Abstract
This article presents the results of a comparative study regarding the impact and contribution of two instructional approaches to formal and informal mathematical reasoning with two groups of Spanish students, aged four and five. Data indicated that for both age groups, children under the ABN method [Open Algorithm Based on Numbers (ABN)] (n = 147) achieved better results than the group under the CBC approach (Closed Algorithms Based on Ciphers) (n = 82), which is the widespread approach in Spanish schools to teach formal and informal mathematical reasoning. Furthermore, the comparative analyses showed that the effect is higher in the group of students who received more instruction on skills considered domain-specific predictors of later arithmetic performance. Statistically significant differences were found in 9 of the 10 dimensions evaluated by TEMA-3 (p < 0.01), as well as on estimation tasks in the number-line for the 5-year-old-group. However, the 4-year-old group only presented significant results in calculation and concepts tasks about informal mathematical reasoning. We discuss that these differences arise by differential exposure to specific number-sense tasks, since the groups proved to be equivalent in terms of receptive vocabulary, processing speed, and working memory. The educational consequences of these results were also analyzed.
Highlights
During the 1st years of their lives, students pay special attention to their environment and innately show curiosity about the quantitative relationships that occur around them, developing informal mathematical reasoning
In order to establish that the Algorithm Based on Numbers (ABN) and Closed Algorithms Based on Ciphers (CBC) groups were equivalent, three control tests were computed
No significant differences were found for the WPPSI [MdnCBC = 27.02, SDCBC = 11.20; MdnABN = 26.79, SDABN = 11.03; F(1,109) = 0.011, p > 0.01] and neither were there significant differences in the comparison between the backward digit test for the 4-year-old group [MdnCBC = 1.48, TABLE 1 | Correlation matrix of the student scores in formal and informal mathematical thinking subtest of TEMA-3
Summary
During the 1st years of their lives, students pay special attention to their environment and innately show curiosity about the quantitative relationships that occur around them, developing informal mathematical reasoning. These skills are the basis for the mathematical concepts taught at school. As students begin receiving formal instruction, mathematical reasoning is developed and refined (Ginsburg et al, 1998). Formal mathematical reasoning requires from students a competent level in the management of symbols and language (Godino and Font, 2003). Formal mathematical reasoning involves conventional knowledge related to number literacy as well as knowledge about the basic concepts
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