This paper describes how instruction based on the observation of actual motion can help students to: (1) develop a qualitative understanding of velocity as a continuous varying quantity, of instantaneous velocity as a limit and of uniform acceleration as the ratio of the change in instantaneous velocity to the elapsed time; (2) distinguish the concepts position, velocity, change of velocity, and acceleration from one another; and (3) make connections among the various kinematical concepts, their graphical representations, and the motions of real objects.
The concepts of velocity and acceleration are often used to introduce various instantaneous rates, not only in physics but also in other disciplines.
The understanding of instantaneous velocity as a limit is approached by approaching a limit through a laboratory ticker tape experiment, and by approaching a limit through zooming in on parts of a s-t-graph. The term limit is not introduced, but in practical terms a limit has been reached when the motion appears uniform within the interval. The advantages of this approach are: (1) the limit process is more concrete by using a perceptual criterium, and (2) the limit does not involve a sequence of ratios.
For distinguishing between position and velocity and for distinguishing velocity, change of velocity and acceleration the authors designed a lot of demonstrations that should be worked through in a specific order. Students first have to reason about things that happen, later they work with different graphs of the situations. For making connections between graphs, concepts, and motions students have to construct graphs of motions and produce motions from graphs.
Effectiveness of these materials is difficult to evaluate. The method for finding the slope could be used in other contexts. The traditional emphasis on algebraic formalism does not lead to conceptual understanding.
Hier worden problemen gesignaleerd, die later met meer nadruk op de betekenis van grafieken in See McDermott, L. C., Rosenquist, M. L., & van Zee, E. H. worden opgesomd en gepoogd om op te lossen. Opvallend is dat je daar weinig terug vindt van genoemde limieten-benadering.
Zie ook See Halloun, Ibrahim Abou and David Hestenes voor vergelijkbare alternative concepts van de leerlingen betreffende het onderscheid tussen positie en snelheid.
Vergelijk het vermijden van het limiet concept via inklemmen en locally straight. (`They' call it not mathematically rigorous, but this depends on what you call mathematical rigorous.)