The circle is probably one of the oldest figures of mathematics (Kasner & Newman, 2001). It has characteristics of a non-straight line and is often regarded as a polygon with an infinite number of sides. Thus, a circle is a limit of the inscribed polygons. There are, in fact, some interesting generalisations of the circle when viewed this way. For example, Greek mathematicians such as Apollonius posed the classical circle problem: given three fixed circles, find a circle that touches them all. Teachers and learners could, through visualisation, try to methodologically fit a circle to touch all circles, starting with the smallest possible radius and using trial and error. The link between visualisation and imagination could offer both teachers and learners a more constructive approach to this problem. Imagination, in light of Wenger’s (1998) definition, could facilitate metacognitive thinking about the experience, to encourage reasoning towards a new knowledge of reality, or at least of circles. As this example suggests, creative problem-solvers engage in metacognitive thinking about strategies and monitor and evaluate their attempts. During this process the faculty of the imagination is employed to facilitate the visualisation of the problem-solving process. The rationale behind this argument is that mathematical models (in the case of Apollonius’s circles) are generated through problem-solving experiences and critical thinking which have become a cornerstone in visualisation. Visualisation then serves as a metacognitive skill (Gilbert, 2005). Yet, problem-solving skills require metacognitive awareness, which, in turn, predicts imagination (Liang, Hsu & Chang, 2013). Learning opportunities that are, therefore, framed around exploring, questioning, understanding and imaginative experiences are exemplary of the role metacognitive awareness plays in the imagination.

Theoretically, Franklin and Graesser (1999) clarify that precision is needed to explain what it means when a mathematical image is brought to mind. The image conjured up about Apollonius’s circles (mentioned above) preceded any explanation about radius, foci, area, circumference or diameter. Yet, the image conjured up by the mention of ‘circles’ is an implicit theory with particular visual and spatial content. Further examples of such an implicit theory of circles include wheel of fortune, circle of life, ring of fire, sphere of philosophy, to name but four. The theory behind these images is that they raise awareness of circle properties (or at least knowledge of its shape). However, when confronted with mathematical ideas, such as the properties of a circle, a learner will typically distinguish between three stages of this imaginative experience. Franklin and Graesser explain these stages as metacognitive awareness of (1) a conjured image, (2) a line drawing not necessarily labelled and (3) attached symbols that serve as a map or a plan with some symbolised spatial relation.

Visualisation, also called the representational view of the mind (Makina, 2010), integrates the mental processes made up of visual imagery, visual memory, visual processing, visual relationships, visual attention and visual imagination (p. 25). The aim, then, of teaching mathematics, with these visual functions in mind, is to support learners in the construction of mental representations of mathematical phenomena such that they develop an awareness of the underlying notions or concepts that the mathematical ideas develop from. Such an understanding not only brings to mind the imagined visual object (such as a circle), but also creates awareness of the knowledge of the person, task and strategies in order to provide a mental space in which these representations can be regulated. Mainly, visualisation should be considered as a fundamental aspect of mathematics learning and understanding (Makina, 2010) as learners learn to demonstrate their thinking and, thus, become metacognitively aware of the experiences this thinking involves. Such experiences could be reading for a particular purpose to select important words or phrases, describing them, providing proof of the knowledge they are accepting, examining or recognising. As a result, these representations can act as ways in which thinking and reasoning can be visualised, and acted out, or embodied, to extend the knowledge of one’s own understanding, the problem or task and the strategies to solve a particular problem. Imagination can therefore serve as a crucial faculty of the metacognitive awareness of one’s own and others’ mathematical understanding. The awareness, then, transposes to other higher order faculties, such as metacognitively planning, monitoring and evaluating one’s attempts in solving a mathematics task, and assists in directing one’s learning and problem-solving attempts.

