Number patterns presented as a sequence of pictorial terms have the potential to open up meaningful spaces for classroom exploration and discussion. However, despite the potential richness of such pictorial contexts, potentially meaningful pattern generalisation activities carried out in the classroom often become degraded to simple rote exercises in which the given pictorial sequence is simply reduced to an equivalent numeric sequence. As such, the generalisation process becomes a somewhat superficial or mechanistic exercise using set algorithmic methods. Whilst such an approach may well be successful in arriving at the correct general formula, the potential for genuine mathematical exploration offered by the context of the question is largely lost, with the generalisation process becoming ‘an activity in its own right and not a means through which insights are gained into the original mathematical situation’ (Hewitt, 1992, p. 7). As Thornton (2001) remarks, the danger with such an approach is that the focus becomes ‘… the development of an algebraic relationship, rather than the development of a sense of generality’ (p. 252), the result being little more than disconnected algebraic formulation. This disconnection from the original context becomes particularly problematic when importance is placed on the justification or validation of the general rule, since algebraic expressions arrived at through this process become ‘statements about the results rather than the mathematical situation from which they came’ (Hewitt, 1992, p. 7). This is unfortunate since not only can generalisation of pictorial patterns lead to different but algebraically equivalent expressions of generality, thereby opening up excellent opportunities to engage with the notion of algebraic equivalence, but, as Orton (2004, p. 114) points out, justifying pattern generalisations provides pupils with legitimate and valuable experiences of proof en route to more formal mathematical proofs.

Number patterns presented in the form of a sequence of pictorial terms are thus far more than simply a visual representation of a given numeric pattern. In essence, the use of a pictorial context aims to exploit the visual decoding of the pictorial sequence to give meaning to the symbolic expressions constructed. Two critical aspects of this process are the ability not only to grasp in a meaningful way the perceived underlying structure of the pictorial context, but also the ability to use this structure to articulate a direct expression for the general term of the sequence.

FIGURE 1: A simple geometrical figure.

If we now take the same image but place it in the context of a pictorial sequence, then the visual scenario is profoundly altered (Figure 2).

FIGURE 2: A simple pictorial sequence.

A further dimension is added to the visual image shown in Figure 2 when the pictorial sequence is used as a referential context for finding an algebraic expression for the number of lines in the nth diagram (or term) of the sequence. Although the context remains the same as that represented in Figure 2, the process of generalisation provides a further layer of complexity as it necessitates not only the perception of the figure within the context of a sequence of similar figures, but it requires the perception of generality, the notion that the figures in the sequence have a related structure. Finally, there is a requirement that this perceived generality must be articulated in such a way that it can be written in the form of an algebraic expression. A critical aspect of the visualisation process thus relates to the usefulness or meaningfulness of the perceived structure of the image in terms of the extent to which this perceived structure supports or hinders the process of generalisation.

Drawing on the nomenclature used by Fischbein (1993), figures such as the image shown in Figure 1 could be said to contain figural properties. What one sees in the image is a result of the Gestalt laws of figural organisation (Helson, 1933; Katz, 1951; Wertheimer, 1938; Zusne, 1970, pp. 111−135). Images such as that shown in Figure 2 could be said to contain both figural properties and conceptual qualities. What one sees in the image is still based on the Gestalt laws of figural organisation, but this is further influenced by the additional conceptual qualities of the image that have been added by virtue of the image being contextualised, in this case within a sequence of similar images. The critical point here is that an underlying tension is likely to pervade visual strategies applied to pictorial pattern generalisation tasks as a result of the relationship between the figural properties and conceptual qualities of the given images.

By way of example, consider Figure 3, which shows Term 3 and Term 5 of a pictorial sequence¹.

FIGURE 3: Term 3 and Term 5 of a typical pictorial sequence.

FIGURE 4: Local visualisation of Term 5.

By way of example (see Figure 5), one could subdivide each term of the given pictorial context into an upper row of n horizontal matches, a lower row of n horizontal matches, and a central row of n + 1 vertical matches, thus yielding the general formula T_n = 2n + (n + 1). Alternatively, each term could be seen to contain n overlapping squares, each made up of four matches, thus giving a count of 4n matches. However, this would result an overcount because there are n − 1 overlaps. Correcting for this overcount gives the final formula T_n = 4n − (n − 1). These two different global visualisations are illustrated in Figure 5.

FIGURE 5: Global visualisation of Term 3.

