The poor performance of learners in mathematics has long been a matter of concern in South Africa. The Annual National Assessment (ANA) results reveal that the problem starts in the foundation phase with number concepts.

This research sought to ascertain how foundation phase teachers used mathematical resources to teach number concepts as this may be one of the contributors to poor mathematics results.

The purposively selected participants included five foundation phase teachers teaching Grades 1–3 at two schools in the Western Cape, in South Africa.

The research was located within the interpretive qualitative research paradigm and used a case study approach. Data were collected through lesson observations and interviews and analysed through the lens of Vygotsky’s sociocultural theory.

The findings of this study revealed that teaching for understanding was often compromised by teaching to enable learners to pass systemic assessments. Teachers are inclined to rote teaching with drill work in preparation for assessments such as the ANA and the systemic assessment. Consequently, manipulatives are not necessarily used optimally or opportunely.

This study recommends that teachers should receive the necessary training to use and follow Vygotsky’s Zone of Proximal Development and also make an effort to follow the guidelines indicated in the Curriculum and Assessment Policy Statement mathematics document in respect of how and when to use practical mathematical manipulatives.

The use of systemic tests and Annual National Assessments (ANA) to measure learners’ performance in Mathematics have put pressure on teachers to reflect on their teaching strategies and look for ways to ensure that all their learners are taught effectively. This article reports on teachers’ use of concrete manipulatives (resources), as opposed to virtual manipulatives (Cockett & Kilgour 2013), in teaching number concepts. As a foundation phase teacher, the first author has always understood the importance of using mathematical manipulatives to support learning in the classroom. She is aware that teaching learners mathematics is a step-by-step process, which requires the selection of appropriate manipulatives to support every concept taught. This is supported by many researchers who believe that concrete materials help learners to learn mathematics concepts (Drews

Various descriptions exist of what concrete manipulatives mean. A more comprehensive description is that of Moyer (

Thus this study explores learners’ understanding of number concepts in foundation phase classrooms, in terms of how concrete manipulatives are applied to facilitate conceptualisation and understanding. Number concepts form the foundation for all other mathematical concepts, and proficiency in this area has long been one of the main objectives of teaching and learning mathematics in both school and university (Engelbrecht, Bergsten & Kågesten

Diagram showing what teachers should do to be able to assist learners within the Zone of Proximal Development.

The poor performance of South African learners in mathematics is a major concern (Siyepu

There are five content areas to be covered in the foundation phase, as stipulated in the Mathematics Curriculum and Assessment Policy Statement (CAPS 2011:10–11), namely number, operations and relationships; patterns, functions and algebra; space and shape; measurement; and data handling. This study only focused on the content area dealing with number, number operations and relationships.

There are several factors that contribute to the poor performance of learners in mathematics in a South African context. These include the tendency to blame poor teaching approaches, learners who study mathematics in a second language, socio-economic factors, qualifications of parents and lack of qualified mathematics teachers in the foundation phase (Siyepu

Answers to the following research question are consequently sought: How do foundation phase teachers use mathematical manipulatives to teach number concepts? The rationale for this research is furthermore strengthened by the following pertinent questions, based on the research findings emphasised by Back (

Performance of learners in mathematics is aligned to understanding of number concepts rooted in the foundation phase. The literature review focuses on pertinent aspects related to the teaching of number concepts as encountered during the course of this study, the different types of concrete manipulatives used in a mathematical classroom and the use of concrete mathematical manipulatives reflected in past research literature.

