Awareness of one’s own strengths and weaknesses during visualisation is often initiated by the imagination – the faculty for intuitively visualising and modelling an object. Towards exploring the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task, four secondary schools in the North West province of South Africa were selected for instrumental case studies. Understanding how mathematical objects are modelled in the mind may explain the transfer of the mathematical ideas between metacognitive awareness and the rigour of the imaginer’s mental images. From each school, a top achiever in mathematics was invited to an individual interview (

There are not many references to imagination in metacognition research, particularly in the context of mathematics education. There are, however, publications that illuminate the importance of imagination in fields related to mathematics education. Some examples of these fields include physical science and technology education (e.g. Nemirovsky & Ferrara,

The scarcity of accepting the applicability of imagination in the field of mathematics could be because imagination received a bad reputation in Western philosophy where some scholars believed that imagination is inferior to reason (e.g. Spinoza’s naturalistic theory of imagination) while others (who agree with Hume’s theory of imagination) argue that imagination constrains metaphysics and makes way for scientific reasoning. We hold on to Wenger’s (

A holistic view of mathematics aspires to a more imaginative approach to teaching and learning mathematics. In the sections that follow, the inclusion of the imagination as part of the holistic view of mathematics education is explored in terms of the role metacognition and imagination play to facilitate visualisation about a Euclidean geometry task. In contrast to the rational view of mathematics, we suspected that learners who follow such a holistic approach would more likely be able to become aware of their own strengths and weaknesses during problem solving. As part of a larger project (Jagals,

The conceptual and theoretical framework on which the study was based and the empirical design of the study that was done follows. The results informed the role of imagination in metacognitive awareness during a mathematics visualisation task. In essence, the findings portray the results of four instrumental case studies and a discussion in view of the theoretical framework. This is concluded by some guidelines for classroom practice and recommendations for future consideration.

Imagination can serve as an indispensable tool for unlocking and discovering reality, and the mathematical ideas that surround reality’s hidden structures. Without knowing how to ‘see’ a geometric point, or without awareness of the fact that a geometrical plane has no thickness, or that a series of points about the same axis can form a circle rather than a straight line, a personal mistrust in one’s thinking and awareness about reality (Schoenfeld,

The circle is probably one of the oldest figures of mathematics (Kasner & Newman,

Theoretically, Franklin and Graesser (

Visualisation, also called the representational view of the mind (Makina,

The cognitive processes related to the expression of metacognitive awareness are embodied, situated and distributed in nature.

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,

This embodiment of mathematics can be harnessed in situated specific contexts. O’Connor and Aardema (

In developing an empirically categorised imagination scale Liang et al. (

Metacognitive awareness refers to reflecting on understanding and regulating knowledge (Schraw & Moshman,

^{2}) 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,

In order to explore the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task, a predominantly qualitative approach with a collective case study design was followed. An interpretivist perspective allowed us to explore the role of metacognition and imagination to facilitate visualisation, by means of the tenets of metacognitive awareness. The tenets show how the conceptual and theoretical frameworks of this study are intertwined and the empirical investigation that follows explores the significance of these tenets further.

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.

The idea of the task was initiated by Jagals (

Suppose a circle’s diameter is 20 cm. This is also four times the radius of a second circle. Calculate the area of a third circle if the third circle’s radius is half of the second one. (p. 158)

The task was inspired by a similar activity by Fortunato and Hecht (

Before solving the problem, each participant received a set of 15 metacognitive statement cards, taken from the idea by Wilson (

Careful observation was used to note any utterances and gestures or movements with hands and arms that can indicate the embodiment of mathematical ideas. Mainly, gestures of a rest position, stroke, pointing, preparation or movement about the work place, retraction and movement of hands, or movement of the lips and other relevant body language which represented the embodiment of cognition of mathematical ideas served as criteria for noting a gesture or utterance.

Since the word problem was meant to elicit imaginative capabilities, it was anticipated that participants will conjure images of the elements of a circles (e.g. radius, chord, circumference and diameter) which, at the end, served as a mathematical model for the word problem. Each participant’s conjured images served as evidence of the situated and distributed cognition instigated by the information provided in the word problem.

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 (

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.

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,

Following is a narrative account on each of the cases’ findings, flowing from the four instrumental case studies. In each case, tenets of metacognitive awareness (by Franklin & Graesser,

Learner A solved the mathematics problem in 9 min and 20 s. She drew three circles as illustrated in

Learner A’s mathematical model for the three circles.

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, ^{2} (Jagals,

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

(a) Learner B’s mathematical model for the three circles, (b) Learner B’s mathematical formula to calculate the area of a circle and (c) Learner B’s mathematical formula to calculate the area of a circle after changes (made to

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,

I wanted to write my formula so I went back to check what are we doing, what area, perimeter or volume, so then I paused to double check; okay what is the area, so I specifically went to look for, if it’s area. (Jagals,

This learner’s completed visual formula is depicted in

She paused, sat back in her chair and then erased the exponent (2) as well as the radius (

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 ^{2} (Jagals,

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

Learner C’s mathematical model for the three circles.

