%0 Conference Paper %1 Li93 %A Li, L. %A Tall, David O. %B the 17th International Conference for the Psychology of Mathematics Education %C Tsukuba, Japan. %D 1993 %K constructionism ijtme2006 infinity limit mythesis process-product programming sequences %P 41-48 %T Constructing different concept images of sequences and limits by programming %U http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1993e-lan-li-pme.pdf %V 2 %X As a transition between an informal paradigm in which a limit is seen as a never-ending process and the formal eﾖN paradigm we introduce a programming environment in which a sequence can be defined as a function. The computer paradigm allows the symbol for the term of a sequence to behave either as a process or a mental object (with the computer invisibly carrying out the internal process) allowing it to be viewed as a flexible procept (in the sense of Gray & Tall, 1991). The limit concept may be investigated by computing s(n) for large n to see if it stabilises to a fixed object. Experimental evidence shows that a sequence is conceived as a certain kind of procept, but the notion of limit remains more at the process level. Deep epistemological obstacles persist, but a platform is laid for a better discussion of formal topics such as cauchy limits and completeness.

@inproceedings{Li93, abstract = {As a transition between an informal paradigm in which a limit is seen as a never-ending process and the formal eﾖN paradigm we introduce a programming environment in which a sequence can be defined as a function. The computer paradigm allows the symbol for the term of a sequence to behave either as a process or a mental object (with the computer invisibly carrying out the internal process) allowing it to be viewed as a flexible procept (in the sense of Gray \& Tall, 1991). The limit concept may be investigated by computing s(n) for large n to see if it stabilises to a fixed object. Experimental evidence shows that a sequence is conceived as a certain kind of procept, but the notion of limit remains more at the process level. Deep epistemological obstacles persist, but a platform is laid for a better discussion of formal topics such as cauchy limits and completeness.}, added-at = {2008-05-30T00:59:52.000+0200}, address = {Tsukuba, Japan.}, author = {Li, L. and Tall, David O.}, biburl = {https://www.bibsonomy.org/bibtex/2b8b457c76c0b838028f27458983addd8/yish}, booktitle = {the 17th International Conference for the Psychology of Mathematics Education}, citeulike-article-id = {532456}, comment = {Difficulty with the functional definition of sequences: For instance, the definition of a sequence as a function from N to R includes the requirement that the function be specified simultaneously for all values on the infinite set N, involving actual infinity rather than potential infinity. Difficulty with the deffinition of limit: The definition of limit is formulated in terms of an unencapsulated process (given e, an N can be found such that !K) rather than being described explicitly as an object. It involves several layers of quantifiers which exceed the short-term memory processing capacity of many students. There is a severe problem of the status of the limit notion ¡V can one define an object linguistically, or does it need to have an independent existence? For example, if a decimal such as 0.9.. (nought point nine recurring) is believed to exist as a number less than one, can it be defined to be something equal to one?}, interhash = {d7742ee305580a7e4873aa164567fb86}, intrahash = {b8b457c76c0b838028f27458983addd8}, keywords = {constructionism ijtme2006 infinity limit mythesis process-product programming sequences}, pages = {41-48}, priority = {2}, timestamp = {2008-05-30T00:59:52.000+0200}, title = {Constructing different concept images of sequences and limits by programming}, url = {http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1993e-lan-li-pme.pdf}, volume = 2, year = 1993 }