As previously stated, the number content area forms the foundation for all the other content areas in the foundation phase. In order for teachers to lay a firm foundation for learners, number concepts should be taught and understood well. Subsequently, learners’ chances of success in other content areas would increase substantially. Locuniak and Jordan (

The selection and effective use of appropriate mathematical resources require careful consideration and planning on the part of the teacher (Drews

it is important that learners do not come to rely on using concrete materials for modelling number but that they develop mental imagery associated with these materials and can then work with ‘imagined’ situations. (p. 10)

The mere use of manipulatives does not necessarily mean that desired outcomes such as promoting understanding and enhancing knowledge of mathematical concepts will be achieved (Van de Walle, Karp & Bay-Williams

Many foundation phase classrooms employ concrete mathematical manipulatives such as number charts, number lines, counters, Dienes blocks and Cuisenaire rods. These mathematical manipulatives are designed to represent and develop mathematical concepts. However, it is not always easy for young children to connect concrete objects such as blocks, beans and sticks, with mathematical concepts (Paek

Manipulatives can be helpful to young children when they are used correctly (Boggan, Harper & Whitmire

Contrasting views, referred to as ‘inconsistencies … within the manipulation-based literature’ by Carbonneau, Marley and Selig (

In contrast, some of the older studies by Heddens (

Later studies corroborate older research findings on the benefits of concrete mathematical manipulative, such as the research project by Cockett and Kilgour (

Research by Carbonneau et al. (

such [as] the perceptual richness of an object, level of guidance offered to [learners] during the learning process, and the development status of [how] the learner moderate the efficacy of manipulatives. (p.391)

This study uses Vygotsky’s (

The ZPD is one of Vygotsky’s (

the distance between the actual development level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or collaboration with the more capable peers. (p. 86)

The ZPD relates to knowledge that a learner is capable of learning with appropriate teaching, support and guidance by a more knowledgeable other (Wright et al.

As mentioned before, Vygotsky (

The ZPD can be divided into four stages (Dunphy & Dunphy

Stage 1 is where learning is assisted by a more knowledgeable other (Dunphy & Dunphy

Stage 2 is called self-assisted. This is when a learner is able to perform and carry out tasks independently and is trying to make sense of tasks independently. This does not mean that the performance is fully developed or internalised. It merely means that the control and direction of the performance has been passed on to the learners (from other-regulation to self-regulation).

Stage 3 is where performance is developed and automatised. At this stage, a learner is fully able to perform on his or her own and has advanced from the ZPD into the developmental stage for the task. The task is now achievable without intervention or assistance from the more knowledgeable other.

Stage 4 is where de-automatisation of performance leads to going back through the ZPD and starting from stage 1 again. Dunphy and Dunphy (

Through their research aimed at improving the mathematics results of some schools, Scott and Graven (

focusing on the learners’ actual interaction during collaborative learning activities enables teachers to describe how learners’ making meaning of conceptual representations emerges in a particular setting and is responsive to the characteristics of the setting. (p. 158)

This suggests that learners’ performance could vary from activity to activity, day to day and under different contexts, depending on the above-mentioned interactional influences. Learners move through the ZPD in ways which are predictable, as a result of making their own sense of what is provided by the teacher. Bliss, Askew and Macrae (

This research is by nature qualitative and located within an interpretive case study approach. The case study is used as an empirical inquiry that investigates an existing phenomenon in depth and within its real-life context, particularly when the boundaries between phenomenon and context are not clearly evident (Yin

This research was conducted in two primary schools in a sub-area of the Western Cape, South Africa. Both schools comprise learners from communities considered low socio-economic status. The schools were referred to as school A and school B. School A where the second author teaches is situated in a Xhosa-speaking area and school B in a previously Afrikaans-speaking school in Athlone.

The participants were purposively selected on the basis of their knowledge and experience in the foundation phase and all had mathematical manipulatives (resources) in their classes as per recommendations from CAPS. The first author worked with five foundation phase teachers, two of whom were from the school at which she was teaching at the time of the study, whilst three were from school B. She worked with one teacher per grade, from Grade 1 to Grade 3 in each school. The teachers were selected according to their experience in the grade concerned.

Information about participants.

