Instructional approaches to foster third graders’ adaptive use of strategies: an experimental study on the effects of two learning environments on multi-digit addition and subtraction

Abstract

The adaptive use of strategies, that is selecting a strategy which allows an efficient solution for a given problem, can be considered as an important individual ability relevant in various domains. Based on models of subjects' skills of adaptive use of strategies, two idealized instructional approaches are suggested to foster students in their strategy development. The explicit approach aims at reducing cognitive load by demonstrating and practicing strategies combined with an explicit identification of criteria for strategy efficiency by contrasting problem solutions. The implicit approach capitalizes on the generation effect and stimulates students to generate their own strategies and efficiency criteria based on the analysis of task characteristics and the comparison of problem solutions. In a 1-week experimental study (16 lessons) with 73 third-graders, we examined the effectiveness of these instructional approaches in the domain of multi-digit addition and subtraction. Results from post-and two follow-up tests after 3 and 8 months did not yield different effects of the two approaches on students' skills in adaptive use of strategies. A comparison of strategies used by the students showed that the students of the explicit approach more frequently applied complex strategies whereas the students from the implicit approach showed a more sustainable use of self-generated strategies. Hence, for the adaptive use of those strategies students are able to generate, the implicit approach turned out to be more effective than the explicit approach. However, this generation effect does not hold for strategies which are too complex to be generated by students.

Interleaved Learning in Elementary School Mathematics: Effects on the Flexible and Adaptive Use of Subtraction Strategies

“…The dominance of the standard written algorithm even increased over time. As a result, our study replicates the findings of previous research regarding the dominance of the standard written algorithm after its introduction (Selter, 2001; Torbeyns and Verschaffel, 2016; Heinze et al, 2018). Compared to this, the students of the interleaved condition used the compensation strategy and the indirect addition relatively often, also after the introduction of the standard written algorithm, even though there was a small increase of the use of the standard written algorithm over time as well.…”

“…To assess the students’ flexibility, their strategy use was coded by four trained coders independently guided by a standardized coding manual. This coding manual had been developed based on the coding manual of the TigeR-study (Heinze et al, 2018). The inter-coder agreement was very satisfying (κ ≥ 0.88).…”

“…To assess whether a specific strategy is adaptive for solving a specific subtraction task, a more precise definition is required. To decide, whether a strategy is adaptive, i.e., appropriate/efficient, for a certain subtraction task, we take a normative perspective following several other studies (e.g., Blöte et al, 2000, 2001; Heinze et al, 2009, 2018; Torbeyns et al, 2009a). Accordingly, we take (1) the number of solution steps, (2) the mental effort, and (3) the error rate into account when assessing the adaptivity of the used strategies.…”

“…There are several different classifications of subtraction strategies in the literature (for an overview, see Threlfall, 2002). For our research, we concentrated on a categorization of four idealized number-based strategies, which are widely known in the context of mathematics education, as well as the standard written algorithm as a digit-based strategy to solve multi-digit subtraction tasks (e.g., Wittmann and Müller, 1990; Threlfall, 2002; Benz, 2007; Heinze et al, 2009, 2018; Verschaffel et al, 2009; Padberg and Benz, 2011; Fierro, 2013; Bassarear and Moss, 2016; Kupferman, 2016; Schipper et al, 2017). The number-based strategies include two decomposition strategies (stepwise strategy and split strategy) and two shortcut strategies (compensation strategy and indirect addition, Table 1).…”

Developing Flexibility in Mental Calculation

“…• The development of flexibility in mental calculation can be supported by special approaches to math education. In this regard researchers have highlighted the problem-solving approach in general Marschick;Lipowsky, 2009;Schwabe;Lipowsky, 2013;Arend;Lipowsky, 2018) combined with specific activities for fostering number sense and metacognitive competencies (Rathgeb-Schnierer, 2006;2010;Rechtsteiner-Merz, 2013). • Students with learning difficulties in arithmetic need special instructional approaches to develop flexibility in mental calculation van Doreen, 2007).…”