The theory of embodied, situated and distributed cognition is applied as a lens to explore the role of metacognitive awareness, imagination and visualisation. This theory suggests we belong within a reality through actual engagement with that reality, and in the activities we engage in we develop the power of our imagination (Murray, 2011), for instance solving a mathematics task. In terms of visualisation, this means that learners might imagine themselves drawing a circle as part of an activity, becoming aware of an (often similar) experience in order to gain knowledge (or cognition) from that experience needed to solve the problem or create a mathematical model. According to theorists of imagination (Murray, 2013; Yueh, Jiang & Liang, 2014) and embodied, situated and distributed cognition (Clark, 1998; Nemirovsky & Ferrara, 2009), cognition becomes embodied when the mental mathematical imaginations (e.g. of a circle) are expressed physically, that is, outside of the mind. This can be through body poise, facial expressions, gestures, utterances or any other motor activity, like drawing a circle, which resembles some form of mathematical intuition.

This embodiment of mathematics can be harnessed in situated specific contexts. O’Connor and Aardema (2005), for example, explain that cognition is imagined interaction with the imaginary world. It is in this world that senses, acts and ideas are shaped to visualise mental mathematical models. There are, however, some doubts as to the applicability of situated cognition to imagination (Jansen, 2013). Reasons for this include the offline nature of metacognition and imagination where the object or model is absent, and needs to be conjured up in the mind, and therefore does not define any particular situation. As an example of the situatedness, Jagals and Van der Walt (2016) developed a mathematics task, based on an example by Fortunato and Hecht (1991), during which the image of circles needs to be imagined and then visualised in the context situation of the problem. Then there is also the argument that situatedness deals with the conscious rather than the imagination which requires a phenomenal consciousness (awareness about a particular phenomenon). Imagination therefore deals with the paradigmatic of higher cognitive levels, beyond awareness, towards intuition. It therefore seems that imagination also associates with the distribution of cognition. In this sense, knowledge of circles’ (mathematical) properties can be applied in calculations involving glass mirrors or Sacrobosco’s sphere, thereby distributing the knowledge from one domain to another. In another thread, an argument can be made that a circle, as a mathematical object, can be used to show the definitions and logic, thoughts and other non-mathematical ideas (as in the case of Venn diagrams), and therefore plays a role, in terms of its imagined image in the metacognitive awareness.

In developing an empirically categorised imagination scale Liang et al. (2013) identified initiating, conceiving and transforming as three types of imagination. In this study, however, to explore the role of metacognitive awareness and imagination in facilitating visualisation in solving mathematics tasks, these types of imagination are aligned against the theoretical framework of embodied, situated and distributed cognition (Clark, 1998). The theoretical framework thereby serves to inform a more conciliatory image of the role of metacognitive awareness and imagination in facilitating visualisation and the consequences for metacognition and mathematics education research.

Metacognitive awareness refers to reflecting on understanding and regulating knowledge (Schraw & Moshman, 1995). When the mathematical model is not immediately tangible and accessible (like the conjured image of a circle) it can be connected with through the power of the imagination (Murray, 2011). This connection, according to Wenger (1998) ‘needs an opening. It needs the willingness, freedom, energy, and time to expose ourselves to the exotic, move around’ (p. 185). When learners engage and experiment with reality, they develop the power of imagination. Wenger’s comments indicate a strong parallel between imagination and metacognitive awareness.

Initiating imagination involves the imaginative capability to explore and produce new, unfamiliar and unique ideas (Lin, Hsu & Liang, 2014). Learners who reflect deeply on themselves are more metacognitively aware. Perhaps this explains why new and novel ideas are often generated in solitude (Lin et al., 2014).

Conceiving imagination is the imaginative capability to understand mathematical ideas through personal intuition and using one’s senses (Lin et al., 2014). Typically, individuals who have a strong sense of conceiving imagination are capable to understand and conceptualise mathematical ideas through rehearsed concentration (and thereby memorise a formula or strategy) and follow logical steps or apply appropriate strategies through metacognitive awareness of the knowledge of these strategies.