The basic tenet of enactivism is that there is no division between mind and body, and thus no separation between cognition and any other kind of activity. Within an enactivist framework, there is a purposeful blurring of the line between thought and behaviour (Davis, 1997, p. 370), and cognition is thus viewed as an embodied and co-emergent interactive process. From this theoretical stance, as Davis (1995, p. 4) points out, language and action are not merely outward manifestations of internal workings, but should rather be seen as ‘visible aspects of … embodied (enacted) understandings’. Radford’s (2008) theoretical construct of knowledge objectification resonates strongly with an enactivist theoretical framework since it foregrounds the phenomenological and semiotic aspects of figural pattern generalisation. Knowledge objectification thus represents an ideal theoretical construct to critically engage with pupils’ whole-body experience and expression whilst they explore the potentialities afforded by a given pictorial pattern generalisation task.

Knowledge objectification is a theoretical construct to account for the manner in which learners engage or interact with a given scenario or context in order to make sense of it en route to a stable form of awareness (Radford, 2006, p. 7). Knowledge objectification is premised on two notions. Firstly, semiotic means such as gestures, rhythm and speech are not simply epiphenomena, but are seen as playing a fundamental role in the formation of knowledge (Radford, 2005a, p. 142). Secondly, in order to study the process of knowledge production one needs to pay close scrutiny to multiple means of objectification, for example words, linguistic devices, gestures, rhythm, graphics and the use of artefacts, where ‘…meaning is forged out of the interplay of various semiotic systems’ (Radford, 2005b, p. 144). It is through this multi-systemic, semiotic-mediated activity that the objects of perception, or rather the objects of knowledge, progressively emerge – a process of ‘concept-noticing and sense-making’ (Radford, 2006, p. 15). Importantly, from an enactivist stance, use of the word ‘object’ by no means suggests that these ‘objects’ are pre-existing properties inherent in the environment. Rather, the ‘objects’ of perception are brought forth through the co-determination of knower and known, the co-evolution of individual pupils and their surroundings.

The data analysis was guided by an enactivist methodological framework in which the researcher and research environment are seen to co-emerge (Reid, 2002). This interdependence of researcher and context was characterised by a flexible and dynamic process of investigation (Trigueros & Lozano, 2007). The iterative and reflexive process of co-emergence was built up over time through the use of multiple perspectives and the continuous refinement of methods and data analysis protocols. Audio-visual data was examined repeatedly in different forms (e.g. video and transcript) and in conjunction with additional data retrieved from field-notes and participants’ worksheets. In addition, nodes of activity that seemed particularly interesting were identified and meticulously characterised with reference to the various semiotic means of objectification in the form of descriptive vignettes.

From a more philosophical standpoint, there is also an important ethical consideration stemming from the enactivist theoretical underpinnings of this study. In enactivist terms we need to be sensitive to the notion that ‘… our actions have the potential to alter the worlds and possibilities of others’ (Simmt, 2000, p. 158). Furthermore, an enactivist stance compels us to see each person’s certainty as being ‘… as legitimate and valid as our own’ (Maturana & Varela, 1998, p. 245). Sensitivity to both of these ethical considerations was maintained throughout the study.

It has been shown that patterning tasks presented with consecutive terms encourage a recursive strategy (Hershkowitz et al., 2002; Samson, 2007) and thus tend to draw attention away from global structural features that could potentially co-emerge through the interaction of a perceiver and the given figural context. Since not all patterning tasks could be presented unambiguously using a single term, it was decided to use two non-consecutive terms for all questions. This purposeful decision was thus not intended to encourage global visualisation per se, but rather to ensure that the potential for global engagement with the pictorial context was not discouraged by virtue of terms being presented consecutively.

In addition, data collection and analysis protocols were sensitive to the enactivist underpinnings of the study, and thus made use of multiple data sources and approaches to data handling (as previously outlined). This process acted as a form of triangulation, which sought to ensure validity.

FIGURE 6: Pictorial pattern presented to Terry.

It seems like it’s basically just adding on the same sort of thing again, every time, and then just finishing it off with that [indicating the rightmost > shape]. (Samson, 2011b, p. 131)

After making this remark he carefully drew Term 4 in a very structured manner. He began by drawing the three matches of the leftmost triangle. Thereafter he drew the middle section of the structure in a very rhythmic fashion: 4,5,6 … 7,8,9 … 10,11,12 … 13,14,15. Interestingly, instead of drawing each group of three matches in the flowing form of a backward C-shape (top match, then vertical match, then bottom match) which would have been slightly more economical, he instead methodically drew each group of three matches by first drawing the top horizontal match, then the bottom horizontal match, and then finally the vertical match. To check that he had drawn the correct number of matches for Term 4, he then carefully counted the four ‘squares’ in the diagram before adding on the two oblique matches on the far right. After completing the middle section of the diagram, he drew a series of inverted V-shapes across the top of the structure (rhythmically drawing them in pairs from left to right) and finished the diagram by drawing a series of V-shapes along the bottom of the diagram, once again in rhythmic pairs from left to right (Figure 7).

