Primary School Children's Counting and Number Composition Processes from Two Pilot Studies in Urban Schools in Zambia

Abstract

One of the biggest challenges for primary school children is moving beyond the use of counting in calculation. The purpose of this study was to investigate how Zambian Grade 1 to Grade 4 children at primary school see a group of 10 as an effective pattern and structure with the given concrete material. The study consisted of three phases consisting of two pilot studies and the main study, during which a total of 146 children from Grades 1–4 were asked during interviews to perform certain tasks. These tasks focused on number competencies related to counting objects, recognising number patterns and structures of concrete objects, expressing numbers of concrete objects, and composing and decomposing numbers. The response categories of the PASA (Pattern and Structure Assessment) framework in number competencies were modified to analyse the data. The aim of the analysis was to identify the degree of the acquired pattern and structural thinking. The results of the two pilot studies showed that, in formal addition with two-digit numbers, all children counted without identifying ‘groups of 10’ while some used concrete materials, some recognised numbers in a pattern and identified a ‘group of 10’, which has not been previously observed in Zambia. The main survey showed that children were able manipulate concrete objects and to recognise a ‘group of 10’ in the given 10-frames. The results confirm that children have the potential to develop the skill of transitioning from manipulating concrete objects for calculations but this is missing in the current Zambian syllabus. The findings offer new insights about positive learning processes in Zambian primary school students. The results can help to provide appropriate support in class and at curriculum level.

Keywords:

• Zambia
• primary mathematics education
• number competencies
• a group of 10
• counting
• PASA

Background of the Study

The mathematics performance by Zambian primary school children is very low compared with other Sub-Saharan African countries (Makuwa, 2010). One of the biggest challenges for primary school children when learning mathematics is moving beyond the use of counting in calculations (Nakawa, 2012). Even Grade 5–6 students (who are approximately 11–12 years old) use counting in calculations, making it difficult to teach advanced mathematical methods and the mathematical thinking necessary for investigative work (Nakawa, 2016; Uchida, 2009). A key concept required for these areas is pattern and structural thinking, which helps children later on in the abstraction and generalisation of mathematical ideas and relationships (Mulligan et al., 2015). Although the importance of the key concept is accepted universally, more research about processes and steps for acquiring these concepts in an African context is required since the socio-cultural conditions of children differ in different countries, especially within the African context (e.g. Mostert, 2019a, 2019b). Accordingly, this article investigates how Zambian children identify a group of 10 as an effective pattern and structure towards addition and other calculations in a broader sense. Our research question is as follows: given concrete materials, how do Grade 1–4 learners relate a group of 10 to pattern and structure? We hope that this study is able to add some knowledge about pattern and structural thinking by Zambian children by focusing on how they operate concrete objects for addition and subtraction.

Literature Review

Many studies have discussed counting numbers and addition calculations (Baroody, 1987; Fuson & Burghardt, 2003; Verschaffel et al., 2007). Comparative research on this topic has also been conducted in Western and East-Asian countries (Murata & Fuson, 2006). Other studies have demonstrated the developmental stages of calculations (e.g. Murata & Fuson, 2006) and stressed the importance of recognising groups of 10 in their theoretical assessment of multidigit addition and subtraction. In addition, Herzog et al. (2017) and some others (e.g. Jordan et al., 2010) have stated that understanding of base-10 numbers is connected to understanding more advanced mathematical problems and the performance of basic operations in mathematics.

The Zambian syllabus includes grouping in 10s and learning the place value system (Ministry of Education, Science, Vocational Training, and Early Education, 2013). However, the syllabus does not emphasise the importance of manipulation of concrete objects before abstract calculations are introduced. For example, for Grade 1, the introduction to mathematics allows children to ‘Recognise, count, read and write numbers from 1 to 100’ and ‘Interpret numbers using ten as a unit’ (Ministry of Education, Science, Vocational Training, and Early Education, 2013, p. 1). It does not emphasise the use of concrete objects before introducing abstract numbers and numerals; thus, the mathematics learning experiences for children is abstract from the beginning. However, the focus of this study is the way of manipulation with concrete objects for calculation, because of its importance in children's development of efficient calculation skills.

