The relationship between processes (which I shall try to use as a generic term, reserving procedure for a specific algorithm for a given process) and objects has been the subject of much scrutiny by a number of researchers in recent years. Skemp’s (1961 p. 47) insight that:
For the algebraist will continue, in due course, to develop concepts of new classes of numbers (e.g., complex numbers) and new functions (e.g., gamma functions) by generalising the field of application of certain operations (taking square roots, taking the factorial of a number); and will study the application of the existing set of operations to the new concepts.
describes the construction of new mathematical objects (concepts) by generalising operations. Davis (1984, pp. 29–30) formulated a similar idea:
When a procedure is first being learned, one experiences it almost one step at time; the overall patterns and continuity and flow of the entire activity are not perceived. But as the procedure is practised, the procedure itself becomes an entity – it becomes a thing. It, itself, is an input or object of scrutiny. . . . The procedure, formerly only a thing to be done – a verb – has now become an object of scrutiny and analysis; it is now, in this sense, a noun.
Describing here how a procedure becomes an object he strikes at a key distinction between the two when he mentions the ‘one step at a time’ nature of procedures when they are first encountered, in contrast with ‘the overall. . . flow of the entire activity’, or the holistic, object-like nature which they can attain for an individual. More recently others have spoken of how an individual encapsulates or reifies the process so that it becomes for them an object which can be symbolised as a procept (Dubinsky & Lewin, 1986; Dubinsky, 1991, Cottrill et al., 1997; Sfard, 1991, 1994; Gray & Tall, 1991, 1994). However, it seems that there are at least two qualitatively different types of processes in mathematics; those object-oriented processes from which mathematical objects are encapsulated (see Tall et al. in press) and those solution oriented processes which are essentially algorithms directed at solving ‘standard’ mathematical problems. An example of the first would be the addition of terms to find the partial sums of a series, which leads to the conceptual object of limit and the second could be exemplified by procedures to solve linear algebraic equations. Much mathematics teaching in schools has concentrated on solution-oriented processes to the exclusion of object-oriented processes. However solution-oriented processes usually operate on the very objects which arise from the object-oriented processes. Hence ignoring these is short sighted and will prove counter-productive in the long term. Failure to give students the opportunity to encapsulate object-oriented processes as objects may lead them to engage in something similar to the pseudo-conceptual thinking described by Vinner (1997, p. 100) as exhibiting “behaviour which might look like conceptual behaviour, but which in fact is produced by mental processes which do not characterize conceptual behaviour.” In this case students may appear to have encapsulated processes as objects but this turns out on closer inspection to be what we will call a pseudo-encapsulation.
For example, in arithmetic students may appear to be able to work with a fraction such as 4/5 as if they have encapsulated the process of division of integers as fractions. However, many have not done so. I am still intrigued by the first time I encountered such a perception. It was in 1986 (see Thomas, 1988; Tall & Thomas, 1991) when I discovered that 47% of a sample of 13 year old students thought that 6÷2 x +17 and 6/7 were not the same, because, according to them, the first was a ‘sum’ but the second was a ‘fraction’. Such students are at a stage where they think primarily in terms of solution-oriented processes, and in this case they had not encapsulated division of integers as fractions. Instead they had constructed a pseudo-encapsulation of fraction as an object, but not one arising from the encapsulation of division of integers. Similarly in early algebra many students may work with symbolic literals in simplifying expressions like 3a+2b- 2a + b as if they view them as procepts. However, rather than having encapsulated the process of variation as a procept, symbolised by a or b, they are actually thinking of them as pseudo-encapsulations, concrete objects similar to ‘apples’ and ‘bananas’. When dealing with expressions such as 2x +1, a common perspective enables students to work with it but see it as a single arithmetic result not as a structural object representing the generalisation of adding one to two times any number x. When they move on to equations these students can use their pseudoencapsulations to solve equations such as 2x +1= 7. They may appear to be using algebraic methods to get their answer but they are actually thinking arithmetically, and given the answer 7, either use trial and error, or work backwards numerically, to find x. This becomes clear when they are unable to solve equations similar to 2x +1= 7x - 3 (Herscovics & Linchevski, 1994).
It is possible that students might be assisted to avoid engaging in the construction of pseudo-encapsulations by approaching the learning of mathematics (in different representations) in the following way.
1.Experience the object-oriented process
2.Encapsulate the conceptual object
3.Learn solution-oriented processes using the object
The versatile approach to the learning of mathematics which is being espoused here recognizes the importance of each of the three stages of the recurring trilogy above and stresses the importance of experiencing each step in as many different representations as possible in order to promote the formation of conceptual or C-links across those representations. Students need personal experience of object-oriented processes so that they can encapsulate the objects. The versatile learner gains the ability to think in a number of representations both holistically about a concept or object, and sequentially about the process from which it has been encapsulated. A major advantage of a versatile perspective is that, through encapsulation, one may attain a global view of a concept, be able to break it down into components, or constituent processes, and conceptually relate these to the whole, across representations, as required. Without this one often sees only the part in the context of limited, often procedural, understanding and in a single representation.
How though does one encourage versatile thinking? What experiences should students have so that they may build such thinking and conceptual links? We have seen that reflective intelligence is a key to the transition from process to object, and hence to progress in mathematics (although this is not the only way objects may be formed in mathematics – see Davis, Tall & Thomas, 1997; Tall et al, 1999) and my contention is that representation-rich environments which allow investigation of the object-oriented processes which give rise to mathematical objects will encourage reflective thinking.
Skemp (1971, p. 32) appreciated that “Concepts of a higher order than those which a person already has cannot be communicated to him by a definition, but only by arranging for him to encounter a suitable collection of examples.” Whilst this is a valuable principle, it somewhat oversimplifies the situation. Many higher order concepts are not abstracted from examples but, as we have discussed above, have to be encapsulated by the individual from object-oriented processes (see Tall et al., 1999). Hence the examples which the student needs are not simply those embodying the finished concept itself, but also the processes which give rise to it.
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