles Saunders Peirce's thoughts, and its similarities and differences from inferential acts used in mathematics, specifically in mathematical reasoning. It also proposes and demonstrates a pattern generalization scheme in which abduction and induction are both required before any final statement of generalization is inferred.

This paper discusses the content and structure of generalization involving figural patterns of middle school students, focusing on the extent to which they are capable of establishing and justifying complicated generalizations that entail possible overlap of aspects of the figures. Findings from an ongoing 3-year longitudinal study of middle school students are used to extend the knowledge base in this area. Using pre-and post-interviews and videos of intervening teaching experiments, we specify three forms of generalization involving such figural linear patterns: constructive standard; constructive nonstandard; and deconstructive; and we classify these forms of generalization according to complexity based on student work. We document students' cognitive tendency to shift from a figural to a numerical strategy in determining their figural-based patterns, and we observe the not always salutary consequences of such a shift in their representational fluency and inductive justifications. [ABSTRACT FROM AUTHOR]

The article introduces various papers published within the issue including research about young students' generalization in algebra and a commentary by Willibald Dörfler to these studies.