Number Line Estimation: another view in the light of the ACE, Arithmécole, and ELFE data.

Background. Numerous experimental or empirical papers using a number line estimation task have already been published. In this task, the participant must indicate the position of a number (e.g., 12) on a line bounded by two numbers (e.g., 0 and 100). However, the authors of these articles mainly sought to model the representation of numbers based on data obtained from participants. Methods. This research draws on data from three French studies, each involving several thousand primary students, with an educational perspective. These data made it possible to investigate questions that have rarely been addressed, such as the singularity of this task and the asymmetry of the influence of the bounds. More importantly, they have enabled a systematic analysis of students' errors with the task presented in the form of multiple choice. Results. Qualitative principal component and automated classification analyzes show some singularity of the number line task (in its multiple-choice exercise, at least). The asymmetry of the influence of the two bounds argues against the two main models in the literature—the logarithmic model and the linear model. The analysis of the approximately 100,000 students' erroneous choices reveals that they do not completely respect their distance to the correct choice: The expected order of choice through genuine estimation, that is first the correct number, then the nearest wrong number, then the middle wrong number and lastly the farthest wrong number, was not respected in more than half of the 46 number line estimation items. Furthermore, young children exhibited under-comprehension by confounding the magnitude of the target number with its position value, and educators or school authorities do not clearly understand the task or its modeling. Conclusions. No model predicts all the observations, often original or curious, that we have made. Moreover, for theoretical reasons, we believe that the search for such a model is somewhat vain. In addition, we are sceptic about using the task for anything other than student assessment, given the distortions and difficulties that emerge from the children's erroneous choices, and from the practices of teachers and their supervisors also discussed in the article.