Transforming imagination is the imaginative capability to identify a pattern, and to transform it into an abstract idea to distribute and apply the conceptualised understanding across different situations (Lin et al., 2014). Individuals who have a strong transforming imagination therefore tend to reflect on mathematical ideas from past experiences, become aware of these knowledge and strategies, and mimic the route of memorised strategies they employed in different situations. They can also transform one abstract idea into another by manipulating the abstract pattern, typically as a learner will manipulate the formula for the area of a circle (A = πr²) to calculate the circle’s radius.

In other words, the role of metacognition and the imagination becomes powerful in an environment which provides learners with personal mathematical independence. When reflecting on a problem-solving experience the imaginer can become metacognitively aware of the imaginings and can express these imaginings in the form of a drawing or written steps (embodying their cognition) as strategies are applied (Nemirovsky & Ferrara, 2009). Utterances such as drawings or written steps allow us to explore the processes whereby learners express their imaginations during visualisation. To do this, these imaginations through metacognitive awareness were explored across a collective case study based on a mathematics word problem in the case of the invisible circle.

Four instrumental case studies were conducted to explore the role of metacognition and imagination, to facilitate visualisation, towards embedded, situated and distributed cognition during visualisation of elementary Euclidean circle geometry. Each case study facilitated an understanding of the construction of metacognitive awareness on three levels, to be triangulated.

Four participants were purposively and conveniently invited to take part in an individual interview and included the top achievers from Grade 8 and Grade 9 in the senior phase having mathematics as a compulsory school subject. Since this was not a comparative study, we wanted to know how academically strong learners think and do mathematics and therefore asked teachers to identify participants who achieved 70% or higher during the previous year for mathematics. In addition to this criterion, teachers were asked to identify those top achievers who would be willing and comfortable to communicate about their thinking. According to literature, learners who perform well in mathematics can provide more accurate information about their thinking and problem-solving behaviour. If learners were identified with poor academic achievement and who could not explain their thinking and reasoning, then it would have limited the opportunity to collect data regarding metacognitive awareness. Teachers at the three participating schools identified these participants mainly because of their achievement in mathematics at the end of the previous school year. To enhance the trustworthiness of this small case study, one main criterion for sampling these learners was that their teachers had to identify them as learners who were not too shy to provide information about themselves, particularly about their thinking. The assumption was that having a high achievement in mathematics suggests they had the necessary knowledge and understanding of mathematical ideas to solve open ended and non-routine mathematics word problems.

Learner A, a 15-year-old girl, has a seemingly quiet nature. She had an average of 70% for mathematics at the end of Grade 8 and took 9 min and 20 s to solve the given word problem. Learner B was 15 years old and from an all-girls school. She smiles a lot and seemed to enjoy the discussion and questions in the interviews. She averaged 80% for mathematics at the end of Grade 7 and solved the word problem in 16 min and 58 s. Learner C was 14 years old. She seemed more interested in the study than were the other participants. She asked questions about why the study was done and whether she was allowed to ask questions during the interview. This learner had obtained an average of 76% at the end of Grade 7 and solved the word problem in the shortest amount of time, only 4 min. Learner D was a 13-year-old boy, the youngest, and from an all-boys school. His teachers considered him a top student, achieving 96% at the end of Grade 8. He took 9 min and 30 s to solve the problem, almost equal to Learner A.

The four instrumental case studies, each instrumental towards the collective case study, were conducted using: (1) a word problem based on the area of circles to transgress the mathematical ideas from invisible to visible, (2) metacognitive statement cards to elicit metacognitive awareness of the mathematical ideas underpinning the visualisation of circles, (3) observations of the utterances and gestures that pose the imaginative capabilities of the participants during visualisation, and (4) a collection of the images of participants’ own conjured mathematical models of the invisible circles on which the word problem was based. A brief outline on each of these instruments now follows.