Focusing on counting, Wittmann and Müller (2009) suggests that counting is a good starting point for determining how young children understand numbers, but it becomes problematic if children continue to count. Children develop and use various counting strategies when adding two numbers or combining sets of objects. Initially, they use a ‘counting all’ strategy. For example, when combining three items with four items, children will count one, two, three, and then one, two, three, four, and then finally, one, two, three, four, five, six, seven. Eventually, children utilise the known facts, along with their knowledge of decomposing numbers, to count more efficiently. In the above example, children may decompose the ‘four’ into ‘three and one’, use a known fact such as ‘three and three makes six’, and then add the leftover ‘one’ to make ‘seven’ (Verschaffel et al., 2007). However, two prerequisite skills, verbal counting and object counting, are necessary for composing and decomposing numbers. Verbal counting refers to saying the number words in the proper order and knowing the principles and patterns in the number system (Baroody, 1987). In relation to this, Papic et al. (2011) argued that the early development of patterning could provide a foundation for successful mathematical development. Pattern and structure are important for understanding the abstractions and generalisation of mathematical ideas and relationships in children's future learning (Mulligan et al., 2006; Mulligan & Mitchelmore, 2019). Thus, pattern and structure are interconnected not only with geometrical thinking but also with computational thinking.

Conceptual Framework: Number Competencies

Mulligan et al. (2006) argue that children's mathematical development depends on being able to recognise structures and that even in calculation, the recognition of mathematical structure helps them grasp the base 10 number system, including grouping in 10s. In investigating young children's early mathematical development in pattern and structure Mulligan et al. (2015) developed a tool to assess children's number competencies, called Pattern and Structure Assessment (PASA), which we draw upon in this study. PASA covers the first three schooling years (Foundation, Year 1 and Year 2) in the Australian context, and contains three types of one-on-one assessments with approximately 15 different tasks. Some of PASA's key features include focusing on children's strengths and weaknesses and providing in-depth diagnostics regarding types of information and students’ mathematical achievements and mathematical thinking processes, instead of just students’ abilities to complete the task (Mulligan et al., 2015). This is a very useful framework for us to identify Zambian children's different stages analytically. Further, Mulligan and Mitchelmore (2013) stated that helping children develop an awareness of mathematical pattern and structure across a wide range of early mathematical concepts will improve children's mathematical competencies. Since children's representations in calculation reflect their level of mathematical understanding (Robert, 2015) it follows that recognising structure is also related to the counting strategy (Baba et al., 2019). Regarding the use of concrete objects, patterns are also an important factor to be considered. Therefore, the idea of PASA is regarded as the basis for identifying the number competencies in our study. In addition, Mulligan and Mitchelmore (2019) described several mathematical structural features in children's representations: counting and symbols, number patterns and sequences, grouping by 10s, and the use of 10s as an iterable unit. Specifically, the base-10 system is a critical aspect of mathematics performance (National Council of Teachers of Mathematics [NCTM], 2000). Laski et al. (2014) found that an understanding of the base-10 structure of the number system significantly contributes to the frequency of students’ use of a base-10 decomposition strategy when performing calculations. Therefore, a greater focus on the base-10 number structure during the early stages of maths instruction is particularly beneficial for the development of computational skills (see also Herzog et al., 2017). From the above discussion, regarding the process that starts from counting, capturing numbers structurally and performing the basic four arithmetic operations using the structure, we can identify the eight crucial competencies based on PASA:

1. counting objects one by one, by groups, and counting forwards and backwards;

2. recognising the patterns and structures of numbers, using concrete objects;

3. expressing numbers using concrete objects;

4. composing and decomposing numbers with concrete objects;

5. identifying numbers in terms of a relevant unit and the relative size of the numbers with several representations such as dots and a number line;

6. understanding the decimal number system;

7. the significance, procedure and proficiency of the calculation with concrete objects; and