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:

Why did you read the question again?

I was looking at this [

What were you doing there?

(Jagals,

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

Learner D’s mathematical model for the three circles.

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

The purpose of this research was to explore the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task. References to metacognitive awareness in mathematics education research often fail to acknowledge the imagination as a key role player in the cognitive processes during visualisation. Some studies have shown that teachers’ metacognition is not adequate to model metacognitive awareness and undertones the vagueness in our understanding of the imagination as a faculty of self-directed learning, as it is rarely promoted in mathematics classrooms (Van der Walt et al.,

We therefore contemplated the visualisation activities learners engage in can prompt their imaginative capabilities. Originating from this investigation, and in line with the model by Lin et al. (

The four instrumental case studies collectively reflect this theoretical stance and support the role which the tenets of metacognitive awareness and visualisation, through embodied, situated and distributed cognition in the imagination, plays in facilitating visualisation in solving a mathematics task. To elucidate these findings,

Summary of findings that reflect the tenets of metacognitive awareness and imagination.

Metacognitive awareness of: | Examples of metacognitive statement cards | Link with theory | Examples of indicators of imagination through utterances and gestures | Learner |
---|---|---|---|---|

Knowledge of the person/self | I thought I cannot do it | Embodied | I went back to the question to read the second part again. | A, B |

Knowledge of the task | I thought I know this sort of problem | Situated | They did not give you the fact, I had to make it up for myself. | A, C, D |

I thought about what I already know | Distributed | …and then I put a question mark. | A, B, C, D | |

I tried to remember if I had solved a problem like this before | Distributed | I think I looked at that because it looked similar. | A, B, C, D | |

Knowledge of the skills and/or strategies | I thought I know what to do | Situated | I was not sure about the whole thing. | A, C, D |

I thought about something I had done in the past that was helpful | Distributed | I could not remember what was the difference between circumference and area, and then I remembered that area has a squared at the end. | A, C | |

I thought about a different way to solve the problem | Distributed | I tried to make other formulas… and do other things as well. | A | |

Regulation of understanding | I read the question more than once | Embodied | I read that like three times. | A, B, C, D |

I drew a diagram to better understand the questions | Embodied | If it is a big circle, I draw a big circle and if it is a small circle, I draw a small circle. | A, B, C, D | |

Regulation of planning | I changed the way I was working | Situated | I then used plan B to do long multiplication. | A, B, D |

I thought about what I would do next | Embodied | …and that gave me a fraction sign so I was confused, I was like what? And then I kept looking for a decimal. | A, B, C | |

Regulation of monitoring | I thought about how I was doing | Embodied | I was checking if I am right because I wrote it down. | A, B, C |

I checked my answer as I was working | Embodied | Here I wondered if this is right or am I doing something wrong. | A, D | |

I thought about whether what I was doing was working | Embodied | I was making a dot in the middle. I do not know why I did that, maybe just to make sure that I was right. | A, B, C | |

Regulation of evaluation | I thought, is this right? | Embodied | I paused … I was checking if I was on the right track. | A, C, D |

Based on these tenets, it seems reasonable to suggest that metacognitive awareness and imaginative capabilities can serve as guiding principles embedded in visualisation tasks. To do so, the tenets need to be incorporated in the development of appropriate activities (such as open-ended non-routine word problems) to elicit the necessary mental images that will conjure metacognitive awareness. Each of these underlying tenets seems to advance, collectively, the rational debate about mathematics, promoting a more imaginative mathematics.

In respect of the role of metacognitive awareness of the learners, all cases were joined and overlapping with Franklin and Graesser (

Visualisation research has recently received increasing attention with a focus on metacognitive awareness and visualisations, including imagination. Since imagination is a scarce topic in metacognition research, a recent emphasis in the South African school mathematics curriculum prompts mental images, predictions and visualising of thoughts and decisions to promote self-directed learning as imaginative states of mind. The conceptual link, therefore, between these imaginative states of mind and the cognitive structures that draw on mathematical ideas seem to relate to a demand for exploring the conceptual understanding of the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task. The significance of this study is the close connection between metacognition and imagination as portrayed through the tenets of metacognitive awareness to illustrate the role it plays during problem-solving tasks. Educators need to design learning environments that support learners’ metacognitive development and encourage them to engage their imaginations in the learning process. In mainstream mathematics, the holder of a rational view will need to reflect on the cognitive demands of the problem and the associated tenets of the imaginative capabilities to solve circle geometry problems that are non-routine and open-ended in nature. The beholder of a more imaginative and holistic view, however, should reflect on at least two kinds of imagination:

The authors acknowledge funding from the National Research Foundation. Views expressed are not necessarily those of the National Research Foundation.

The authors declare that no significant competing financial, professional or personal interests might have influenced the performance or presentation of the work described in this manuscript.

D.J. conceived of the presented idea, developed the theory and performed data collection and analysis. M.v.d.W. verified the analytical methods, and the tenets that emerged from them, and supervised the findings of this work. Both authors discussed the findings and contributed to the final manuscript.