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 reform-based 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.

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Notes

  1. 1.

    Mental effort in this respect means the expected mental effort an idealized person of the considered target group (third graders or adults, etc.) has to spend (in a comparative estimation from an expert perspective).

  2. 2.

    There is a discussion whether the split strategy is useful for subtraction problems with regrouping. Some of the German textbooks introduce this strategy but avoid the notation of intermediate (negative) results, others state or explain that this strategy is inappropriate for subtraction.

  3. 3.

    It has to be mentioned that the 1-year implementations of the gradual program design and the realistic program design cover more content than the aspects mentioned here. For details, see Klein, Beishuizen, and Treffers (1998).

  4. 4.

    We refer to the edition from 2002 for Bavaria (Southern Germany). The new edition of “Das Zahlenbuch” from 2007 shows a stronger emphasis on exploring strategies instead of presenting strategies in the first phase.

  5. 5.

    The results of Selter (2001) which are comparable with our results show a much higher proportion of items (about 50%) solved by mental computation, i.e. only the solution was given. Since we do not know exactly the strategies that were used to solve the items, a precise comparison is difficult.

  6. 6.

    Indirect addition was used 19 times to solve these two subtraction items, this means in 3.87% of the corresponding 2 × 245 = 490 solutions. The other nine solutions based on indirect addition were used for the remaining three subtraction items.

  7. 7.

    We used the Scheffé procedure as a post hoc test of an ANOVA.

  8. 8.

    Selter (2001) tested his sample three times. Here, we chose his first measurement which took place at the same time in the school year as our data collection.

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Acknowledgments

This research was funded by the German Research Foundation (DFG), AZ HE 4561/3-1 and LI 1639/1-1. We like to thank the reviewers for their valuable comments which helped to improve the manuscript.

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Correspondence to Aiso Heinze.

Appendix

Appendix

Table 7 presents the addition and subtraction items and the solution strategies which were considered as efficient strategies. All other appropriate strategies were rated as inefficient. Mental computation was always rated as efficient since we assumed that students in this case chose an efficient way to process the item (only 6.3% of all processed items were solved by mental computation). For the two cases of pair-items, the rating of the strategy of the corresponding second item took into account the strategy used for the first item (cf. Table 7).

Table 7 Computation items and corresponding strategies considered as efficient

For the test, we chose items which could efficiently be solved by the specific strategies simplifying, compensation or indirect addition. Accordingly, the frequently used strategies, stepwise and split, are considered as inefficient (with one exception, see the remark below). The items were distributed in a test with 18 items. Except the pair-items (773 − 49 and 462 + 258), all computation items were separated by other items and did not occur together in the test.

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Heinze, A., Marschick, F. & Lipowsky, F. Addition and subtraction of three-digit numbers: adaptive strategy use and the influence of instruction in German third grade. ZDM Mathematics Education 41, 591–604 (2009). https://doi.org/10.1007/s11858-009-0205-5

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Keywords

  • Adaptive Strategy
  • Instructional Approach
  • Split Strategy
  • Strategy Execution
  • Subtraction Problem