FIGURE 7: Terry’s drawing procedure for Term 4.

So you started off with your little triangle [indicating the leftmost < shape] so that’s obviously +2, then you finish it off with a little triangle again [indicating the rightmost > shape], plus another 2, so it’s +4; and then however many things in between, just work out how many it is for that [indicating the 7-match additive unit] (…) well how many it is for the top triangle, bottom triangle and then plus that one [indicating the vertical match connecting the top and bottom triangles in the 7-match additive unit]. (Samson, 2011b, p. 132)

At this point Terry wrote down the formula 7n + 4. However he quickly realised that he had not taken into account the first vertical match from the left:

Shape 3 is Term 3, it’s got 3 of these little squares like that [pointing to the three central squares of Term 3] and Shape 5 has 5. So then you’ve got these 2 [indicating the leftmost < shape] so you start off, there’s +2, there’s another 2 [indicating the rightmost > shape] plus 4, then you’ve already got [points to the leftmost vertical match and realises he has missed it out] – oi! (…) With the front triangle [indicating the triangle formed from the leftmost < shape and the leftmost vertical match] it’s the full triangle that you’re starting off with, so it’s 1, 2, 3 matchsticks. With the end one you’ve already got the (…) base of the triangle coming from the previous square. (Samson, 2011b, pp. 132−133)

Terry thus gave his final formula as 7n + 5 which he subsequently altered to 3 + n(3 + 3 + 1) + 2 as being more representative of how he was visualising the pictorial context (Figure 8).

FIGURE 8: Two different formats of Terry’s initial visualisation.

An interesting aspect of Terry’s discussion is his frequent reference to squares in the pictorial terms. He makes express reference to the fact that the nth term in the sequence would contain n ‘little squares’. In addition, when he initially drew Term 4 he did so by drawing the central structure first and then checking that he had drawn the correct number of matches by quickly counting the four squares. However, these squares do not feature anywhere in either of the two versions of his initial formula. In fact, the horizontal matches from these squares are seen to form part of an upper and lower triangle. Thus, after visually deconstructing the diagram into triangles, the squares become negative space as the matches that originally formed them have been apportioned to different component parts. Nonetheless, Terry continued to refer to them as a helpful structural unit. There are two possible reasons for this that are worth considering. Firstly, the ultimate aim of the patterning task from Terry’s perspective is to arrive at an algebraic expression for the general term through a process of visualisation. It is possible that this goal had an unconscious influence on the visualisation process since some visualisations would be algebraically more useful than others – for example, squares would overlap and a correction would thus be necessary for the resulting overcount. A second possibility is that there is a tension between local and global aspects of the pictorial context. Local considerations focus on the additive unit by virtue of attention being focused on the step-by-step process of constructing the next term from the previous one. It is possible that this local focus obscured a more global outlook where the structural unit of a square could be properly incorporated into the general expression.

What I’m trying to do now is almost use the squares. So now instead of having that sort of backwards C, actually have a full-on square (…) that gets connected to another square [gesturing to the right with his pencil], that then, just got to take out that one [indicating the overlapping match between two squares], and gets connected to another square [making multiple gestures further to the right with his pencil]. (Samson, 2011b, p. 134)

Whilst Terry was explaining his strategy, he made use of a number of crucial semiotic means of objectification. The first of these was his gesturing to the right whilst saying the words ‘that gets connected to another square’. This indexical or deictic gesturing was specifically related to Term 3, which Terry had in front of him. He thus used the gestures to signify existing physical structures in the particular diagram he was looking at. His second set of gestures, accompanying the words ‘and gets connected to another square’ mark a transition from existential signification to what Sabena, Radford and Bardini (2005, p. 134) refer to as imaginative signification. This second set of gestures moves from indicating materially instantiated aspects of the pictorial term to miming an ongoing sequence of connected squares, squares that are not yet materially present. We thus see a progressive distancing from the physical referent. Another important aspect of Terry’s objectification process is his use of the words ‘another square’. These words serve an important generative action function in terms of objectifying the generality of the interconnecting squares through an imaginative conception of iterative potential action. This linguistic device supports the process of objectification by allowing the recursive addition of squares to be ‘… repeatedly undertaken in thought’ (Radford, 2000, p. 248). However, and critically important in terms of the local-global visual tension, the words ’and gets connected to another square’ foreground an iterative process. Thus, whilst supporting an important generative action function in terms of objectifying the generality of the structural unit of the square, they also tend to focus attention on the recursive nature of step-by-step construction, thus drawing attention away from a more holistic view of the overall general structure.

