Flexibility and inflexibility in accurate mental addition and subtraction: two case studies

The relationship between mental computation and relational thinking in the seventh grade

“…In addition, students' scores on the relational thinking assessment improved immediately following the mental mathematics intervention. We speculate that this improvement can be explained by the overlap in cognitive processes that are implicated in both mental mathematics and relational thinking [13,14,[45][46][47][48]. As indicated previously, at the heart of relational thinking are transformations and substitutions of equivalent numbers and expressions, which are contingent not only on a relational conception of the equal sign, but also on a solid understanding of number properties, such as commutativity and distributivity.…”

“…Irwin and Britt [16], for instance, described the Numeracy Project as an effective curriculum for improving relational thinking in New Zealand. The Numeracy Project is a mathematics curriculum designed for primary-school students (ages [5][6][7][8][9][10][11][12][13][14] to improve their understanding of operations and to assist them to be flexible in their problem-solving strategies. Students participating in the Numeracy Project were encouraged to use a variety of mental strategies to solve mathematics problems such as 27 + 15 or 34 + 19, rather than rely on computation or paper-and-pencil algorithms.…”

Enhancing Arithmetic and Word-Problem Solving Skills Efficiently by Individualized Computer-Assisted Practice

“…Working memory can best be relieved by the automatization of skills, which requires practice (Tronsky, 2005). On the other hand, rote learning of procedures promotes the development of buggy algorithms (Brown & VanLehn, 1981) and leads to inflexible application (Heirdsfield & Cooper, 2002;Lovett & Anderson, 1996;Luchins, 1942;Ohlsson & Rees, 1991).…”

Profiles of Cognitive Flexibility in Arithmetic Reasoning: A Cross-Country Comparison of German and American Elementary Students

“…Another pattern reported in the literature is that students who exhibit flexible and adaptive expertise in mental calculation also show deep understanding of number and operation relationships as well as knowledge of basic facts and fact families. These students also possess high self-confidence and a positive attitude towards mathematics (Heirdsfield & Cooper, 2002, 2004Threlfall, 2002). Additionally, they show recognition and use of number patterns and relationships to solve problems (Macintire & Forrester 2003;Rathgeb-Schnierer 2006, 2010Schütte 2004;Threlfall, 2009).…”