8. reading and expressing mathematical sentence skills,

PASA's five individual structures (sequences, shape and alignment, equal spacing, structured counting and partitioning) are restricted to the first four components, because those four components are believed to form the foundation for formal calculation. The first component is essential for children to experience the world of numbers. This component relates to the structured counting in PASA. The second one is related to children's pattern recognition with given objects, which relates to equal spacing, sequences and shape and alignment in PASA. The third one is related to the abstract form of numbers and the formal calculation. The fourth one influences addition and subtraction. The fifth to eighth competencies are connected to formal calculation with numerals. In other words, the first four components are for basic number competencies, and the latter four competencies for formal calculation. Regarding pattern and structure, in this study, we focused on making a group of 10 with concrete objects made up of local material available to government primary schools with a limited financial cost. The objects that were used included bottle caps, and a so-called 10 frame (a frame of 2×5 squared boxes). Addition and subtraction are also discussed in terms of the fourth competency, but the related tasks are only used as a means to establish the level at which learners are seeing the structure of numbers. This article reports the assessment of the first four competencies.

Methodology

Data Collection Method

In our project, we undertook two pilot studies before finalising our research tool for Zambian children. For the article, we prepared tasks and interview schedules in a local language, Chinyanja. We conducted three different sets of interviews: Pilot study 1, Pilot study 2 and Main survey. We confirmed the children's calculation ability—especially for formal addition, which is the most fundamental operation—through the two pilot studies described below.

Samples for the Various Interviews

The Ministry of General Education randomly selected two academically average government schools in Lusaka for this study. We chose governmental primary schools in Lusaka because they were representative of the average academic strength of urban schools.

In Pilot study 1, 96 Grade 4 students in one of the two schools were randomly selected for the calculation interviews in order to investigate the feasibility of the survey strategy and the range of the children's competencies. In Pilot study 2, 18 students from Grades 1–4 were selected from the same school to identify the children's cognitive levels, as well as to judge how the interview schedule could work. For the Main survey class teachers randomly selected at least four students from each grade from both schools—a total of 32 students from Grades 1–4 were interviewed for the main data collection. Owing to the time needed for children to concentrate on the given tasks, we decided to decrease the number of tasks and target these per grade. Children from Grades 1 and 2 answered tasks focusing on Competencies 1–4, whereas children of Grades 3 and 4 answered only tasks focusing on Competencies 3 and 4 (see Table 1).

Table 1. Competency and question allocation for different grades

Competencies	No. of question items	Grades for question items regarding each competency task				No. of students
		Grade 1 (N = 8)	Grade 2 (N = 8)	Grade 3 (N = 8)	Grade 4 (N = 8)
1. Counting objects one by one, by groups, and counting forward and backward	3	✓	✓			16
2. Recognising patterns and the structures of numbers	4	✓	✓			16
3. Expressing numbers with bottle caps	2	✓	✓	✓	✓	32
4. Composing and decomposing numbers	4	✓	✓	✓	✓	32

Japanese and Zambian researchers administered the interviews together in different pairs. During the observations of the interviews, each researcher independently classified and verified each child's response.

Research Ethics

The Ministry of General Education and our research institution gave the research team ethical clearance to conduct interviews, and government officials in the Curriculum Development Centre under the Ministry of General Education worked with the researchers.

Assessment Tools to be Used: Modified PASA Response Categories

Considering the results of the pilot studies and children's information from the Main study, we referred to PASA (Mulligan et al., 2015) for the development of an assessment tool. In PASA, five different response categories, which are assigned to each pattern- and structure-related task, were used to assess the children's awareness: pre-structural, emergent, partial structural, structural and advanced structural. As it is important to consider the children's different stages and their cultural characteristics, we decided to modify the response categories of PASA. As illustrated in Table 2, we added ‘unknown response’ to the response categories of the PASA framework because of the results of previous research in Zambia (e.g. Uchida, 2009). Uchida (2009) noted that some children with learning disabilities who neither understood nor wrote with mathematical symbols and logics often made unusual mistakes in their mathematical work, which inhibited researchers from inferring their intentions. The procedure for producing Table 2 was described in detail in Baba et al. (2019).

