This article uses parts of qualitative data from the first author’s study that focused on exploring Pirie and Kieren’s process of

A few studies have focused on geometric reasoning (Battista,

Nevertheless, there are difficulties learners encounter in geometry (Fujita, Yutaka, Kunimune, & Jones,

This article is guided by Pirie and Kieren’s (

Primitive knowing refers to the knowledge that learners bring to their learning context as prior knowledge.

Image making is the level where a learner engages in activities intended for developing ideas and images for a concept (Martin et al.,

Image having is a layer where learners can use an understanding of a topic to help them to make mental plans that can be used when working on mathematical tasks (Martin et al.,

Property noticing deals with learners’ examination of an image for relevant properties.

Formalising is a level where learners consciously notice properties and work with them.

The nesting illustrates that the growth in mathematical understanding is neither linear nor unidirectional (Pirie & Martin,

The framework for describing

Each of the three elements explains some practice of

The second of the three acts of

This knowledge and understandings are used to re-view and re-read concepts to meet the needs of the task at hand. Furthermore,

While working at an inner layer using existing understanding, collecting at an inner layer, and moving out of topic and working there result in thickening or further and enriched understanding of an existing concept, the act of causing a discontinuity in

The three forms of

A teaching experiment methodology research design (Steffe & Thompson,

Data collected were conversations between learners. In some instances the teacher was part of the conversations. The conversations were analysed through rewriting them as stories – narrative analysis (Polkinghorne,

To ensure quality criteria, transferability, confirmability, credibility and dependability were employed (Bitsch,

The participants and their parents completed the informed consent forms. Informed consent acknowledged the protection of the participants’ rights (Creswell,

We organised this section by drawing extracts from the main study’s data and only tracking the two groups of students in order to demonstrate instances of conversation where

The learners were given a learning activity to prove that ∆

Corresponding angles of similar triangles.

Learners were expected to show that since lines

1.1 | Koena: | Here we can use |

1.2 | John: | Look … okay … you cannot just say |

1.3 | Koena: | ∠_{1} is equal to ∠ |

1.4 | John: | F shape this way? |

[ |
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1.5 | Koena: | Yes. |

1.6 | John: | Yes, it is correct. |

1.7 | Koena: | If we can put it this way, can’t we take it out? |

[ |
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1.8 | John: | Yes, it is correct, they are this way, and then ∠_{1}. They are corresponding angles. |

[ |
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1.9 | John: | Then we can say in triangle _{1} is equal to |

1.10 | Koena: | Corresponding angles are equal. |

[ |
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[ |
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1.11 | Koena: | ∠_{1} is equal to ∠ |

1.12 | John: | They are corresponding angles. |

1.13 | John: | Therefore, we can say triangle |

1.14 | Koena: | Yes, they are similar. |

[ |
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[ |

The two learners in Extract 1 were given a task that required them to prove that the two triangles are similar. On the model for growth in mathematical understanding, this task is pitched at the level of formalising understanding in that it required learners to notice and work with the property of similar triangles that corresponding sides have equal ratios. Furthermore, for the learners to be able to provide the required proof, they should have the image that when two parallel lines are cut by a transversal, the resulting corresponding angles are congruent. The extract begins with Koena’s intentional intervention that they work with side

All the same, from the given information of the task, John noticed that lines _{1} and ∠_{1} and ∠_{1} and ∠_{1} and ∠

In Extract 2 the learners must find the magnitudes of

Proportionality in similar triangles.

Extract 2 shows an instance where moving out of topic and working there resulted in a discontinuity. With critical sharing of ideas on what caused the discontinuity, which emerged after Extract 2,

2.1 | Lesiba: | Here we can start by finding the sides that are in proportion. |

2.2 | Lebogang: | Here we can separate the triangles [ |

2.3 | Lesiba: | Yes, that is correct. Then we can say side |

2.4 | Lebogang: | Okay, it will be this way |

2.5 | Lesiba: | Yes, let us continue. |

2.6 | Lebogang: | We can say |

2.7 | Lesiba: | We can now cross multiply. |

2.8 | Lebogang: | It is going to be 9 multiply |

2.9 | Lesiba: | Two unknowns, how? [ |

2.10 | Lebogang: | Yes |

Similar to Extract 1, Extract 2 began with peer intervention by Lesiba. The intervention was intentional and explicit as he suggested that they find the sides of the triangles that were proportional. In order to do that Lebogang drew ∆

The extract that follows continues Extract 2. It shows how the learners re-started the process of

2.11 | Lesiba: | Actually can’t we have … let me ask … on |

2.12 | Lebogang: | |

2.13 | Lesiba: | 15 + |

2.14 | Lebogang: | |

2.15 | Lesiba: | |

2.16 | Lesiba: | Therefore we say |

2.17 | Lebogang: | The whole of side of |

The extract shows that, regardless of Lebogang’s confusion in lines 2.2 and 2.4 of Extract 2, ultimately the students realised that

