Ethical clearance was obtained from the National Science Centre, Ministry of General Education of Zambia (MoGE) (NSC/10/7/4). The study conformed to ethical requirements, and permission was obtained from National Science Centre under the ministry. The author agreed to respect the rights of teachers and children and ensure the participants’ privacy, confidentiality and anonymity. The author also followed the guidelines of the affiliated university’s ethical rules for research.

The material that the author utilised was first developed by Matsuo (2010). She created an embedding and disembedding activity in shape, taking the idea of learning trajectory proposed by Clements and Sarama (2009), who demonstrated the learning trajectory for the disembedding of geometric shapes. Clements and Sarama (2014) explained that embedding refers to creating a complex diagram combining simple shapes, rather than only a piece of a shape such as a triangle, circle and rectangle. Furthermore, they explained that disembedding two-dimensional shapes is meant to separate structures within embedded figures and to find ‘hidden shapes’ within more complex diagrams. According to the learning trajectory concept, many 4-year-old children in the United States could not find embedded circles or squares in some structures of shapes, while, on the other hand, 5–6-year-old children could embed shapes into other shapes (Clements & Sarama 2014). Children’s ability to embed and disembed shapes can be strengthened in early childhood mathematics education lessons before entering primary school, which will be useful for their later development of geometrical perspective. In Matsuo’s (2010) geometrical programme, she proposed an activity called ‘Making a fish’ and created a teaching plan. As Matsuo (2010) did not implement the lesson planned, the current author developed the teaching plan described below and shared it with the local teacher:

After conducting the first trial lesson in School A, the present author intended to include guided play in the form of clear, short instructions given by the teacher for physical activities that the children could engage in before introducing the fish activity, because of the children’s readiness, which will be discussed in more detail later. The physical activity should be similar to the fish activity; however, it should be easier for children to understand its rules (e.g. Müller & Wittmann 2004a, 2004b) and should respect their cultural background (Worthington & Van Oers 2016). For children to work more subjectively, it was decided that the teacher should not constantly talk and control the children. It was also decided that a set of small paper shapes would be distributed to each child to enhance the activity because some studies have noted that play should be implemented through interactions with the environment via a concrete object (Matsuo 2010; Mielonen & Paterson 2009; Jemutai & Webb 2019).

In School A, Ms. Sakala and the author had a planning session the day before the implementation to ensure that the teacher could confidently carry out the activity. In the class, children were seated in a semicircle manner on the floor. Ms. Sakala first pointed to one pupil and asked him to come to the front of the classroom; he was shown the fish like the ones in Figure 2, and he then started embedding the shapes by looking at the fish. Alongside the child, she also started to make several fish for demonstration purposes, and everybody could see what he was doing. When they finished making two fish, she explained to the children what they were supposed to do. She also asked them the different shapes’ names in Nyanja and English, and they repeated her all together. However, she did not understand the meaning of embedding the shapes, and instead showed varying sizes of fish. During the planning stage, for the children to think about embedding and disembedding, we had agreed to let them make the same size of fish with different patterns. It seemed that she did not understand the importance of using the same size of fish during our planning discussion; moreover, she did not understand the mathematical meaning of embedding.

