Teacher to Teacher: Figural and Numerical Modes of Generalizing in Algebra

Abstract

Induction Plays A Central Role In Performing generalization and abstraction, two important processes that are necessary and highly valued in all areas of mathematics (Kaput 1999; Mason 1996; Romberg and Kaput 1999; Schoenfeld and Arcavi 1988). From 2000 to 2004, at least 30,000 eighth-grade students in northern California were tested on algebra tasks that asked them to construct linear patterns of the form y = mx + b. The students were expected to generalize using explicitly defined functions, including selecting, converting flexibly among, and employing various representations for, the patterns. Five years of data collection and analysis of students' work have shown that only three-fourths of the eighth graders tested could successfully deal with particular cases of linear patterns in visual and tabular form, and that less than one-fifth could use algebra to express correct relationships or to generalize to an explicit, closed formula (Becker and Rivera 2004).