## Abstract

This study aimed at analysing traditionally taught children’s acquisition and use of shortcut strategies in the number domain 20–100. One-hundred-ninety-five second, third, and fourth graders of different mathematical achievement levels participated in the study. They were administered two tasks, both consisting of a series of two-digit additions and subtractions that maximally elicit the use of the compensation \(\left( {45 + 29 = \_;45 + 30 - 1 = 75 - 1 = 74} \right)\) and indirect addition strategy (\(71 - 68 = \_;\,68 + 2 = 70,\,70 + 1 = 71\), so the answer is 2 + 1 or 3). In the first task, children were instructed to solve all items as accurately and as fast as possible with their preferred strategy. The second task was to generate at least two different strategies for each item. Results demonstrated that children of all grades and all achievement levels hardly applied the compensation and indirect addition strategy in the first task. Children’s strategy reports in the second task revealed that younger and lower achieving children did not apply these strategies because they did not (yet) discover these strategies. By contrast, older and higher achieving children appeared to have acquired these strategies by themselves. Results are interpreted in relation to cognitive psychological and socio-cultural perspectives on children’s mathematics learning.

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## Notes

- 1.
Baroody (2003) distinguishes among four qualitatively different approaches to mathematics instruction, namely the skills, conceptual, investigative, and problem-solving approach. The (traditional) skills approach focuses on the procedural content of the mathematics curriculum (i.e.,

*how*to execute a specific strategy) and the memorization of basic skills by rote, fostering the development of routine expertise. In contrast, in the investigative and problem-solving approaches, instruction does not merely focus on the procedural content of the mathematics curriculum, but also extensively addresses the conceptual content and the processes of mathematical inquiry (i.e., problem solving, reasoning, etc), stimulating the acquisition of adaptive expertise. - 2.
The percentiles of the mathematics achievement tests for second-, third- and fourth-graders are based on a representative sample of Flemish children from, respectively, the second, third and fourth grade.

- 3.
It is important to note here that all children who applied the compensation and indirect addition strategy in the SST on other item types than Co respectively IA

^{-}items also used these strategies on the respective item types. In other words, the data from the SST do not indicate that children who did not apply the compensation and indirect addition strategy on Co respectively IA^{-}items knew these strategies. - 4.
A strategy was scored as a different strategy if it could be classified as another type of strategy than the strategy or strategies reported on the previous trial(s) in our classification scheme. So, a child who first used the standard version of decomposition strategies (40 + 20 = 60, 5 + 9 = 14, 60 + 14 = 74) and afterwards applied the standard version of sequential strategies (45 + 20 = 65, 65 + 9 = 74), solved 45 + 29 with two different strategies. Likewise, a child who first used the standard version of sequential strategies (45 + 20 = 65, 65 + 9 = 74) and afterwards applied another type of sequential strategies (45 + 9 = 54, 54 + 20 = 74), answered this item with two different strategies. But a child who applied the same type of strategy on consecutive trials (45 + 20 = 65, 65 + 9 = 74, and 45 + 20 + 9 = 74), was scored as using the same, and thus not a different, strategy on these trials.

- 5.
It is important to note here that the compensation and indirect addition strategies are not easy strategies to discover and apply in comparison with the standard sequential and decomposition strategies. First, the discovery of the compensation and indirect addition strategy requires a good conceptual understanding of the numbers and number relations up to 100 (e.g., 19 is one less than 20; the difference between 68 and 71 is small). Moreover, once discovered, the application of these strategies might induce an increased cognitive load, because it is important to ascertain that you add and subtract the numbers in the correct order (i.c., adding 20 and subtracting 1) and keep track of the counting process (i.c., count on from 68 to 71). Consequently, the fluent application of these strategies on additions and subtractions that seem to maximally elicit their use, requires a sufficient amount of practice.

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## Acknowledgements

Bert De Smedt and Joke Torbeyns are Postdoctoral Fellows of the Fund for Scientific Research–Flanders (Belgium). This research was partially supported by Grant GOA 2006/01 “Developing adaptive expertise in mathematics education” from the Research Fund K.U.Leuven, Belgium. The authors would like to thank Evelien Goethals for her assistance during data collection.

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Torbeyns, J., De Smedt, B., Ghesquière, P. *et al.* Acquisition and use of shortcut strategies by traditionally schooled children.
*Educ Stud Math* **71, **1–17 (2009). https://doi.org/10.1007/s10649-008-9155-z

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### Keywords

- Strategy development
- Strategy diversity
- Strategy discovery
- Shortcut strategies
- Compensation strategy
- Indirect addition
- Multi-digit addition and subtraction
- Elementary school children