%0 Journal Article %1 Cottrill96 %A Cottrill, Jim %A Dubinsky, Ed %A Nichols, Devilyna %A Schwingendorf, Keith %A Thomas, Karen %A Vidakovic, Draga %D 1996 %J Journal of Mathematical Behavior %K apos function ijtme2006 limit mathematics mythesis sequences weblabs %P 167--192 %T Understanding The Limit Concept: Beginning With A Coordinated Process %U http://citeseer.nj.nec.com/cottrill92understanding.html %V 15

@article{Cottrill96, added-at = {2006-03-07T17:56:12.000+0100}, author = {Cottrill, Jim and Dubinsky, Ed and Nichols, Devilyna and Schwingendorf, Keith and Thomas, Karen and Vidakovic, Draga}, biburl = {https://www.bibsonomy.org/bibtex/2d9668db3124a919ff9f86a6e302b96b3/yish}, citeulike-article-id = {532426}, comment = {Cottrill et al (1996) put forth an extensive theory of understanding of the limit concept. They base their analysis on APOS theory (Dubinsky, 1992), an extention of Piaget's theory. APOS stands for "Action, Process, Object Schema". The idea, which is central to the theory, is that conceptions evolve from action to process to object and schema. Actions are reminiscent of Piaget's "senomotoric" phase. It is defined as a "phsyical or mental transformation of objects to obtain other object", i.e., a procedure to obtain a goal with no reflection on the procedure. Once the individual begins to reflect on the action, he or she may develop control over it. This is what the theory calls the process view. An object is constructed by encapsulation of a process. The learner can reflect on and mentaly manipulate it. Finally, a schema is defined as "a coherent collection of actions, processes, objects and other schemas, which are linked in some way and brought to bear upon a problem situation". The authors do not explain what coherence means, or elaborate on the "some way" in which the constituents of the schema are linked. However, in their analysis they use the term schema to refer to mental toolboxes that proficient problem solvers draw upon when tackilng a problem. APOS theory sees the progression from action to schema as strongly directed. Constructing scehmata is the panacea of learning. However, this view is in a way inconsistent with the theory itself: once a process or a schema has been internelized and encapsulated into an object, we stop reflecting on it, and it becomes as action. Consider the act of braking a car. At first, this is done by delibiration, and with great control. Luckily, over time, we automate this procedure into a thoughtless action. The authors mention that in mathematics, it is important to be able to shift back and forth between process and object views. This observation is surely true in the case at hand. I would add that in life it is important to be able to move between all four modes as neccesary. The crux of the analysis provided for the learning of the limit concept lies in the claim that it involves the construction of a schema of two coordinated processes: convergence in the function domain leads to convergence in the function range. This observation is in agreament with Confrey and Smith's (1994) notion of co-variation, allthough they do not find it neccesary to embed their idea in a grand theory of learning. Following others (Tal l\& Vinner, 1981; Sierpinska, 1990, 1987) the authors highlight the difficulty of understanding the quantification in the formal concept of limit. In my view, this is a general difficulty in mastering formal mathematical concepts, which I attribute to the difficulty in shifting from a narrative to a propositional form of language. In their genetic decomposition, Cottrill et al. identify 6 steps in the construction of the limit concept. These are validated, elaboraed and refined through a teaching experiment and a set of 25 interviews. It is hard to argue against the possibility of a learning path which follows their description. Nevertheless, the claim that this is the unique manner in which learners construct the unique correct concept of limit seams a bit extreme. Furthermore, apart from providing a desciption that appeals to common sense, and some data which can be matched to it (but may fit another model just as well), the authors provide no cognitive explanation of why learners go through this set of 6 steps and why the transitions from one level of conceptualization to the next appear to be so hard. As for the data itself, it is arguable that the learners confusion is a result of the specifc examples used in class, rather than of an inherent epistemological phenomena.}, interhash = {4410c5905aded4709f09b4eab2ae5114}, intrahash = {d9668db3124a919ff9f86a6e302b96b3}, journal = {Journal of Mathematical Behavior}, keywords = {apos function ijtme2006 limit mathematics mythesis sequences weblabs}, pages = {167--192}, priority = {2}, timestamp = {2008-04-27T15:58:16.000+0200}, title = {Understanding The Limit Concept: Beginning With A Coordinated Process}, url = {http://citeseer.nj.nec.com/cottrill92understanding.html}, volume = 15, year = 1996 }