Table 2. A framework for the six response categories assessing the number competencies

Response category	Characteristics of the response
Competency	Unknown response (C0)	Pre-structural (C1)	Emergent (C2)	Partial structural (C3)	Structural (C4)	Advanced structural (C5)
1. Counting objects one by one, by groups, and counting forwards and backwards.	Actions or words are not understandable, or the student is silent	At most, shows limited and mistaken counting	Shows partially correct counting	Shows partially correct counting that is more than ‘emergent’	Accurate counting	Accurate, efficient counting
2. Recognising the patterns and structures of numbers, using concrete objects		Shows the limited and disconnected features of the pattern and structure	Shows some relevant features of the pattern and structure but in an inaccurately organised manner	Shows most of the relevant features of the pattern and structure but in an inaccurately organised manner	Correct but limited use of the underlying pattern and structure	Accurate, efficient, and generalised use of the underlying structure. Student expresses what they did verbally
3. Expressing numbers using concrete objects		At most, shows limited and disconnected features regarding a number	Shows some relevant features related to a number but inaccurately	Shows the same number of bottle caps when given a certain number	Identifies the number and writes it down	Accurate, efficient, and generalised use of the underlying structure. Student expresses what they did verbally
4. Composing and decomposing numbers with concrete objects		At most, shows limited and mistaken action	Shows some relevant features of the pattern and structure but in an inaccurately organised manner	Shows most of the relevant features of the pattern and structure towards a group of 10 but in an inaccurately organised manner	Shows correct but limited use of the underlying pattern and structure of the group of 10, not explaining their actions verbally	Accurate, efficient, and generalised use of the underlying structure. Student expresses what they did verbally

Modified from Mulligan & Mitchelmore (2019).

As shown in Table 2, when children are rated as C4, we assess that the children grasp the pattern of a group of 10 and the structured way of seeing bottle caps. Further, children are rated as C5 when they successfully explain these patterns and structures verbally.

Data Analysis

For each question item, we used the modified Mulligan and Mitchelmore (2019) framework to set up six different categories to thematically judge the children’s responses. During the observations of the interviews, each researcher verified each child's response and marked their responses. After the interviews, each team gathered, reflected on and confirmed the scores of each student's performance. The different teams maintained the same assessment guideline.

The Results for Pilot Studies 1 and 2

The pilot studies consisted of two interviews that needed to be refined for the main survey. Firstly, formal interviews were conducted with 96 Grade 4 children who spoke Chinyanja, one of Lusaka's most commonly spoken languages. In the first pilot study, the children were asked to solve three simple addition problems (8+7, 13+8 and 17+14) and explain their calculation strategies. We deliberately chose Grade 4 students because previous research indicated that these children still use counting for formal calculations. This would imply that children in lower grades would also use counting in basic number competencies such as composing numbers and calculations, numbers. We found that all Grade 4 children in the sample (N = 96) counted during calculations, and none recognised a ‘group of 10’.

In the second pilot study, we set out to investigate more profoundly children's calculation thinking processes in the same school. For this phase, we asked the class teachers to randomly select 18 children from a mix of academically excellent and academically poor Grade 1–4 students. We examined how children deal with concrete objects, such as bottle caps, to see a certain group of numbers of objects and connect them to perform addition and subtraction tasks. Out of 18 students, only one Grade 4 student was perfectly able to explain how he had identified these groups of bottle caps. However, while 12 students were seemingly able to recognise groups of bottle caps when given its structural arrangement on the 10 frame at C4 in Table 2, very few could verbally explain their actions of identifying the number of the bottle caps, which is at C5.

The two pilot studies showed that, first, for formal addition, all of the children from Grade 4 counted and did not identify any ‘groups of 10’ and, second, while using concrete materials, some students from Grades 1–4 recognised groups of 2s and 5s of bottle caps in the 10 frame and identified a ‘group of 10’ in the frame.

Results of the Main Survey

We further investigated the positive outcomes from Pilot study 2, using interview items and examining the children's performances on them. The Zambian researchers asked students to perform simple calculations with bottle caps and recorded each child's actions. As shown in Table 3, for each of the items the responses of the children were coded using C0–C5, which represent the categories of unknown response, pre-structure, emergent, partial, structure and advanced structure, respectively. We now present more detail of the results according to the four competencies.

