Abduction–induction (generalization) processes of elementary majors on figural patterns in algebra

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

“…Such results could be tested and used as a guide during the interviews. According to the researchers, these types of strategies allow local generalizations in linear and quadratic patterns, but do not allow global generalizations KREBS, 2003;RIVERA, 2007;AMIT;NERIA, 2008). In addition, all of the students using recursive (additive) strategy benefited from numerical representations, but when the students use counting strategy then can get better support from pictorial representations.…”

“…Besides, the students using recursive, counting, whole-object and multiplying with difference strategies had, generally, the tendency to convert shape pattern problems into number sequence problems. So, it seems that they want to use only numerical and pictorial (geometric) representations, apparently because they were negatively influenced while reaching global generalizations and because, in the process of the solution of complicated problems and especially shape and quadratic pattern problems, pictorial (figural) representations help them at the beginning of the problem and at the subsequent process, in order to organize the information (KREBS, 2003;RIVERA, 2007). S10 7 and S15 8 could reach L 4 level by using numerical and pictorial (table) representations, and guess and check strategy in linear pattern problems.…”

“…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.…”

“…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).…”

Strategic vs Definitory Rules: Their Role in Abductive Argumentation and their Relationship with Deductive Proof

“…In mathematics education, the term "abduction" is used to refer to a number of different processes and in relation to different activities. Abduction has been considered in relation to mathematical activity in general (see, e.g., Cifarelli & Sáenz-Ludlow, 1996;Krummheuer, 2007;Mason, 1996;Rivera, 2007), and it has been studied to explore ways in which it may be implemented in the mathematical classroom (Rivera 2017). Many researches have considered abduction in relation to mathematical proof (see, e.g., Antonini & Mariotti, 2003, Arzarello, Micheletti, Olivero, & Robutti, 1998Knipping, 2003, Pedemonte & Reid 2011Pedemonte, 2007Pedemonte, , 2008 to compare it with other forms of reasoning like deduction or induction.…”