Terry then went on to draw Term 4. Interestingly, the order in which he drew the various lines did not seem to correlate with his description of overlapping squares. Instead, his drawing process seemed to suggest a subdivision into a triangle at either end, two rows of horizontal matches, a row of vertical matches, and V-shapes at the top and bottom.

After completing his drawing of Term 4, Terry sat staring at it for just over a minute before commenting: ‘I had something and now I’ve, I had something else but now I’ve lost it.’ It thus seems that his initial idea of using overlapping squares came from a flash of insight that has since receded. It is possible that this may have been at least partially precipitated by Terry’s drawing of Term 4 in a manner which did not mimic his initial visualisation of overlapping squares. In this instance, it is possible that the drawing process itself obfuscated the visual apprehension. However, another interpretation of the data could suggest that the drawing process actually reflects a competing, albeit unconscious, visualisation of the pictorial terms, thus suggesting an underlying visual tension between two different apprehensions of the pictorial context.

Terry then came up with the formula 3 + n(4 − 1) + 4n + 2. The ‘3’ at the beginning of the formula represents the starting triangle on the far left whilst the ‘+2’ at the end of the formula represents the > shape at the extreme right of each term. Terry described the n(4 − 1) portion of his formula as representing ‘each square minus the one that’s being taken up by either the previous one or the next one’ whilst the 4n is required for ‘the triangles above and below it’. Although he specifically refers to ‘squares’, these structural features are not reflected in his present visualisation. Although his initial visualisation was suggestive of overlapping squares, he has in essence reverted to a previous visualisation in which the central structure is seen not in terms of overlapping squares but rather in terms of a series of backward C-shapes (Figure 9).

FIGURE 9: Terry’s change in visual apprehension to a local visualisation.

At this point I asked Terry what had happened to his initial idea of focusing on the squares:

I don’t know, I had something … I was busy looking at it and something hit me and then I lost it. I noticed something to do with n minus, open brackets n minus 1, and then that in brackets [i.e. (n − (n − 1))], that had something to do with it, but I cannot for the life of me remember what it was. (Samson, 2011b, p. 137)

Terry’s reference to his noticing something to do with (n − (n − 1)) does not initially seem to make any sense as the expression simplifies to +1. However, it retains an interesting remnant of his initial visualisation in which there are n overlapping squares with n − 1 overlaps. At my suggestion he continued to pursue his initial idea. After staring at the diagrams for about half a minute he commented:

I think I might have found it … So what I’m trying to do is now, is almost separate it so you’ve got, you just put all the squares together (…) and then take out this extra match right at the end [pointing in turn to each of the three overlaps between the four squares in Term 4]. (Samson, 2011b, p. 137)

This marks the crucial moment when Terry changes from a local to a global visualisation (Figure 10) and is thus able to make sense of, and articulate, his initial fleeting visualisation. After trying to incorporate (n − (n − 1)) into his general expression, he eventually abandoned it and came up with the final formula 2 + 4n − (n − 1) + 4n + 2:

That works, then you’ve got your 2 that starts it off [indicating the leftmost < shape], your 2 that finishes it off [indicating the rightmost > shape], you’ve got your four for each square, then the − (n − 1) (…) for each square there’s an extra line except for the first (…) then + 4n for each triangle above and below it. (Samson, 2011b, p. 137)

FIGURE 10: Terry’s final global visualisation of overlapping squares.

Although competing visualisations have been shown to cause tension, the crucial aspect relates to the process of coming to realise how the visualisation is regular, and how this regularity can be expressed in an algebraically useful manner. Thus, although both local and global visualisations can be useful in their own particular way, it is likely that the process of objectifying and articulating an appropriate algebraic expression for the general term is complicated when tension exists between these two modes of visualisation. An awareness of and appreciation for these subtle tensions has the potential to provide an added depth of engagement for a sensitive practitioner.

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