“Rising to the challenge”: using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students

Abstract

This study focuses on the generalization methods used by talented pre-algebra students in solving linear and non-linear pattern problems. A qualitative analysis of the solutions of three problems revealed two approaches to generalization: recursive-local and functional-global. The students showed mental flexibility, shifting smoothly between pictorial, verbal and numerical representations and abandoning additive solution approaches in favor of more effective multiplicative strategies.

Comparison of 6th-8th graders' efficiencies, strategies and representations regarding generalization patterns

“…Some direct quotations from the dialogues between the interviewer and the interviewed have been presented here, in terms of enabling the reader to make their own comments and see a descriptive and a more realistic picture of what and how interviews happened. The data gathered is classified and discussed by taking into consideration pattern generalization strategies in previous researches KREBS, 2003;LANNIN, 2005;BECKER, 2005;EBERSBACH;WILKENING, 2007;AMIT;NERI, 2008). These strategies are summarized in Table 4.…”

“…Smith (2003), stressing on the relationship between patterns, functions and algebra, claimed that these three components must be integrated in the curriculum. In addition, many types of patterns -like linear and non-linear (quadratic and geometrical sequence), numerical, pictorial, arithmetical, geometrical and repetitive patterns -can be themes in the studies about pattern generalization including different education levels, from elementary schooling to pre-service school teachers, or from primary to secondary school (ORTON; BISHOP, 2000;LANIN, 2005;RIVERA, 2007;AMIT;NERIA, 2008). In literature, great numbers of different types of patterns are studied using different strategies in order to generalize them.…”

“…In these studies, researchers define many strategies to deal with generalization in problems related to patterns such as recursive, common ratio, counting, additive, explicit or non-explicit, whole-subject, linear, guess-andcheck, trial-and-error and contextual strategies, for instance (STACEY, 1989;RIVERA, 2004RIVERA, , 2005AMIT;NERI, 2008). Besides, all these studies, especially the one in which Stacey (1989) defines strategies in generalizing pattern problems,give foundation to many other studies.For instance,in a study about linear patterns, Stacey (1989) put forward some strategies about linear patterns including near and far generalization for 9-13 years students.…”

“…Besides, most of the studies examining strategies of generalization of patterns study, generally, linear patterns. Only few of these studies examine non-linear patterns (KREBS, 2003;EBERSBACH;WILKENING, 2007;AMIT;NERIA, 2008). As an example, we can point out that according to the studies related to non-linear patterns (KREBS, 2003;EBERSBACH;WILKENING, 2007), additive strategies are mostly common and it has been stated that there has been a great tendency towards linearity, although the patterns are obviously non-linear.…”

“…As an example, we can point out that according to the studies related to non-linear patterns (KREBS, 2003;EBERSBACH;WILKENING, 2007), additive strategies are mostly common and it has been stated that there has been a great tendency towards linearity, although the patterns are obviously non-linear. Moreover, while additive (expansive) strategies used in linear pattern problems are much more employed then a global generalization -because of the constant and obvious difference between each two successive patterns -, in non-linear patterns it can be said that this approach involving the use of visual approaches can prevent students from understanding the global structure (KREBS, 2003;RIVERA, 2007;AMIT;NERIA, 2008).…”