The aim of the study is to investigate teachers’ perceptions about peer collaborative work in designing lessons as a team helped them to identify threshold concepts in the teaching and learning of foundation phase mathematics in Motheo District of Education.

A qualitative approach, with a case study design, was used to combine data from observation and focus group discussions, interviews and group task sheets.

Classroom observation was conducted during a workshop conducted by a subject advisor from the Motheo District of Education in collaboration with the researcher. Teachers were purposively selected from seven schools in the Motheo District of Education based on cluster sampling as a way of reviving their professional development through acquisition of mathematical teaching skills involving innovative approaches to teaching and learning of early childhood mathematics. Seven mathematics teachers, one from each school, were interviewed during the workshop.

Underpinned by a collaborative theory, the findings of the study revealed that peer collaboration in mathematics teaching was key to helping them (participant teachers) identify threshold concepts in mathematics that they had initially found difficult as individual teachers. This assisted them in teaching the subject effectively at the foundation phase level. The study, furthermore, established that collaboration by mathematics teachers was necessary in order to overcome the paucity of global mathematics teaching skills for early childhood mathematics, to foster learners’ knowledge of mathematical concepts and to stimulate their interest in the subject.

It is recommended that more structured collaborative work amongst teachers in general should be encouraged to enable teachers overcome the problem of content gap in their area of specialisation.

The dawn of democratic governance in South Africa in 1994 has been followed by a series of reforms in the education system (Khuzwayo

Shepherd (

Newborn (

[

However, it seems that in many countries, teacher preparation programmes for FP learners for the rural areas sometimes delay for no apparent reasons and do not facilitate the acquisition and development of the necessary content knowledge required by teachers to teach the curriculum to perfection. Some of these programmes are not implemented early at preschool for the learners to acquire skills but put off until they attained the age of 6 years or more, depending on the child’s ability to manipulate objects (Preston & Haines

The current study was triggered by a request by a senior education specialist in the Motheo education district of the Free State province of South Africa to organise an intervention programme to support mathematics teachers in the district with the ultimate aim of improving the quality of teaching at FP level and learners’ performance in mathematics in the early stage of FP level. Prior to this intervention programme, the researcher realised that there was a deficiency in teachers’ professional learning and development, through collaboration in the district, as a way of increasing interest to support the progressively complex skills learners need from teachers to learn mathematics in preparation for further education and work in the 21st century. Darling-Hammond, Hyler and Gardner (

How can collaborative work on designing lessons as a team help teachers identify threshold concepts in the teaching and learning of early childhood mathematics?

Peer collaboration and co-operation amongst teachers are regarded as key factors in improving teachers’ PD. International researchers such as Ni Shuilleabhain and Seery (

Studies have shown that the concept of teamwork is pervasive within the United States Army but found to be limited in the world of academia (Charbonneau et al. 2010). A review of related literature in international contexts has revealed that effective collaboration amongst peer teachers for lesson planning is a form of development of teacher knowledge. Furthermore, collaboration encourages a more learner-centred approach to teaching and learning of mathematics (Dudley

In this study, teachers’ perceptions about collaboration were introduced as a new school-based teacher model of PD facilitated through a workshop. Teachers of a particular community met and collectively discussed and identified key mathematical concepts that needed to be taught but with which they were not familiar. Doing so facilitated and improved the teaching of mathematics and further assisted teachers to improve their understanding of certain mathematical concepts.

This study is underpinned by Vygotsky’s sociocultural theory which states that because learning takes place amongst individuals, it is an inherently social process activated through the zone of proximal development (Dillenbourg

The aim of every teacher is to establish a professional learning community conducive for his or her learners as one of the effective means for enhancing teachers’ PD effort in the teaching environment. Studies have shown that different methods or forms such as theory-driven approach and Manabu Sato’s learning community theory can be applied to structure all components of teacher PD workshops that impact positively on teachers’ teaching beliefs, knowledge and skills acquisition for better teaching and learning (Darling-Hammond et al.

