Armyanskii ln., 4-2,
Moscow, 101000, Russia


Anna Shvarts1,2
  • 1 Lomonosov Moscow State University, 1 Leninskie Gory, Moscow, 119991, Russia
  • 2 Utrecht University, 5 Princetonlaan, 3584 CC, Utrecht, Netherlands

Does the Real Form Interact with the Ideal Form? A Study of the Teaching-Learning to Count on the Number Line by Means of Eye-Tracking

2019. Vol. 16. No. 1. P. 145–163 [issue contents]

The article investigates acquisition of mathematical knowledge in collaboration with an adult as it is exemplified by preschoolers’ learning to count on the number line. A qualitative analysis of the eye-movements reveals the diversity of possible strategies in determination of a number on the number line. The developmental experiment discloses the mechanisms of emergence of these strategies in children. The quantitative comparison of the adults’ strategies, the strategies, which are involved in the teaching-learning process, and the strategies that the children used after the learning stage demonstrates the process of development (χ2= 44, 936; p<0,001). We distinguished the statistically significant differences between the stages in the ratio of counting up versus down along the number line and in the ratio of counting from versus towards the target point. The results demonstrate that children’s strategies after the learning stage are more similar to the adults’ inherent strategies than to the strategies that were introduced by the adults during the teaching stage. The analysis of the videos of shared activity that was synchronized with the eye movements showed that the adults demonstrated the basic strategy to the children at the teaching phase as they guided children’s perception by their pointing gestures and speech. However, the adults did not expose the ideal form, namely the diversity of their own strategies during their teaching. Nevertheless, the children were able to supplement the given teaching/learning formof counting from zero up along the number line to the target point with a variety of strategies by themselves, relying on their coherent notion of the number concept. The strategy that required the sequence-to-proportion shift was the only one that children were not able to constitute by themselves. According to our results, the ideal, cultural form of perception exists in the latent form, and a child needs to re-constitute it in their own practice. The children rely on the basic strategy and enrich this strategy as they include it in the integral conceptual knowledge about numbers. The results enrich our understanding of microgenesis of mathematical knowledge during the collaboration with an adult and open perspective on learning as an active reinvention of ideal form on the ground of cultural practice.

Citation: Shvarts A. (2019) Vzaimodeystvuet li real'naya forma s ideal'noy? Issledovanie ovladeniya schetom na chislovoy pryamoy s pomoshch'yu zapisi dvizheniy glaz [Does the Real Form Interact with the Ideal Form? A Study of the Teaching-Learning to Count on the Number Line by Means of Eye-Tracking]. Psychology. Journal of Higher School of Economics, vol. 16, no 1, pp. 145-163 (in Russian)
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