Table 3. Data collection results

S/N	Task	Ten frames used in the question	Inidividual student responses
			Grade 1								Grade 2
			S1-1	S1-2	S1-3	S1-4	S1-5	S1-6	S1-7	S1-8	S2-1	S2-2	S2-3	S2-4	S2-5	S2-6	S2-7	S2-8
1.1	The interviewer places 20 bottle tops randomly and asks the students: ‘Count these and tell me the number’		C4	C3	C4	C4	C4	C5	C5	C4	C5	C4	C3	C4	C5	C5	C4	C4
1.2	Ask the students to ‘count up to 20’ in English or their local language. Record which language the children want to use. If it is too hard for them, the interviewer can say, ‘You can use bottle tops for the counting’. For the students who reach the fourth response category, ask them to ‘count in 2s and 5s up to 20’		C4	C2	C4	C3	C4	C5	C5	C4	C4	C4	C4	C5	C5	C5	C4	C4
1.3	Ask the students to ‘Count down from 20’ in English or their local languages. Record which language the children want to use. If it is too hard for the students, then the interviewer can say, ‘You can use bottle tops for counting’. For the students who reach the fourth response category, ask them to ‘count in 2s and 5s up to 20’		C4	C0	C4	C0	C4	C4	C4	C2	C4	C4	C0	C0	C4	C4	C4	C4
2.1	Ask the students to make shape patterns on their own by recognising patterns, The interviewer will say ‘Choose any colour of your choice, make any shape or pattern you like’, preparing 10 red and 10 white bottle tops, in total 20 bottle tops on the desk		C4	C1	C3	C1	C5	C5	C5	C2	C3	C4	C4	C2	C5	C4	C5	C3
2.2	To understand and identify the number of bottle tops in a structured way. The interviewer shows 10 bottle tops in a 10 frame vertically and four bottle tops in the other 10 frame and asks ‘How many are there altogether?’		C5	C1	C3	C3	C3	C5	C5	C5	C4	C3	C3	C5	C5	C5	C5	C5
2.3	To understand and identify the number of bottle tops in a structured way. The interviewer shows 10 bottle tops in a 10 frame and eight bottle tops in the other 10 frame vertically and asks ‘How many are there altogether?’		C5	C0	C3	C3	C5	C5	C5	C4	C5	C4	C3	C5	C5	C5	C4	C5
2.4	To imagine the number 10 in their minds and place the bottle tops in this order. The interviewer says, ‘Arrange these bottle tops as you imagine the 10 frame’ without showing the 10 frame		C5	C0	C1	C2	C4	C5	C5	C4	C4	C4	C3	C4	C2	C4	C5	C5
3.1	To identify seven bottle tops in a structured way and express the total number of the bottle tops in numerals. The interviewer places seven bottle tops (five on the first line and two on the second line) on a 10 frame and asks, ‘How many bottle tops are there at the 10 frame?’		C5	C0	C3	C3	C3	C5	C5	C5	C3	C3	C3	C5	C5	C4	C5	C5
3.2	To identify 18 bottle tops in a structured way and express the total number of the bottle tops in numerals. The interviewer places 10 bottle tops on a 10 frame and eight on the another frame and asks, ‘How many bottle tops are there at the 10 frames?’		C4	C0	C4	C3	C4	C5	C5	C4	C3	C3	C2	C5	C5	C5	C5	C5
4.1	To perform the calculation 3+9 by identifying the bottle tops in a structured way and writing down the operation in a mathematical sentence. The interviewer places three bottle tops on a 10 frame and nine bottle tops on the other 10 frame and shows them separately and asks, ‘Can you get the sum altogether?’		C5	C1	C5	C3	C5	C3	C5	C4	C4	C4	C2	C4	C4	C5	C5	C5
4.2	To perform the calculation 13+19 by identifying the bottle tops in a structured way and writing down the operation in a mathematical sentence. The interviewer places 13 bottle tops on two 10 frames and 19 bottle tops on the other two 10 frames and show them separately and asks, ‘Can you get the sum altogether?’		C5	C0	C5	C3	C5	C3	C5	C3	C3	C3	C2	C2	C5	C5	C5	C5
4.3	To perform the calculation 12+8 =20 by identifying the bottle tops in a structured way and writing down the operation in a mathematical sentence. The interviewer puts 12 on two 10 frames and 8 on the other two 10 frames, puts them separately and asks, ‘Can you get the sum altogether?’		C5	C0	C4	C4	C4	C4	C3	C5	C3	C3	C3	C5	C3	C5	C5	C5
4.4	To perform the calculation 27+13=40 by identifying the bottle tops in a structured way and writing down the operation in a mathematical sentence. The interviewer puts 27 on three 10 frames and 13 on the other two 10 frames, puts them separately and asks, ‘Can you get the sum altogether?’		C4	C0	C4	C4	C5	C5	C2	C3	C4	C3	C3	C5	C5	C5	C5	C3
S/N																	C0: Unknown Response	C1: Pre-structure	C2: Emergent	C3: Partial	C4: Structure	C5: Advanced structure
	Grade 3							Grade 4
	S3-1	S3-2	S3-3	S3-4	S3-5	S3-6	S3-7	S3-8	S4-1	S4-2	S4-3	S4-4	S4-5	S4-6	S4-7	S4-8
1.1																	0	0	0	2	9	5
1.2																	0	0	1	1	9	5
1.3																	4	0	1	0	11	0
2.1																	0	2	2	3	4	5
2.2																	0	1	0	5	1	9
2.3																	1	0	0	3	3	9
2.4																	1	1	2	1	6	5
3.1	C5	C5	C5	C5	C5	C5	C5	C5	C3	C4	C5	C5	C3	C4	C5	C5	1	0	0	8	3	20
3.2	C5	C4	C4	C5	C5	C5	C2	C5	C5	C5	C5	C5	C3	C4	C5	C4	1	0	2	4	8	17
4.1	C4	C2	C4	C5	C5	C4	C5	C5	C5	C5	C5	C5	C3	C5	C4	C4	0	1	2	3	10	16
4.2	C3	C2	C3	C5	C5	C4	C5	C4	C5	C4	C5	C5	C3	C5	C2	C5	1	0	4	8	3	16
4.3	C3	C5	C0	C5	C5	C5	C5	C4	C5	C2	C3	C3	C3	C5	C3	C5	2	0	1	10	5	14
4.4	C5	C5	C0	C5	C5	C5	C5	C5	C5	C5	C5	C5	C5	C5	C2	C5	2	0	2	4	4	20

