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

Abstract

Empirical findings show that many students do not achieve the level of a flexible and adaptive use of arithmetic computation strategies during the primary school years. Accordingly, educators suggest a reformbased instruction to improve students' learning opportunities. In a study with 245 German third graders learning by textbooks with different instructional approaches, we investigate accuracy and adaptivity of students' strategy use when adding and subtracting three-digit numbers. The findings indicate that students often choose efficient strategies provided they know any appropriate strategies for a given problem. The proportion of appropriate and efficient strategies students use differs with respect to the instructional approach of their textbooks. Learning with an investigative approach, more students use appropriate strategies, whereas children following a problem-solving approach show a higher competence in adaptive strategy choice. Based on these results, we hypothesize that different instructional approaches have different advantages and disadvantages regarding the teaching and learning of adaptive strategy use.

Dutch sixth graders’ use of shortcut strategies in solving multidigit arithmetic problems

“…There is a general consensus that there are three types of mental, 1 number-based solution strategies to solve multidigit addition and subtraction problems: (a) sequential strategies in which the subtrahend is decomposed: e.g., solving 45 − 29 via 45 − 20 = 25; 25 − 9 = 16, (b) decomposition strategies in which both operands are decomposed: e.g., solving 45 − 29 via 40 − 20 = 20; 5 − 9 = − 4; 20 − 4 = 16, and (c) varying (or shortcut) strategies: e.g., the compensation strategy 45 − 29 = 45 − 30 + 1 = 15 + 1 = 16 or the indirect addition strategy (also called subtraction by addition) in which one adds on from the subtrahend: e.g., 29 + 1 = 30; 30 + 15 = 45; so the answer is 1 + 15 = 16 (for overviews, see for instance Beishuizen et al 1997;Heinze et al 2009;Peltenburg et al 2012;Peters et al 2013).…”

“…In the domain of multidigit addition and subtraction, studies with German third graders (Heinze et al 2009;Selter 2001), Dutch second graders (Blöte et al 2001), Flemish second to fourth graders (De Smedt et al 2010;Peters et al 2013;c), and recently, a cross-national study of Dutch and Flemish third to sixth graders showed that students used the shortcut strategies indirect addition, compensation, and other simplifying strategies rather infrequently, usually below 20% (Torbeyns et al 2017). In subtraction problems, compensation strategies seem to be used somewhat more often than indirect addition.…”

“…The following general patterns of factors impacting shortcut strategy use emerge. The first factor is the instructional setting: generally speaking, students following a more reform-based investigative or problem-solving program focusing on a diversity of strategies were more likely to use shortcut strategies than children in a more traditional skills-oriented program focusing on the acquisition and mastery of standard strategies (Blöte et al 2001;De Smedt et al 2010;Heinze et al 2009). The second factor is problem characteristics: besides the impact of number characteristics, the presentation format has also been found to have an effect, with word problems eliciting more shortcut strategies than symbolically presented problems (De Smedt et al 2010).…”

“…In contrast to the importance attached to flexibility, adaptivity, and clever strategies, empirical results have shown that primary school students do not use shortcut strategies very frequently in Flanders (e.g., De Smedt et al 2010;Torbeyns et al 2009a, b), Germany (e.g., Heinze et al 2009;Selter 2001), or The Netherlands (e.g., Blöte et al 2001;. The current study aims to extend these results by investigating the use of shortcut strategies in very favorable conditions, that is, by studying older students (with higher levels of conceptual and procedural knowledge) who have received years of reform-based mathematics instruction focusing on fostering adaptivity and clever strategies.…”

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

“…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).…”