This research followed a qualitative approach with a case study, as the research sought to understand the experiences of teachers who collaborated as a way to identify and understand certain mathematical concepts that were perceived by the participating teachers to be difficult to teach at the FP level as well as the relevant principles that enhance effective teaching and learning of ECE. Following a discussion with the mathematics district curriculum specialist (DCS), mathematics teachers from certain schools especially where we have FP classes were invited to participate in the research. Participants (30 mathematics teachers) were purposefully selected from various schools in a Free State Education District by means of a cluster sampling technique. At least one teacher was selected from each cluster, with a total of 15 clusters based on FP levels. The aim was to have at least one representative from each cluster, whereby he or she would share the skills and strategies acquired through this collaboration with his or her cluster members at a convenient time in teaching of ECE learners. The purposive sampling technique was used to select 30 FP level mathematics teachers for the purpose of this study. Ten mathematics teachers each from the following three categories were used for the selection of the participant teachers: 10 mathematics teachers from high-achieving schools either in the current or previous teaching experience (five from urban schools and five from rural schools), 10 mathematics teachers from average-performing schools (five from urban schools and five from rural schools) and 10 mathematics teachers from low-performing schools (five from urban schools and five from rural schools) in the Free State province. These schools were identified by relying on the Annual National Assessment results for the subsequent 3 years in the province and have experience of teaching FP levels, skills and strategies used in teaching these children for meaningful understanding, as most of their parents do not have time for their children at home. The purpose of the selection strategy was to share a variety of opinions from different teaching and learning environments.

Observation, focus group discussions, group task sheets and in-depth interviews were used as data collection strategies. Evidence was gathered by observing a group of 30 mathematics teachers who participated in a workshop (see

Objectives regarding the collaboration to identify threshold concept effective for ECE.

Action plans to solve problems under discussion using physical or real objects of ECE.

Reports and feedback pertaining to the resolution for the problem (P1) elaborated on by individual teachers in the study.

Final reports on the resolutions of subsequent problems to be discussed or solved with learners in the class regarding the actual teaching of learners.

Contingency plans that show teachers’ reflective practices used to identify other threshold concepts in early childhood-level mathematic teaching and learning and to assist learners further to solve problems where possible.

Reports from individual teachers indicating the success or failure of a collaboration in identifying certain mathematical concepts.

The way forward for identifying more threshold concepts for future discussion.

Activity 1 on mental maths.

Peer collaboration during the workshop.

Peer collaboration during the workshop.

In addition to the data listed above, the researcher requested information from the participant teachers on the performance of their learners in their various schools prior to the collaboration, the mathematics curriculum and textbooks used in teaching and learning. The researcher also used his field notes report collected to assist in triangulating the data gathered from the teachers. Teachers were later interviewed for further details to augment their outputs.

Observation and focus group discussions, interviews and group task sheets were used for data collection (see

Before any discussion of threshold concepts took place, three of the participant teachers taught three different lessons, with the group observing. An observation schedule, designed according to a strand or strands of teaching mathematics for proficiency, was used to identify particular concepts and how these were presented to the group. The lessons were also videotaped for the purpose of reflection and discussion. Doing so assisted the researcher in critically analysing teachers’ mathematics knowledge.

The video recording assisted the group in successfully identifying the listed threshold concepts for group discussions. During the course of the discussions, equal opportunities were given to each participant teacher to contribute, share their opinions and clarify points in relation to the identification of threshold concepts that assist learners with problem-solving. This was followed by discussion and in-depth interviews. Throughout the study, participants were recorded and analysed who, when teaching and during -in-depth interviews, actually demonstrated interesting mathematics knowledge for identifying key concepts in their presentations and in the task sheet.

Based on the purpose of this study, relevant participants and suitable instruments were selected for this study. The data collected in this research were recorded electronically and transcribed for analysis. The participant teachers were given the opportunity to review the transcriptions to ensure that they were accurate. The researcher used semi-structured questions to guide the interviews, video recorded the participants as they presented their views based on the questions asked or under discussion. The researcher also studied the lesson plans for lessons presented during this study. The researcher also ensured descriptive validity which actually goes into the details of what actually had been gathered at the field; hence he used open and transparent procedures in gathering the raw data without any fabrication of any part of information (Maree