The numbers indicated in the table represent the numbers of students.

Competency 1: Counting Objects One by One and by Groups and Counting Forwards and Backwards

Children were required to identify the total number of bottle caps that they were given (20), count from 1 to 20, and count backwards from 20 to 1. In Q1.1, 14 students out of 16 (87.5%) whose answers were rated as C4 or C5 were able to correctly count up to 20, but only five students (C5) were able to identify groups of numbers. This result relates to Q1.2 and Q1.3. In Q1.2, most of the students were also able to count to 20 without using objects, while two Grade 1 students were not able to count backwards in Q1.3. It seems that a few children found it difficult to count backwards, whereas any forward counting with or without objects was successful. On the other hand, some of them had difficulty identifying groups of 2s and 5s.

Competency 2: Recognising the Patterns and Structures of Numbers Using Bottle Caps

In Q2.1 and Q2.4, the children's responses varied, while in Q2.2 and Q2.3, they performed relatively well. Q2.1 required them to create any pattern with the red and white bottle caps. Students rated from C1 to C4 showed the shapes that they created, or tried to create, without any patterns. These questions were intended to assess whether children were able to manipulate bottle caps with appropriate equal-spacing and to check their geometrical pattern recognition before identifying a ‘group of 10’ on the box instantly. The results demonstrated that more than half of the students reached the C5 rating level (56.25% for both Q2.2 and 2.3); thus, many children were able to identify the ‘group of 10’ when concrete objects were arranged in a structured manner. In contrast, six children were rated below C4 in Q2.2. The C3 rating indicated that the relevant children counted one by one, and thus the results revealed that some of the students could not identify the ‘group of 10’ with concrete materials. Q2.4 asked students to imagine groups of 10 in the 10 frame and to place 20 bottle caps on them. The question intended to assess whether students could place things in an organised manner in groups of 10 without the 10 frame. The C5 rating implied that the students placed the bottle caps in two groups of 10 and explained why the action was mathematically correct, while a C4 rating implied that they placed the bottle caps correctly without any verbal explanation or placed them with a focus on groups of 5. Eleven students out of 16 (68.8%) who were rated C4 or C5, saw the groups of 5 and 10, that is, they identified the pattern and structure of the 10 frame.

