Published in Proceedings of PME 17, Japan, vol. 2 , pp. 41–48, 1992. Constructing Different Concept Images of Sequences & Limits by Programming Lan Li David Tall (formerly from) Mathematics Education Research Centre Chengdu University University of Warwick Chengdu Coventry, CV4 7AL P. R. China United Kingdom As a transition between an informal paradigm in which a limit is seen as a never-ending process and the formal ε–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. The difficulties faced by students in coming to terms with the limit concept are well-documented (e.g. Cornu, 1991), from the coercive effects of colloquial language (where words like “tends to”, “approaches” suggest a temporal quality in which the limit can never be attained) to difficulties coping with the formal definition and the quantifiers involved. Here we investigate the effects of introducing a computer environment allowing the student to construct some of the concepts through programming. Following Dubinsky (1991), Sfard (1991), we formulate our observations using a theory whereby mathematical processes are subsequently conceived as objects of thought. We use the term procept (Gray & Tall, 1991) for the amalgam of process and concept where the same symbol is used both for the process and the output of the process. We hypothesise that different mental structures for the limit concept lim n→∞ s n are produced by different environments, both for the term s n as a computational process and a mental object, and also the limit itself, as process and object. These contain potential conflicts which require cognitive reconstruction to pass successfully from one paradigm to another. We consider three separate paradigms: (I) a (formula-bound) dynamic limit paradigm, (II) a functional/numeric computer paradigm, (III) the formal ε–N paradigm. Paradigm (I) occurs in UK schools for students aged 16 to 18. In this curriculum the limit of a sequence (s n ) is studied only briefly in a dynamic sense as n increases, the main focus being on arithmetic and geometric progressions and convergence of the latter. Many students use the words “sequence” and “series” interchangeably in a colloquial manner. This approach emphasises the potential infinity of a process which cannot be completed in a finite time. The terms are seen as being given by formulae and the specific geometric (and arithmetic) progressions studied also have partial sums which can be