2.18 | Lesiba: | |

2.19 | Teacher: | Yes, that equation is correct. |

2.20 | Lebogang: | Cross multiply. |

2.21 | Lesiba: | Cross multiply. |

2.22 | Teacher: | Okay before we cross multiply. |

2.23 | Lesiba: | We can say |

2.24 | Teacher: | Okay continue. |

2.25 | Lebogang: | Write |

2.26 | Lesiba: | Then we say times, then 4 goes on top … |

2.27 | Lebogang: | I don’t understand you … |

2.28 | Lesiba: | You don’t understand me, like, you see now is |

2.29 | Lebogang: | Yes, write them, but let us first replace |

2.30 | Lesiba: | Okay I see it since |

2.31 | Lebogang: | Yes this line here [ |

2.32 | Lesiba: | Okay, then we will write it this way |

2.33 | Lebogang: | Then we can start substituting. |

2.34 | Lesiba: | Here is going to be this way |

2.35 | Lebogang: | Yes, we change it to |

2.36 | Lesiba: | Then it will be this way |

2.37 | Lebogang: | And then now we can change it by using multiplication. |

2.38 | Lesiba | |

2.39 | Lebogang: | Here |

2.40 | Lesiba: | Then this side |

2.41 | Lebogang: | Then we can cross multiply. |

The conversation began with Lesiba rewriting the equation

It can thus be concluded that this was an instance where effective

In this activity learners were expected to prove from the given diagram (

Proving a quadrilateral to be a parallelogram.

Since the task required them to notice properties of the given diagram and to work with them to prove that

3.1 | Lebogang: | ∠ |

3.2 | Teacher: | Which parallelogram are you looking at? |

3.3 | Lebogang: | |

3.4 | Teacher: | And then what are you supposed to do? |

3.5 | Sipho: | Prove that it is a parallelogram [ |

3.6 | Teacher: | So, are you saying ∠ |

In this extract (Mabotja,

Lebogang’s confusion probably resulted in the teacher’s intentional and explicit intervention. The teacher wanted Lebogang to specify what the task required her to do (line 3.4). This request could have made her realise that her intervention was unfocused. She did not respond; instead Sipho did. When the teacher wanted the learners to clarify if ∠

The vignette that follows shows how

3.7 | Lebogang: | Is ∆ |

3.8 | Lesiba: | Yes. |

3.9 | Lebogang: | Then it means … we can say ∠ |

3.10 | Sipho: | Where is an isosceles triangle here? |

3.11 | Lebogang: | Is not this ∆ |

3.12 | Lesiba: | Two sides are equal [ |

3.13 | Teacher: | Yes that will be true. |

3.14 | Sipho: | Let us find all the angles, in that way we can see if it’s a parallelogram. |

3.15 | Teacher: | You are thinking of finding all the angles? |

3.16 | Sipho: | Yes … and in that way we will see whether our opposite sides are equal or not, and if they are equal and then this means it is a parallelogram and if they are parallel. |

3.17 | Lebogang: | How are we going to use the angles? |

[ |

The extract above could have been triggered by the teacher’s intervention when seeking clarification on what Lebogang had said. The peer intervention is explicit and intended to show that ∠

At this stage the learners could have either used the same procedure used for finding the size of ∠

3.18 | Teacher: | Okay let us look at all the angles at our disposal. If we add |

3.19 | Lebogang and Lesiba: | It’s 180. |

3.20 | Teacher: | This means that those co-interior angles are supplementary. If we have the co-interior angles that are supplementary, what can we say about the opposite lines? |

3.21 | Lebogang: | Ohoo, then opposite sides are parallel. |

3.22 | Lesiba: | Ah! Sir, this means we need two pairs of opposite sides that are parallel. |

3.23 | Sipho: | And we can find the other pair by [ |

3.24 | Lebogang: | We can find other co-interior angles. |

3.25 | Lesiba: | Yes, write. |

3.26 | Lebogang: | [ |

3.27 | Sipho: | Then we must also give a pair of opposite sides. |

3.28 | Lesiba: | It will be line |

3.29 | Teacher: | Okay good, so since we have two pairs of opposite sides that are parallel, what can we generally say? |

3.30 | Lebogang: | This means we can say that this is a parallelogram. |

3.31 | Sipho: | Yes, write there. |

In this vignette, the source of intervention was the teacher who was a participant-observer in this study. His intervention was intentional because the question he asked drew the learners’ attention to co-interior angles. The intention was explicit because he pointed learners to ∠

In line 3.24 Lebogang intentionally intervened and suggested that the same way of thinking be followed to prove that the other pair of opposite sides are parallel. In line 3.26 she writes that ∠

The analysis of the extracts presented in this article show that

Effective

In this article we pursued the question: how does

Learners’ self-questioning and reflections on their thinking in response to questions posed by peers offered them opportunities to revise, modify and extend their initial ideas and ultimately build a connected understanding through

It was the nature of the teacher’s interventional decisions that led learners to

Samuel.M. acknowledges a sponsorship grant from the Research Office of the University of Limpopo through the internal Research Chair in Quality Teaching and Learning.

The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.

Samuel.M. carried out the research project as a master’s student supervised by K.C. and Satsope.M. was the co-supervisor. Samuel.M.’s initial draft of the manuscript was reconceptualised by Satsope.M. and K.C. and thereafter Satsope.M. wrote the introduction, theoretical framework, research methodology, quality criteria, ethical consideration and conclusion sections. K.C. wrote the analysis, aligned and enriched it guided by the theoretical framework. I.K. played a role of a critical reader, improved on the logical flow of ideas and filled in the gaps to thus improve the quality of the manuscript.