Competency 3: Expressing Numbers by Using Concrete Objects

Q3.1 and Q3.2 were similar questions, asking about the total number of bottle caps shown in the 10 frames. For Q3.1, 20 students were rated C5, and for Q3.2, 17 students were rated C5 (N = 32), as they identified a group of either 5 or 10 or instantly recognised a provided number with a verbal explanation. Even those who were rated C4 (three and eight students in Q3.1 and Q3.2, respectively) were able to identify groups of numbers but did not provide a clear explanation. This means that, out of 32 students, 23 (71.9%) and 25 (78.1%) provided answers rated as C4 or C5 for Q3.1 and Q3.2, respectively. They were able to identify and use groups of 5 and 10 to recognise a certain number of bottle caps. Among Grade 1 children, four out of eight scored C5 on Q3.1 and two out of eight scored C5 on Q3.2. Two children among them reached C5 in both questions. This implied that even some lower-grade children were able to recognise and explain the pattern and structure of the 10 frame.

Competency 4: Composing and Decomposing Numbers with Concrete Objects

In Q4.1 and Q4.2, 16 out of 32 (50.0%) were rated C5. These 16 students were not identical. Combining C4 with C5, 26 and 19 students in Q4.1 and Q4.2 (81.3% and 59.4%, respectively) enabled the stage of being able to identify groups of 5 and 10, which is the abstract stage shown in Table 2. Thus, more than half of the children could identify groups of 5 and 10. In Q4.3 and Q4.4, the numbers of those rated C5 on questions were 14 and 20 (43.8% and 62.5%) respectively, and the numbers of those rated C4 or higher were 19 and 24 (59.4% and 75.0%) respectively. Therefore, similar to the results in Q4.1 and Q4.2, children added up two numbers to reach the given number when they were asked to obtain the sum of the bottle caps. Some children moved the bottle caps to create a group of 10 by manipulation. Only one child gave an explanation for their subtraction for Q4.3.

Discussion

The results of the first competency task showed that most of the children received C4 or C5 ratings. Therefore, they demonstrated that they had the necessary counting skills, but counting backwards remained a challenge for a few Grade 1 children. Chinyanja has a relatively complicated number system, so while speaking it, children use English to express numerals. Thus, the language factor does not seem to be influential in this question item. In contrast, the lower primary-level curriculum as well as early childhood mathematics textbooks do not emphasise counting backwards, which may explain the low levels of skills in counting backwards. The second competency task's results showed that many children were able to identify a ‘group of 10’ in the 10 frame, exceeding our expectations; however, some were unable to do this. Thus, the results indicated that some of the Zambian children had the capacity to recognise a ‘group of 10’ when concrete objects were used. The skill of identifying a group of numbers is not culturally embedded but an educationally developed skill, and the material, the 10 frame, may help some learners notice these important groups. This is also related to the pattern recognition of a group of 10, as well as structural thinking. Jansen van Vuuren et al. (2018) also discussed that children constructed a set-based representational concept towards a number and that the conceptual understanding was an important milestone for the development of arithmetic concepts to recognise that the group of 10 can be included in the concept they mentioned. Notably, seven Grade 1–2 students obtained C5 on both Q2.2 and Q2.3, suggesting that once children acquire the ability to recognise a group of 10, they are also able to see the structure of a group of 10. This is a remarkable result compared with our pilot study 1 and previous research, which reported that even Grade 3–4 Zambian students counted one by one (Uchida, 2009). This suggests that helping children develop skills in recognition of patterns and structures will improve their skills in making groups of 10 and making generalisations of the process. The results of third competency task showed that about half of the children, including Grade 1–2 students, could explain their actions, showing that they possess explanation skills, as well as skills in recognising the pattern of groups of 10 and their structure within 10 frames. Furthermore, ranging from Grade 1 to 4 pupils explained the recognition of the pattern verbally when asked the number of bottle caps in the third competency, which is a positive result that has not been reported previously for Zambia. However, other children remained at the stage of counting, showing that they do not notice the pattern of 10s and the structure of the 10 frames. It was also found that those who were rated C3 or below in the first competency remained at C3 in the second competency, showing that these skills are linked. These results suggest that it is advisable for teachers to first start using concrete objects such as bottle caps to help children become familiar with the structure of a group of 5 or 10, as well as composing and decomposing numbers, before the abstract calculation is introduced. Laski et al. (2014) support this idea, stating that children who practised composing and decomposing 10 using blocks tended to use those strategies in addition.