Participants continuously referred back to the video recordings and reflected on their actions and the statements they made. In this sense we consider their statements as valid and trustworthy. The metacognitive statement cards also validated interview responses. To ensure trustworthiness the guidelines by Elliott, Fischer and Rennie (1999) were followed. These included: checking the interpretations of participants’ utterances with the participants and cross-checking the findings with the conceptual-theoretical framework of this study. In line with Curtin and Fossey (2007), we also offer a thick description of the empirical investigation and findings, and employ within-method triangulation where selection of metacognitive statement cards, the individual interviews on the problem-solving experience, observations, and images serve as different methods within the same methodological approach.

Permission and ethical clearance was sought and obtained from the Department of Basic Education, and the university’s ethics committee (reference NWU-00043-11-A2) in which the initiated study was proposed; learners’ parents and school principals also gave consent. Participants were informed about the aim of the investigation and could have withdrawn at any moment, although none did. Participants’ identities were protected by using pseudonyms (e.g. Learner A).

Data were collected for each instrumental case study through four research instruments. First, inductive analysis was conducted regarding the selection of metacognitive statement cards after problem solving. A priori analysis was then conducted on the transcriptions at interview level with a focus on what the learners were thinking, and why they did what they had done. This was followed by identifying the utterances or gestures that relate to imaginative capabilities as observed through video playback. Together with these recordings, we also collected particular ‘screen shots’ of the learners’ mathematical models to represent the circle images of learners’ imaginations, to showcase how they have conjured and imagined distributed mathematical ideas regarding the elements of a circle. Together, the following instrumental cases were chosen because they are thought to be instrumentally useful towards the aim of this investigation.

Each statement card represented a metacognitive awareness statement relating to either embodied, situated or distributed cognition. The video recording was played back to identify the metacognitive statements that were used during particular moments where mathematical ideas were expressed through gestures and utterances. Imaginative capabilities and metacognitive awareness were identified by visible (observable) regulatory actions such as starting the next step, getting an answer on the calculator or rereading the question.

Interviews were first transcribed verbatim, then entered into the computer as a Word document and saved using pseudonyms for the participants (Learners A, B, C and D). Second, a priori codes were identified based on the conceptual-theoretical framework and categorised in the transcriptions. We read and reread the transcriptions and labelled related sentences or words according to the a priori codes. Particular sentences, words or phrases were then cross tabulated according to the a priori codes and were subsequently compared to determine if there was a pattern or structure in participants’ visualisation, metacognitive awareness or imaginative capabilities. Codes represented in the transcriptions (in terms of responses and phrases) were summarised and then tabled in four comparing columns. The columns included the identified codes for each participant. Patterns emerging from the data were compared between participants’ overall visualisation. Care was taken to identify which metacognitive awareness and imaginative capabilities the participants exhibited. Similar patterns for the different themes were joined together and comparisons were made between all participants and relevant themes.

Learner A solved the mathematics problem in 9 min and 20 s. She drew three circles as illustrated in Figure 1, and labelled them in the centers as A, B and C. She circled the middle circle, labelled A, again and reread the question before making a gesture by pointing with her pen from one side of circle A to the other, as if drawing a line through A, and in doing so embodied the mathematical idea of the circle’s diameter.

The image that Learner A conjured portrays her mathematical model for the three circles. Even though she labelled them clearly, she did not attach any symbols to suggest possible mathematical ideas about each circle. She read and reread the problem again, stopped and then drew her own version of what she had read through her conceiving imaginative capability. She also applied appropriate strategies by rereading the question, concentrating more and reading longer, while looking up at her written work and drawings. She paused looking at her work and then looked away from her work, staring in the air. She pointed with a pen towards her written and sketched work while reading the word problem. After every sketch or step, she paused and scanned her page from top to bottom. A seemingly major challenge was when she became aware that she ‘couldn’t remember the formula’ to calculate the area of a circle (Jagals, 2013, p. 127). Trying three different strategies, she reflected on her knowledge and practices from experiences. The three strategies included: trying half of the diameter, then drawing a big circle with a shaded area, and eventually writing the word area on a piece of paper and underlining it. This provided the situated cognition through which she then remembered the formula and wrote it down. She explains that ‘I think that’s why I underlined the word area. … I then remembered the formula for the area’ (p. 127). Almost immediately after getting the answer the learner made another attempt to distribute her cognition by using the formula known to calculate the circumference of a circle. She compared the answers and where deemed necessary tried another way to solve the problem. She identified the formula as a pattern and adapted it to calculate the area. She then wrote her final answer for the area of circle C as 78,5 cm² (Jagals, 2013, p. 127).

