Understanding the limit concept: Beginning with a coordinated process scheme

Students’ attitudes to mathematics and performance in limits of functions

“…But one can also think of the limit as a static entity, or object, to which functions can be compared. Sfard (1991) and Cottrill et al (1996) have described theories of object formation from processes. A notion is first regarded as a process and when the characteristics of the notion become clear the notion can be seen as an object upon which other processes can be applied.…”

“…Such schemas constitute individuals' concept images. Cottrill et al (1996) concluded that the formal concept of a limit is an intricate dynamical schema and not a static one. It is important to have a strong dynamical conception of the notion before it is possible to embrace fully a more formal interpretation where the limit is seen as an object.…”

The transition to formal thinking in mathematics

“…Cottrill et al, 1996) based on the theories of Piaget and was used mainly in programming mathematical constructs in a symbolic development. Later in this paper we will return to APOS theory to show how a blending of embodiment and symbolism gives a more complete way of developing sophistication in mathematical thinking.…”

“…Another mode of compression, described by Dubinsky and his colleagues (Cottrill et al, 1996), occurs in APOS theory where an ACTION is internalised as a PROCESS and is encapsulated into an OBJECT, connected to other knowledge within a SCHEMA; they also note that a SCHEMA may also be encapsulated as an OBJECT.…”

“…Cottrill et al, 1996) focused initially on a symbolic approach by programming a procedure as a function and then using the function as the input to another function. The data shows that, while the process level was often attained, encapsulation from process to object was more problematic.…”