Tall (2004) described the development of mathematical thinking from infant to adult in terms that accounted for the differences experienced from individual to individual. The focus is on three cumulative and interrelated modes of thought, or “worlds of mathematics”, rather than hierarchical stages. The first is the “conceptual-embodied world” and is concerned with sensory perceptions and associated mental reflections. The second is the “proceptual-symbolic world”, concerned with the computation and manipulation of conventional mathemati- cal symbols. The third is the “formal-axiomatic world” which relates to the use of formal definitions within mathematics. It is the second, proceptual-symbolic, world that is of interest here.
The starting point for understanding symbolic mathematics is the notion of “en- capsulation” by which a mathematical process (such as counting to 5) becomes a mathematical concept (such as the number 5). A concept, or encapsulated process, can then become the input to further, higher-level processes, which in turn become encapsulated as concepts, and so on. Encapsulation, some- times known as “entification” or “reification”, is well evidenced throughout the mathematics education literature (e.g. Dubinsky, 1991; Sfard, 1991). However, whereas other theorists emphasise a hierarchy of processes preceding mental objects (concepts), Gray and Tall (1994) emphasised the importance of flexibly switching between process and object when thinking mathematically (Gilmore & Inglis, 2008). This flexibility of thought correlates to employing the “simple device of using the same notation to represent both a process and the product of that process” (Gray and Tall, p.119). Gray and Tall give several examples of this, including:
• The symbol 5 + 4 represents both the process of adding by counting all or counting on and the concept of sum (5 + 4 is 9).
• The symbol +4 stands for both the process of ‘add four’, or shift four units along the number line, and the concept of the positive number +4.
• The algebraic symbol 3x+ 2 stands both for the process ‘add three timesxand two’ and for the product of that process, the expression ‘3x+ 2’ (p.120).
Gray and Tall referred to such a duality of process and concept represented by a single symbol as anelementary procept (a concatenation of the wordsprocess and concept). In the minds of flexible thinkers any given symbol, such as “6”, is one of many names for a single mathematical object, along with 3+3, 8−2, 2×3 and so on. Aprocept, in contrast to an elementary procept, includes a single mathematical object, a collection of symbolic representations for that object, and the processes for generating that object. The termproceptual thinking is used to characterise viewing a symbol as a duality of process and concept, as well as making interchanges of different symbolic names for the same mental object.
Gray and Tall argued that proceptual thinking distinguishes those who enjoy success at mathematics from the unfortunate majority who do not. The latter become bogged down in the procedures of mathematics, developing problem solving efficiency without developing problem solvingflexibility. Consider the case of 2 + 3. A pupil who invariably uses a counting procedure may move from “counting all” (1, 2, 3, 4, 5) to “counting on from first” (2, 3, 4, 5) to “counting on from largest” (3, 4, 5), thereby becoming more efficient. In the latter case “3” is being seen as a procept: the process of counting threeand the outcome, “3”. For some pupils, however, the entire symbol 2 + 3 becomes both a process of counting fiveandthe outcome, “5”; that is, 2 + 3 is seen as another name for 5. It becomes a known fact, not memorised by rote, but meaningful as part of a concept image (Tall & Vinner, 1981) that includes the symbol “5”, word “five”, the image “ ”, the symbol “2 + 3”, and so on. Now consider the case of 12 + 3. Some pupils would use a counting strategy, with varying degrees of efficiency depending on the strategy used. Others would make use of the (meaningful) fact that 2 + 3 is 5, and then add 10. Such qualitatively different ways of seeing symbols makes a marked difference when unfamiliar types of problems are encountered. In new contexts, old ways of working, which are highly situated, can become obstacles to further development. Tall (2004) calls an old, situated
way of working a “met-before” (a pun onmetaphor) and provides the example from arithmetic that “every sum has an answer, for instance, 2 + 3 is 5” (p.286). This can be a problem when algebraic expressions, which contain letters and so lack an “answer”, are encountered in later schooling because there is no answer to be processed. For the proceptual thinker, however, 2 + 3 is both a process and the encapsulated object of that process, and so subsequent encounters with irreducible expressions, such as 2 + 3x, are less daunting.