After 5 minutes, the teacher pointed to two other pupils, and they came forward and began the activity next to the first pupil. The teacher showed a sample fish by putting some shapes on the floor, so the children also tried to create different-sized fish using the material mentioned, imitating the teacher’s fish. The teacher pointed to two pupils every 5 minutes, and soon everybody had started the activity, which lasted for 30 minutes. While they made fish, the teacher sat next to each child, using one example to assist and encourage them to make different fish. Thus, they ended up making different types of fish with different patterns and sizes. All of the children successfully made different sizes of fish, as shown in Figure 3, and only four of them succeeded in making the same size of fish using embedding. Everyone was engaged in the activity and Ms. Sakala reflected on the lesson, saying that it was good for the students to touch and arrange the shapes on their own, experiencing the shapes. However, the aim of the lesson was not achieved because of the misleading lesson conducted by the teacher. This was caused because, during the planning, the author did not let the teacher play with the paper shapes but instead simply verbally explained the process. The children’s readiness was also an issue that should be considered for an improved lesson. This was the first time they manipulated concrete two-dimensional shapes in the class, so the provided materials were very new to the children. Ms. Sakala also mentioned that the children had never engaged with such shapes on their own, so this embedding activity seemed too advanced for them as the majority of the children could not make congruent fish with different patterns with the given shapes. In the process of the children’s work, although the objective of the activity was not achieved, the pupils were rotating different shapes, and manipulated triangles and quadrilaterals of the same length. This showed that the children, without being told by the teacher, were interested in manipulating shapes on their own. It could be assumed that there was a need to include extra time for them to get use to the materials before introducing the activity.

After reflecting on the first lesson in School A and planning with Ms. Mukonka in School B, the lesson objective was changed to an activity based on creating instead of copying the fish. It was decided that children should engage in more subjective play that helped them acquire mathematical skills at the same time, such as seeing shapes from different viewpoints and manipulating the paper shapes, like the pupils’ actions observed in the demo lesson in School A, which would help them develop a good foundation in shapes. Manipulating shapes allows children to build ideas based on shapes (Sarama & Clements 2014) as they gradually begin to recognise shapes and understand how each shape is formed through this manipulation.

The new goal was to examine what kind of figures children would create, what kind of mathematical properties they would discover during the activity and what they would learn from one another in the class.

For this second experimental lesson (the main lesson), the activity in the demo lesson in School A was modified, and it was called ‘Making figures’:

It was decided that the set of paper shapes described in the first lesson would also be distributed to each child to enhance his or her subjectivity. Furthermore, it was agreed that Ms. Mukonka would provide children with short, clear instructions for the play at the beginning of the lesson, encourage them no matter what they created through the play and respect whatever they discovered through the play.

Children sat next to each other with enough space between them, which was demarcated with three big groups of desks. The teacher explained in Nyanja what the different shapes were, and the children understood what they were supposed to do after a few minutes, because suddenly, after the teacher’s explanation, almost all of them (except for a few children with learning disabilities) opened the plastic bag and started to arrange the paper shapes on their own. They concentrated on making different figures for 40 minutes. When the children finished a certain shape, they called the teachers to come and assess the shape. Aside from those interactions, the teacher did not mention anything. Table 1 shows the figures made by the children (N = 35). Three of the children (nos. 9, 11 and 32 with daggers shown in Table 1) were mentally disabled. Except for the one of those three (no. 11), who bent every piece of the shape that was given to her, all of the 34 pupils made one or more than one figure during the activity. Another one (no. 22) showed a figure that was not understood by Ms. Mukonka, but he also made something. A few children (nos. 1, 2, 5, 12, 15, 17 and 21) made more figures than those mentioned in Table 1. There were not big differences based on the different ages of children; the majority of the children were 5–6 years old. Of course, a few 7-year-old children made several figures (e.g. no. 23); on the other hand, a 4-year-old child (no. 15) and a 6-year-old child (no. 21) created unique figures. The most popular shape the children made was that of a house, which was made by 19 out of 35 pupils during the activity. Children made houses with different patterns and sizes, as shown in Figure 4. All of them made congruent houses. The second most popular shapes were squares, as shown in Figure 5. Six children made squares, combining right-angled triangles with the same length of hypotenuses together. A few pupils produced some mathematically interesting shapes, as shown in Figure 6. Figure 6 shows a variety of figures as well as geometric patterns. Pupils first looked at each other’s creations and tended to imitate others’ work. Table 1 also shows that 16 pupils only made one kind of figure, such as a house, squares, a robot, circles, stairs, planes and so on, and they produced many examples of the one type of shape with different or congruent figures. Geometrical patterns (1) and (2), ‘b’ and ‘e’ in Figure 6 do not express a certain figure, but it can be inferred that children are very keen about the size of triangles, especially their colours, congruency and similarity. Robots, faces and handles have symmetrical features, as do the rockets and patterns in Figure 6. Twelve pupils successfully made more than one kind of figure, which was their intention and hope. One of these nine pupils (no. 12) made six kinds of figures – faces, birds, houses, humans, rockets and masks – and reported their work to the teacher, expressing their satisfaction and sense of accomplishment. In the second part of the lesson, the teacher showed the children the shape of a fish; because the pupils were already involved in the activity, they were allowed to make a fish, connected to the activity in the demo lesson in School A. Fourteen pupils, ranging from 4-year-old to 7-year-old children, succeeded in making one fish; however, they did not change the arrangement of the shapes from the first one they made because of time limitations.