Learner B solved the problem in 16 min and 58 s. Unlike Learner A, this participant first read the whole problem three times before drawing the circles as illustrated in Figure 2a. She seemed to have a calm and relaxed approach, doing what she does slowly and double-checking her work regularly. She first drew circle A and then labelled it. Then she drew circles B and C and labelled them.

After drawing each circle, she read the question again and attached symbols to the circles by using information. She appeared to read some parts more and longer, stopped, read again. When the video recording was played back, the researcher asked at this stage what the learner was doing, and she commented: ‘I was putting like into a picture format, yes, like individual, like a, b, c’ (Jagals, 2013, p. 133). She moved closer to the question on the desk. She read it, pointing at some parts of the question and looking up at her drawings, then back to the question. She moved the calculator slightly out of the way and began writing A =. She paused while looking at circle C and then finished the first step with a substituted value for π. When the researcher asked the learner why she was pausing, she said:

This learner’s completed visual formula is depicted in Figure 2b.

She paused, sat back in her chair and then erased the exponent (2) as well as the radius (r) in the formula (refer to Figure 2b), wrote times r, hesitated (paused) and placed the square at π as illustrated in Figure 2c.

At this stage, the learner commented: ‘I was very uncertain there’. She paused again, pen between the lips, and after some time corrected the formula. After substituting the value for π and the radius into the second step and using a calculator to get the answer, the learner evaluated her answer. She recalculated using other strategies and became aware that π was wrongly substituted. She corrected this and wrote the final answer as 19,625 cm² (Jagals, 2013, p. 134).

Learner C solved the problem effortlessly and, almost, mechanically, in 4 min. She read the question only partly, concentrating on some particular aspects at a given moment. The steps were taken without doubting the formula, units or substituted values – unlike Learner A or Learner B. After reading the question, she drew three circles of similar sizes as depicted in Figure 3.

Without showing lines to picture radius or diameter, she wrote down the given and mentally calculated information next to the matching circles. The learner calculated or deduced the information for circle B and circle C mentally, not showing any written work for her conclusions about the diameter or radii. She read information every time before she wrote something, thus breaking the question up into smaller manageable parts. While referring to the video recording afterwards, the learner had difficulty describing what she had done. She kept her answers short and sounded uncertain. The dialogue between the learner and researcher explains this:

Her response had a futile motive. She commented on a particular step: ‘I was writing and then I closed the pen and then I wanted to work out the sum’. Learner C was not aware of what she had done nor could she provide clear reasons for doing what she did. She monitored her work less often than the other participants did, and only reread the question once. She also did not evaluate her answer. She did, however, explain that ‘it was bothering me that there were no labels’ suggesting that she was metacognitively aware of the imaginative capability that the problem requires of her.

Learner D started drawing the circles almost immediately after being given the word problem as illustrated in Figure 4. Although all four participants drew the three circles and filled in information about each circle, Learner D only included information for circle A. He also did not make use of algorithms or so-called steps; instead he conjured a picture, wrote descriptive words as labels and did mental calculations.

Circles B and C were just drawn and labelled but no symbols were attached inside or outside these circles. He claims that: ‘I drew it, what they said it is, I drew it on. So if it is a big circle, I draw a big circle if it is a small circle I draw a small circle’. After drawing the circles, he started calculating the area by first making sure he understood the given information, which was not written routinely. He wrote the formula for the area of a circle and continued to substitute the values for π as well as the predetermined mentally calculated value for the radius of circle C. Learner D seemed surprised when he wrote down the answer as shown on the calculator’s screen. This fraction led him to solve the problem, manually, by doing long division.