Teacher | Age | Number of years teaching mathematics in the foundation phase | Qualification | Total number of years teaching |
---|---|---|---|---|

Teacher 1A | 28 | 3 | BEd (foundation phase) | 4 |

Teacher 2A | 57 | 35 | HDE | 35 |

Teacher 1B | 28 | 5 | BEd (foundation phase) | 5 |

Teacher 2B | 35 | 5 | HND (senior phase in Zimbabwe) ACE (FP) | 13 |

Teacher 3B | 40 | 5 | BEd (foundation phase) | 5 |

HND, Higher national Diploma; HDE, Higher Diploma in Education; ACE, Advanced Certificate in Education; FP, foundation phase.

Data were collected through document analysis, non-participatory observation and semi-structured interviews. The five selected teachers’ lessons were observed whilst teaching number concepts in their classrooms, using available concrete manipulatives. An observation schedule was used. The researcher (second author) recorded the events that happened during course of the lessons in terms of how the manipulatives were used to enhance learners’ understanding in terms of number concepts.

Data analysis was an on-going process whereby the first author extracted meaningful conclusions from the data collected (Patton & Cochran (

The observation and interview findings are presented as integrated discussions to present a more holistic picture. The focus of analysis involves the following: teachers’ teaching strategies in terms of Vygotsky’s ZPD, teachers’ perspectives on the role and use of concrete mathematical manipulatives and essentially a critique of incidents reflecting how teachers actually went about using these manipulatives.

During the observations, it was established that the teachers were aware of what the CAPS document stipulated about mathematics lessons in the foundation phase. All their lessons started with a whole class activity before the teacher engaged with a small group on the mat, leaving the rest of the learners working independently at their tables. The mat work was identified as stage 1 within Vygotsky’s ZPD. This is the stage during which the more knowledgeable other assists by scaffolding the learning process. Teachers engaged learners through asking probing questions and using various manipulatives. The teachers supported the learners through modelling, guiding and discussion in order to achieve the desired learning outcomes.

Stage 2 in Vygotsky’s ZPD was visible at the learners’ tables after the lessons on the mat. Learners were then given activities to complete independently at their tables. Learners were expected to do the activities on their own but could still ask for help if necessary. Teachers differentiated the learning activities to suit the levels at which the learners were competent. Blue books were used because these books are all the same and do not differentiate amongst learning activities.

Stage 3 of the ZPD, when performance is developed and automatised, took place right at the beginning of the lessons when the teachers allocated activities to the class before taking a small group onto the mat. Learners were allocated activities to complete on their own. These were activities relating to concepts taught and already understood. This stage was more visible in some classes than others. In the Grade 1A and Grade 3B classes, learners constantly came to the mat where the teacher was busy with a small group to ask for assistance with the activities assigned to them. In the rest of the classes, the teachers could go through the whole lesson without interruptions from the other learners.

When asked what the purpose of manipulatives in teaching number concepts was, teachers gave similar responses, namely that manipulatives were important to promote understanding in the foundation phase. Teacher 2A replied:

‘If I look at my children, manipulatives are

Teacher 1B answered:

‘Definitely, like with studying also

The above-mentioned responses and those by other teachers clearly suggest that teachers seem to know the value of mathematical manipulatives, even if it may not always have been in a deeper sense of the word. Teachers also reveal some understanding as to the importance of teaching number concepts development. Words and utterances such as ‘definitely’, ‘have to show them’, ‘have to have’ and ‘I need manipulatives’ reveal a sense of dependency on manipulatives and a belief that it has a powerful role to fulfil in helping learners to understand, which are in line with Drew’s (

Upon probing deeper, it became clear that participant teachers realised the potential didactical value of concrete manipulatives and that using manipulatives was an important component of teaching. This viewpoint matches the claims by Cockett and Kilgour (

‘Manipulatives are

Also, this teacher expresses a firm belief in the importance of manipulatives as fundamental in laying the basic foundations in terms of developing, understanding and conceptualising number. This belief is in line with Phillips’ (

Observation of lessons indicated that teachers tended only to resort to resources when they saw that the learners did not understand a concept, which may not be the appropriate way to use manipulatives a pointed out by Back (

The teacher then looked around and saw that some of the learners had made more hops than they were asked. Some learners grasped the concept and others did not. Realising this, the teacher then told the learners to count the hops that were made on the board and then count their hops to see if they had made the correct number of hops. The teacher tried to help the learners who had more hops by counting and demonstrating on the big number line on the board (see

A learner shows how to hop from 2 to 4 on the number line.

