Kinder auf dem Weg zum flexiblen Rechnen: Eine Untersuchung zur Entwicklung von Rechenwegen bei Grundschulkindern auf der Grundlage offener Lernangebote und eigenständiger Lösungsansätze

Addition and subtraction of three-digit numbers: adaptive strategy use and the influence of instruction in German third grade

“…In Germany, case studies of Schütte (2004) and Rathgeb-Schnierer (2007) show that children learning by an implementation of the problemsolving approach can achieve a high level of flexible strategy use. Mathematics educators such as Threlfall (2002) and Schütte (2004) are convinced that such a reform-based constructivist-oriented approach is most beneficial for all students and that an explicit teaching and practicing of strategies will run the risk of preventing an adaptive strategy use.…”

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

“…Moreover, when students learn through examples, they tend to learn specific procedures rather than general rules or relationships, which in turn leads to a negative impact on the development of mental flexibility (Heirdsfield & Cooper 2004;Klein, Beishuizen, & Treffers, 1998;Schütte 2004). Others have reported that student strategies are related to the context of usage, as occurs with different target operations , specific problem characteristics (Blöte et al, 2000;Torbeyns et al, 2009), and student-initiated recognition of problem characteristics (Rathgeb-Schnierer, 2006, 2010. 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).…”

“…In this regard researchers have highlighted the problem-solving approach in general (Heinze, Marschick, & Lipowsky, 2009;Heinze, Schwabe, Grüßing, & Lipowski, 2015) combined with specific activities for fostering number sense and metacognitive competencies (this special approach is called "Zahlenblickschulung") (Rathgeb-Schnierer, 2006, 2010Rechtsteiner-Merz, 2013). Furthermore, students with low achievement in mathematics need special instructional approaches to develop flexibility in mental calculation (Verschaffel, Torbeyns, De Smedt, Luwel, & van Dooren, 2007).…”

“…In this context, students' ability to solve multi-digit arithmetic problems, without using paper and pencil computing algorithms, has come under increasing scrutiny among researchers (see e.g., Blőte, Klein, & Beishuizen, 2000;Heirdsfield & Cooper, 2004;Rathgeb-Schnierer, 2006;Threlfall, 2009).…”