3.3.1
A proceptual view of the equals sign
The place-indicator view of the equals sign (see Section 2.2) might be described as the “every equals sign is followed by an answer” met-before, which arises from the exclusive presentation ofexpression =numeral forms in typical classrooms (Section 2.4). As such, the statement 8 = 4 + 4, may seem strange:
[like saying] ‘cake the I ate’ instead of ‘I ate the cake’, the order is ‘not natural’, at least, not for the child who is thinking in terms of procedures rather than in terms of flexible procepts (Tall, 2008a).
For the proceptual child, however, the symbol 8 = 4 + 4 can be accepted as a statement of identity: the “every equals sign is followed by an answer” met- before has not caused an obstacle, because “answers” and “sums” are different symbols for the same object anyway.
From this vantage point, presenting varied statement forms throughout pri- mary schooling avoids the “every equals sign is followed by an answer” met- before in favour of an “is the same as” meaning (or “basic relational” view — see Section 2.3). However, researchers “must be careful not to conclude simply that making sure children are exposed to the various forms of equality sentences will remove [misconceptions of the equals sign]” (Behr et al., 1976, p.10). Just as there are qualitatively different ways of thinking about 2+3 and 12+3 (Gray, 1991), so contrived equality statements designed to appeal to structural read- ings are viewed in different ways by different pupils. For example, Gilmore and Bryant (2006) presented problems of the type 15 + 12−12 =,+ 7−7 = 13 and+ 14−9 = 18 to primary pupils and discovered use of the inversion prin- ciple (i.e. a and−acancel in the statements presented) was widespread, even amongst young and low-achieving pupils, but there was significant variation in how appropriately they applied it to different statement forms. Similarly, for
the case of long division, Anghileri (2006) found that primary pupils used a variety of strategies despite an emphasis in the National Numeracy Strategy on place-value partition.
Even when pupils do employ a given strategy they do not necessarily grasp the underlying arithmetical principle (Tall, 2001). Baroody and Gannon (1984) reported on a child who was asked whether presented pairs of terms, such as 6 + 3 and 3 + 6, produce the same result. The child performed a calculation for each term but, interestingly, used a “count-all starting with the larger term” strategy (p.81). On the one hand his use of a “start with the larger” strategy suggested a conception of commutativity (he viewed the order of addends to be irrelevant to outcome); on the other hand his need to compute both suggested not (he needed to compute results to satisfy himself of numerical sameness). In a follow-up interview, in which he was asked whether pairs of terms (such as 7 + 2 and 2 + 7)would produce the same result, he answered no, and so did not have a conception of commutation. However, when then asked to compute the same pairs of terms he again used a count-all strategy starting with the larger numeral. Baroody and Ginsburg argued this demonstratesprotocommutationin which the order of addends is sometimes ignored simply to save computational effort and with no regard for consistency of outcome.
To summarise, some pupils make use of computational shortcuts without seeing symbols, such as 2 + 4, as manipulable entities in their own right. They think in terms of processes rather than in terms of the encapsulation of processes. The “is the same as” meaning for the equals sign (see Section 2.4) does not explicitly promote working with encapsulated symbols, and so researchers present pupils with statement examples that appeal to non-computational readings. However, these appeals allow proceptual thinking but do not necessitate it. One problem is that the task of assessing or establishing numerical sameness promotes seeing equivalent symbols as different processes that generate the same object; but encapsulating processes and using those encapsulations as inputs to further thinking are optional. For some pupils this is a step too far and despite making increased use of notational shortcuts they continue to see symbols as processes. As such, tasks in which pupils establish the numerical sameness of contrived statements provide limited insights because they do not promote all the elements of a proceptual-symbolic equivalence conception. This issue is discussed further in the next section.