The results of the fourth competency task also showed that several children noticed a ‘group of 5 or 10’ when the addition questions were asked by moving the bottle caps to the other frame to create a group of 10. Seven children who were assessed on the four questions at C5 were consistent in their recognition of pattern and structure with verbal explanations, which is also a positive result. The responses of several other children were inconsistent, in that they were not able to explain their actions although they seemed to grasp the structure of the 10 frame. Fritz et al. (2020) mentioned that ‘breaking down’ of a number should be used for mental arithmetic and the manipulation of bottle caps can contribute the skill of breaking down. It is therefore recommended that Zambian teachers should encourage children to explain their findings of operations in the mathematics classroom more often to help them think, express and reflect on the abstract form. This may enhance children's consistency in explanations about grouping, composition and decomposition.

The analysis revealed that there was a gap between formal calculation and children's number recognitions and operations with concrete objects. The importance of this latter skill is not recognised in the Zambian syllabus, even though children's understanding of pattern and structure might help them improve the calculation strategies, overcoming counting. The importance of pattern is also mentioned by Afonso and McAuliffe (2019) in South Africa and it shows that young learners have the potential to think algebraically. Thus, the process of moving from manipulating concrete objects and making a ‘group of 10’ to performing calculations with numerals is an important part of children's development. The result shows the significance of dealing with concrete objects before/in parallel with calculations. Throughout Grade 1 learning, these objects can be used for supporting their calculations to understand a group of 10 and its pattern and structure. Mulligan et al. (2010) commented that the empirical research from the PASA studies provided impetus for the current discussions about the centrality of patterns and relationship in the curriculum. Similarly, in Zambia, more empirical studies should be conducted and they could inform revisions of the Zambian mathematics curriculum.

Conclusion

This study reported the results of a tool that was adapted to assess children's performance in four key number competencies, providing analytical insight into the assessment of these competencies. It was found that a significant number of children, including Grade 1 and Grade 2 students, acquired the pattern and structure of the 10 frames, as well as groups of 10. This was the first time that young children from Zambia were recorded as being able to group 10 with concrete objects. The skill is connected to addition and subtraction, as well as other domains of mathematics (Mulligan et al., 2006). This is an important contribution, showing that Zambian children are able to see the numbers of objects effectively without counting. On the other hand, other children remained at the stage of counting, not yet noticing the structure of the 10 frames. The pilot study showed that in the formal calculation with two-digit numbers, children kept to the previous counting strategies that they used with concrete objects. Therefore, in making connections to calculations in numerals, concrete manipulations can support children to realise a group of 10 while also helping them to consistently recognise patterns of 10s in classroom tasks. The study also shows that the skill can be developed pedagogically and hence identifying a group of numbers is not a culturally embedded skill, so teachers need to address this. While Zambian children showed positive results, we recommend that the Zambian mathematics curriculum further emphasises smooth transitions from counting to recognising objects to perceiving numbers. It is also recommended that Zambian teachers should encourage students to reflect on and explain their ideas, as well as to use concrete objects to bridge the concrete operation and abstract thinking with numerals. In terms of limitations, it is important to note that the small sample size limits generalisability. Furthermore, the research was performed in an urban area, and the results may differ if a similar study was carried out in a rural context, which would be worth investigating.

Disclosure Statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The article reports part of the research project funded by JICA (Japan International Cooperation Agency). The authors are fully responsible for the content of the article.