Prior to the analysis, all recordings were transcribed. The researcher analysed the data collected, that is, raw data captured from the responses of the participants who were asked the same set of questions, including some body language as well as facial expressions exhibited by the participants, which were recorded in field notes. An inductive analysis approach was used, and the data analysis was guided by specific evaluation objectives, which involved a detailed reading of the raw data to derive the concept, themes or models. This understanding of inductive analysis in research starts with an area of study and allows the theory to emerge from the data (Miles & Huberman

Ethical clearance was obtained from the University of the Free State, Ethical Clearance Number: UFS-HSD2018/0395. 17/07/2018

The purpose of the research was to establish mathematics teachers’ perceptions about peer collaboration amongst teachers as a way to address difficult key concepts or identify the threshold concepts in early childhood mathematics teaching and learning. The data analysed in this research were gathered through observation, focus group discussion, interviews and task sheets. After teachers engaged in collaborative discussion and shared ideas with one another and with the researcher (see

The teachers formed groups comprising at least seven teachers per group of four (see

‘Actually, in our group, we were really impressed by the way various teachers solved problems on fractions. For the fact that we are mathematics teacher does not mean that we know everything. Some of the skills, methods, strategies demonstrated by some teachers through this research has been overwhelming in dealing with ECE mathematics. We are really blessed to have collaborative work like this that exposed us to different opinions of solving mathematics problems or different ways of identifying mathematical concepts which makes teaching of mathematics in FPs very ease.’ (Group 3, teacher 2)

‘We need to embark on this kind of project very often whether we like it or not because we get to know many things during group discussion which is really difficult for most of us to understand when we plan alone as individual teachers in our respective schools. We could now see different ways of addressing mathematical problems or identifying mathematical concepts that will definitely help us to guide our future learners by showing different skills, activities, concepts and models to make our teaching enjoyable and understandable to our learners which we never knew at the beginning (See

‘Unlike the teacher who only challenged the learner by mere talking without any illustrations is not a good way or procedure of teaching FP learners. The reason why most teachers cannot develop their skills and strategies by applying practical work in their teaching is that, in most cases, the problem may come from the Department of Education, whereby you are being forced to complete the syllabi at all cost without taking into consideration the cognitive level of the learners. When this happens, you will be forced to teach abstractly without doing illustrations of this nature and this does not help you as a teacher teaching FP learners. This must in fact, be looked into by those at the management level in order to recruit teachers with relevant skills and strategies to improve teaching and learning of FP mathematics.’ (Group 1, teacher 1)

‘There no doubt that anybody here will oppose collaborative work looking at what we have acquired here through challenging, discussion, probing and demonstration. We need to advise the department to make provision for this type of workshop which we hardly get so that we will be able to share our ideas. We are lucky that DCS for mathematics is here with us and we hope to see him taking this request to the provincial manager.’ (Group 4)

During classroom observation of lessons presented by individual teachers, the focus was on the way teachers presented their lessons in relation to their content knowledge and PCK so as to identify key concepts based on the topic presented in line with what Darling-Hammond et al. (

This is what some teachers had to say:

‘This topic is about ‘factors’ and ‘multiples of 2, 4 and 20.’ (Teacher C)

‘This particular example is very easy but at times, you find it very difficult to identify some key concepts in topics like fractions, 3D shapes and naming of intercepts in geometry, so we always need to do collaborative work like this to empower us to overcome any barriers in teaching of early childhood mathematics of this nature.’ (Teacher A)

‘Even though collaborative work is good for teachers for PD; however, it needs time and commitment to make it work effectively. But looking at our time schedules nowadays, it’s not easy to have teachers come together to have this kind of engagement. I can’t imagine having a collaborative work like this in my lifetime.’ (Teacher C)

During group discussions, comments and questions raised by the teachers on the task sheet provided opportunities to understand the content and produce golden rules for effective teaching and identification of mathematical concepts. This form of providing coaching and expert support teachers, which involved the sharing of expertise about content and evidence-based practices, focused directly on teachers’ individual needs as well as their learners for PD, as indicated by Darling-Hammond et al. (