The activity lasted for 40 minutes, 10 minutes longer than a normal lesson; a few pupils wanted to continue with the activity for longer. Thus, they concentrated well during the activity.

The flow of the two lessons was written down on an Excel sheet. The teacher’s actions and verbal explanations, as well as the children’s actions and responses to the teacher’s questions, were also written on the Excel sheet. Pupils’ outcomes on the given task were also summarised based on the author’s memo, in the form of, for instance, Table 1. In comparing the two lessons, differences and similarities were readily apparent. Regarding the difference, the importance of identifying the pupils’ readiness towards learning something new is emphasised. In the demo lesson, the planned activity, which was focussed on embedding and disembedding, seemed mathematically important; however, as compared to the first lesson, the children in the second lesson better understood the meaning of the activity and enjoyed engaging in mathematical play more than the children in the first lesson. As the children from the second lesson were able to get used to manipulating the shapes, perhaps, in the future, they may be able to progress to the originally planned activity. Clements and Sarama’s (2009) findings might hold true if children have a rich mathematical background that involves manipulating concrete shapes that have been given in an activity. By providing Zambian children such an experience beforehand, their educational gaps can be better addressed, and this guided play approach would be more effective for them. Therefore, the lesson applied in this study should be implemented again after children are first given the chance to become accustomed to manipulating concrete shapes in a guided play approach. Regarding the similarities, although the objectives of the two lessons were not the same, children were engaged in manipulating colour boards in the process of learning. The children’s work from both groups also suggested that they were drawn to symmetry, which was also noted by Clements and Sarama (2014). Thus, Zambian children also liked manipulations that have been emphasised in other studies (Linder, Powers-Costello & Stegelin 2011).

The result of the demo lesson can be used to infer that the learning trajectory by Clements and Sarama (2009) may not always be appropriate considering Zambian children’s sociocultural background, such as in the case that their experiences with shapes outside of school were not very well connected to learning at school. In Zambian children’s sociocultural setting, some types of play, such as with plastic balls, making wire cars and so on, allow them to implicitly manipulate, rotate or flip over concrete shapes. However, there is a big gap between their life experiences and school mathematics. In fact, the textbook that has been approved and published by the Ministry of General Education of Zambia (MoGE) only shows two-dimensional shapes, such as rectangles and circles, and does not encourage children to manipulate shapes physically (MoGE 2016). Thus, children may be given opportunities to see geometrical shapes in their lives; however, there are not many chances in which this primitive sense about shapes can be explicitly elicited and discussed with a focus on shape properties. Thus, this type of lesson (the second lesson) can be a bond between experiences outside the school and fundamental mathematics.

In the second lesson, the pupils enjoyed handling different shapes and were engaged in the activity as shown in Table 1 and Figures 4–6. This could be because the purpose of the second lesson was not as challenging for the children as the first one, because they were asked to create what they liked. Everything the children produced fully depended exclusively on their creativity and motivation. This also meant that there was no correct or incorrect way of completing the task. The activity was crucial for children to implicitly learn what shapes are. Furthermore, without the teacher’s verbal support, a good number of children learned from their friends and successfully created a subjective play atmosphere. This shows that activities focusing on playing will only function if the contents and aims are appropriate for children’s sociocultural background.

