In order to obtain a more complete survey of students' understanding of trajectory motion, students were asked to respond to six multiple choice questions. The questions dealt with freely falling objects and objects moving along a trajectory, and were based (in part) on Galileo's thought experiments described in his Dialogue Concerning the Two Chief World Systems.
The data is summarized in tables. The authors also collected reasons for answers and present a selection of those reasons.
From the results the author presents the hypothesis that the concepts involved in the questions are relatively independend of mathematical background. Ze verwijzen hierbij ook naar een studie van Trowbridge en McDermott die vonden dat leerlingen wel een correcte definitie van snelheid hadden, maar het concept (relatie snelheid/afgelegde weg/versnelling) onvoldoende beheersten om te gebruiken in "real physical situations for deciding if and when two objects have the same speed." En bovendien verwarden ze versnelling en snelheid, en "while some demonstrated understanding of the differences between velocity and change in velocity, they neglected the time interval over which the velocity takes place." (zie ook See Halloun, Ibrahim Abou and David Hestenes )
The full meaning of the concepts involved is not yet assimilated into their experiences with the phenomenon of position, velocity and acceleration. The author quotes Renner: "There is, however, a necessity (...) to interact with concrete objects, events, and/or situations if they are to progress from the concrete to the formal thought stage." And quotes Arons: "all too frequently they are expected to be purged of their natural Aristotelianism simply by listening to a lecture, working a few sterile, end-of-chapter examples (not problems) that contribute neither insight nor understanding, and then regurgitating the verbal statement memorized as Newton's first law."
The conclusion is: many students still use concrete situations when approaching these concepts. Students must be given the opportunity to examine and be aware of the ways they think about these concepts prior to the introduction of a formal analysis of them. This was clearly the approach of Galileo, through the voice Salviati in the Dialogue.
Conclusie: meer relatie met concrete situaties en meer tijd nemen voor een (expliciete) ontwikkeling van concepten vanuit preconcepties van de leerlingen. Op de relatie van de wiskundige achtergrond wordt niet verder ingegaan, ook niet op de mogelijkheden van verschillende aanpakken van het ontwikkelen van die concepten (alleen dat het uitvoeriger en explicieter moet).