Peer collaboration amongst mathematics teachers is a way to address difficult key concepts or to identify the threshold concepts in early childhood mathematics teaching and learning, which threaten teachers’ feelings and confidence. The solutions presented for questions revealed the kind of mathematical knowledge teachers possessed and how that knowledge paved the way for them to identify certain concepts in some aspects of their teaching and learning of mathematics regarding both content and method. Teachers shared ideas through group discussion, and it revealed what was happening in their classes to promote the development of mathematical proficiency. By identifying mathematical concepts during teaching and learning, examining how teachers present their lessons to their learners and linking ideas in different contexts produce meaningful learning. How the concepts could be linked to real-life situations were checked in line with the kind of concept being taught. The categories and themes identified based on knowledge of mathematics (content) and knowledge of instructional practices (method) were used to compare what Kilpatrick et al. (

A fraction question was posed to the teachers so that they could demonstrate their skill in identifying mathematical concepts to facilitate the understanding of their learners.

Question 1: If the answer to a sum of a particular problem is

The following were the responses by the groups:

During focus group discussions, an attempt was made by the various group members to answer this question; however, some only provided single solutions without explaining their chosen answer. Group A, for instance, simply wrote

In actual fact, establishing that the sum of the numbers is

Initially, the teachers believed the sum involved two numbers being added; however, as discussions continued, it was evident that the participant teachers knew the sum was not just a mere adding of only two numbers, but addition of different numbers. It was clear that teachers found engaging in such a collaborative discussion was useful because skills for identifying mathematical concepts could be acquired easily. However, time and commitment are needed to make this method or approach work effectively.

Extensive research into teacher communities is not common in mathematics education; therefore the need for collaboration is worth considering (Gellert

Results discussed here form part of a larger qualitative study that investigated difficulties experienced by mathematics teachers in teaching mathematical concepts in schools. The study revealed that peer collaboration in early childhood mathematics teaching is key to helping teachers identify threshold concepts in mathematics that they had initially found difficult as individual teachers (See

The study, furthermore, established that collaboration by mathematics teachers of different calibres is necessary to overcome the paucity of global mathematics teaching skills for childhood-level mathematics, in order to foster learners’ knowledge of mathematical concepts and to stimulate their interest in the subject.

Sarason (

It is believed that mathematics is a sequential process or development, fixed to a certain person, topic, environment or idea that changes or influences the life of that person through thinking and doing. Researchers such as Chamoso, Cáceres and Azcárate (

[

This indicates the vital importance of what the teacher knows, how much he or she knows of the content and that he or she knows how to present it so that learners understand. Peer discussion, thinking, doing and sharing of ideas should always go hand in hand in mathematics learning, as it is these activities that help stimulate learners’ creativity in divergent ways through coaching (Vygotsky

From the findings of this study, it can be concluded that teachers demonstrated different kinds of mathematical knowledge, knowledge of instruction and knowledge of curriculum to identify threshold concepts in mathematics. Through extensive collaboration, teachers can develop and acquire knowledge and skills relevant to tracking unnecessary misconceptions amongst learners in the mathematics classroom and hence develop an interest in understanding mathematical concepts in everyday life. The study concluded that collaboration was beneficial for teachers in the following ways: it helps in providing coaching and expert support for teachers, which involves the sharing of expertise about content and evidence-based practices, and it focuses directly on teachers’ individual needs, as well as their learners, for PD through content-focused discussion. It further incorporates active learning amongst teachers whereby they share their problems and find solutions through collaborative support. They make use of models of effective practices that offer sustainable feedback and reflection, which provide teachers with adequate time to learn, practise, implement and reflect upon new strategies to facilitate changes in their teaching practice. Based on the results, it is recommended that a teacher collaboration network should be organised for teachers. Teachers demand PD programmes such as workshops and in-service training to be fully implemented to assist the teachers to grow and develop professionally. Teachers also indicated that there is a need to enforce team-teaching amongst mathematics teachers, which encourages monitoring of the progress of all the mathematics teachers in the schools in the province. It is further recommended that collaborative class observation, discussion and mutual result reflection should be engaged in on a regular basis.

The author would like to acknowledge the teachers who voluntarily contributed to the success of this study.

The author declares that they have no financial or personal relationships which may have inappropriately influenced them in writing this article.

I declare that I am the sole author of this research article.

The researcher received no specific grant from any funding agency.

Data sharing is not applicable to this article as no new data were created or analysed in this study.

The views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author.