The teacher attempts to intervene after learners showed signs of not understanding.

The teacher gave the learners more problems to solve using the number line so that they could master using the number line (3+3, 0+6, 6+0). They were then asked to complete one more problem on their own. This did not mean that the learners had reached stage 2 of the ZPD, although that was what the teacher wanted. One lesson for the concept of number lines was not enough. It was almost certain that if the teacher had given the same learners more of the same type of computations to do that same day or even the following day, they would not have been able to complete them without her assistance.

The teachers observed had access to the required mathematics teaching manipulatives described in CAPS. Each classroom observed had a number chart, number lines, number tracks and strings of beads. At school B, the teachers had their own learner teacher support material (LTSM) kit which consisted of all the required materials recommended by CAPS. School A had one kit to share amongst the whole school. At school A, teachers did not use the LTSM kit because it was an effort to go fetch the resources they needed from the learner support teacher responsible for them and they did not even know what it contained.

Teachers at both schools used the number lines but did not use them with the concrete bead string as CAPS recommends. Learners were not given enough time to engage and explore with the manipulatives because these were kept in closed boxes. These materials should not be in the boxes in which they originally came. They should be placed in the mathematics corner for anyone who comes into the class to see so that the learners can use them. When learners went back to their tables to work independently, there were no manipulatives available for them to use. If and when they used manipulatives, it was only on the mat with the teacher. At their tables, they only engaged in written work. This was because they had already practised the work on the mat with the teacher and the teacher probably saw no need for manipulatives to be used again. However, if learners need to assist themselves as stage 2 as the ZPD suggests, the required manipulatives should be available for them to do so.

Teachers used mathematics manipulatives to teach computations (number sentences), that is, they used manipulatives to help the learners get to the answers. This points to ‘rote learned procedure[s] without a sense of the ways in which the apparatus reflects mathematical structures’ (Back

Vygotsky (

Most of the teachers were able to speak English and Afrikaans but not the African languages. This made learning a challenge, especially in the Grade 1 classes, because most of the isiXhosa learners were being taught in English for the first time. By the time they got to higher grades, they would be able to communicate in English, but by then they would have missed out on the basis of mathematics taught from Grade 1. Teachers said that it became an even bigger challenge when learners came from isiXhosa- or Afrikaans-medium schools and only started at an English school in Grade 3. The content to be covered in Grade 3 is much greater than in previous grades and communication lines are effectively closed.

There are clear commonalities in terms of how participant teachers view, use and apply concrete mathematical manipulatives. They all exclaimed appreciation for the inherent usefulness of manipulatives because they believe such materials assisted learners’ thought processes and also that manipulatives served as catalysts for decomposing intricate concepts to learners’ levels of understanding (Martin et al.

From the research findings, it is evident that mathematics teachers in the foundation phase are in need of training of how to use and follow Vygotsky’s ZPD and how to make sense of and follow the CAPS mathematics guidelines in respect of when and how to use mathematical manipulatives.

There were instances where teachers used a number of different manipulatives, such as beads, the number line and flash cards jointly to help develop and reinforce the development of a particular number concept or procedures related to number operations such as addition, but that practice was not used consistently. At the same time, however, teachers should heed Anghileri’s (

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

L.M.M. was responsible for research design, data collection, and analyses. S.A.A. and S.W.S. refined the article, assisted and reworked the article according to the reviewer reports.