In Zambia, as mentioned earlier, children in ECE learn to fold paper, trace the outline, create and colour shapes; however, they are not proposed any activities involving handling a variety of shapes by hand (MESVTEE 2013). In the Western context, as Williams-Pierce et al. (2018) mentioned, artificial playing is possible in and out of the classroom, with block puzzles or games aimed at increasing children’s spatial reasoning and their ability to composition and decomposition of shapes, as well as Rubik’s cubes to develop students’ ability to understand algebraic group structure. In the Zambian context, the vast majority of children who commute to public primary schools do not have access to this kind of artificial toys. Both lessons, in this case, could offer an opportunity for children to acquire geometrical skills, such as the embedding of shapes (in the first demo lesson), and implicitly notice geometrical properties for congruency, similarity and symmetry. In the second lesson, the pupils were more engaged in the activity than in the first lesson, and their resulting work could be related to their daily lives. Fish are something familiar in a child’s life, but some children in Lusaka have never seen living fish before because they have never been to the sea; some of them have seen cooked fish, but this also depends on their family. Many children created familiar objects from their daily lives, such as houses, people and faces. It can be inferred that this is the beginning of the moment when they were trying to connect the semi-abstract paper shapes with what they usually see; this connection can be more heavily emphasised in early childhood mathematics education to make the lessons more effective. This result could also be connected to various studies, for example, the enactive, iconic and symbolic (EIS) principle (Bruner 1966), the structure of observed learning outcome (SOLO) taxonomy (Biggs & Collis 1982) and the concrete-pictorial-abstract (C-P-A) sequence (Chang, Lee & Koay 2017). These are different models to explain how children’s development occurs, but each one of them emphasises that children’s development starts from the concrete to abstract. Thus, for pre-school-aged children, the opportunity to learn in the concrete environment where they can touch, feel and experience is crucial to allow them to move onto abstract mathematics in which symbols and mathematical terms are used.

Moreover, except for a few pupils with learning disabilities in the class, the children learned together without much support or advice from the teacher. This represents an ideal situation of guided play. At first, a few children started to make houses, and their classmates noticed and imitated their work. Later, some still continued copying their friends’ work for the whole lesson, while others gradually started to develop their thinking in different ways. For example, some of them made bigger houses, some made many of the same type of houses in a line and others made random houses. Thus, there was a wide variety of houses in the class. Moreover, mathematically, their ideas were connected to the fundamental idea of congruency, similarity and symmetry without them explicitly knowing these concepts at this stage through exploring and investigating shapes with their autonomy (Fisher et al. 2011; Vogel 2013; Weisberg et al. 2013). Thus, imitation is a natural first response when children learn, and they subsequently become able to develop their ideas through play.

Taking into consideration the pupils’ challenges, in the first lesson, as well as previous research, the lesson plan was developed and implemented in the class. However, the lesson was unsuccessful. This was because of the lack of understanding of the children’s characteristics in a sociocultural context, as well as the lack of the accumulation of practical knowledge regarding how children learn in the class in the context of Zambia, including their readiness for learning foundational mathematics. After the modification of the objective of the lesson based on the pupils’ actions in the demo lesson, the second lesson’s mathematical activity, which was keenly focussed on a manipulative activity, became easier, and the teacher (along with the author) focussed on the children’s fundamental ability to relate to the crafting of shapes. Embedding and disembedding activity seemed very advanced to children; however, the analysis revealed that handling concrete shapes will be a good starting point for children to learn to be familiar with the mathematical shapes and their mathematical properties. Consequently, the modifications were based on the children’s processes of actions and readiness; this led to the success of a guided play activity that met children’s sociocultural conditions.