Two Pathways into Number Work for Primary Teachers: A Counting Pathway and a Measurement Pathway

Abstract

Teacher quality in primary schools in South Africa is a serious concern, and how best to prepare teachers in initial teacher education—with regard to mathematics—remains an open and vexing question. There is a confluence of challenges facing the mathematics education community in South Africa: poor conceptual understanding of mathematics amongst teachers; poor attainment in Primary Teacher Education mathematics tests at both first and fourth year levels; and few initial teacher education programmes having evidence of improving knowledge for teaching mathematics. In addition, few lecturers in initial teacher education make explicit the theoretical tenets informing their course design for mathematics. In response to the latter challenge, this paper makes explicit the design choices made in the Emergent Number Sense first year mathematics module in Bachelor of Education degrees, used across six universities in South Africa. It then theorises the choice to commence the B.Ed mathematics programmes with two pathways into number knowledge: (1) a counting pathway, making use of discrete objects or events which can be quantified by counting in ones; and (2) a measurement pathway, making use of continuous extents which can be quantified by introducing a unit of measurement. Each has a distinct metaphor, strategy for the reification of the 10 and supporting representation. This offers the lecturers using the course in their own design trials, and initial teacher educators beyond the Maths4Primary Teachers collective, a conceptual framework for this design choice, which can now be critiqued and further improved.

Keywords:

• Initial Teacher Education
• Mathematics Education
• primary/elementary
• PrimTEd curriculum standards

Introduction

Low performance in mathematics and mathematical alienation is reproduced in mainstream primary schooling. How this occurs is multifaceted, rooted in historical apartheid narratives, histories of neglect, inequitable distribution of resources and a bifurcated system of education with too little research serving mainstream teachers to leverage home language resources for conceptual understanding in mathematics. By ‘mainstream’ we draw on Ramadiro and Porteus (2017) and refer to the 70% of primary schools that charge no school fees and are poorly resourced and African language dominant, which are systematically demonstrating little progress in mathematics by Grade 3.

While recognising the complex nature of poor performance in mathematics across the South African schooling system, a long recognised and key contributing factor to the low performance in early grade mathematics is primary teachers’ ‘fragile mathematical knowledge and their poorly connected instructional narratives’ (Venkat & Roberts, 2022: 213). Research has consistently affirmed teachers’ lack of both conceptual and instructional competence in mathematics (Bowie et al., 2019; Taylor, 2021; Venkat & Spaull, 2015). Most teachers have learned early mathematics through a combination of memorisation and harsh discipline (Roberts, 2017). The research suggests that many teachers (not only children) do not have a mental map for numbers beyond a metaphorical line of literal and discrete ones. Recent work deeply embedded in schools suggests the depth of alienation of foundation phase teachers from the conceptual grounding of early number sense (Porteus & Mostert, 2022).

Within this context, how we prepare primary school teachers who exit the mainstream schooling system and enter initial teacher education programmes to become primary school teachers becomes pertinent. In this regard, the broad accreditation framework for initial teacher education offers little guidance as to what should be taught. A large-scale effort by Primary Teacher Education (PrimTEd) to assess student teachers in their first and final years of study concluded that Bachelor of Education (B.Ed) programmes across a range of South African institutions are currently not significantly improving students’ performance in mathematics (Alex & Roberts, 2019; Bowie et al., 2019; Roberts & Moloi, 2022). Given that coursework is individualised to specific universities, there has been little open sharing of coursework, course assumptions or the theoretical basis of instructional choices made. As suggested by the work of Bowie and Reed (2016), there is not a strong evidence base through which to answer the big instructional questions, ‘how much of what?’ and ‘where should we start?’ with regard to mathematics in initial teacher education.

In this paper we make explicit the design choices made for the Maths4Primary teachers ‘Emergent Number Sense’ module, which was used as the first module for BEd programmes at foundation and intermediate phases, in six South African universities. Having made the key choices explicit, we then theorise the choice to commence the BEd mathematics programmes with two pathways into number knowledge—a counting pathway and a measurement pathway— each framing a different representational trajectory to build early number sense.

A Confluence of Challenges in ITE

This theoretical paper is motivated by a confluence of challenges facing the mathematics education community in South Africa, in reference to early grade teaching of mathematics. We summarise the confluences below.

Foundation Phase Teachers and Mathematics

Early grade mathematics researchers have long been concerned about the poor conceptual understanding of mathematics amongst teachers (Taylor, 2021; Venkat & Spaull, 2015). Research has been concerned by an over-reliance of learners on unitising (or counting in discrete ones), without consolidating one of the first key early abstractions of a reified 10 (Ensor et al., 2009; Graven et al., 2013; Hoadley, 2012; Roberts, 2019). As described by the work of Venkat (2013), notions of number remain temporal and literal rather than generalisable and abstractable. Emphasising instructional talk as one of the most important tools for semiotic meaning making in classrooms, Naidoo and Venkat (2013) suggest that many teachers’ instructional discourse is simply not coherent enough to leverage learner meaning making. They talk about the ‘ahistoric’ nature of early grade mathematics instruction whereby there is a ‘complete delinking of subsequent and prior examples, and a lack of “building” unknown knowledge from known information’ (Naidoo & Venkat, 2013: 73).

Schollar (2008) was one of the first to isolate the tendency of learners, well into the intermediate and senior phase, to solve higher number additive problems by counting in ones, drawing small circles (and crossing out to subtract). Far into their mathematical careers, learners have a mental imagery of numbers as a long line of discrete ones (often drawn as small circles). The work of Porteus and Mostert (2022), however, extends this concern. They suggest that this mental imagery (an endless line of discrete ones) is not only common among learners, but also shared by some teachers.

Mathematics Performance in Initial Teacher Education

Since the closing of the teacher education colleges in South Africa in the mid 1990s, initial teacher education has taken place in the context of institutions of higher education. Over the past 10 years, the education research community has become more focused and concerned about the quality of initial teacher education (see the review in Fonseca et al., 2018). The Initial Teacher Education Research Project reviewed mathematics and language courses in the BEd programmes in five universities. Deacon (2012) provided an important critique of the overall quality of what was offered in these programmes. In 2014, Taylor’s review of initial teacher education concluded that the poor quality of teaching lies ‘not with teachers but with the teacher education system that produced them’. Bowie and Reed (2016) called for a critical review of BEd programmes, with a special reference to mathematics.

In 2016, the PrimTEd assessment working group developed instruments designed to assess BEd students’ mathematical knowledge. While the administration of the assessments has not been consistent across universities, and was largely interrupted under Covid-19, one consistent result stands out (Bowie et al., 2019; Fonseca et al., 2018). The mean result of all first-year students in institutions participating in the three year project period (2017–2019) was 47.6% (SD = 15.6%). The mean result for fourth year students was 52.5% (SD = 16.5%). As such, the improvement in mathematical knowledge between first- and fourth-year students was only 5 percentage points. What was most striking about these results was that they were common to a great extent across all institutions, as is evident in Figure 1.

Figure 1. Primary Teacher Education (PrimTEd) Mathematics test—mean results (2017–2019)

Source: PrimTEd assessment workstream (2019)

What do these results mean? There are several possibilities. The first is that the standards (and/or related assessment items) established in the first phase of PrimTEd are too high. This could produce these results in two ways. If the standards are too high, lecturers may be leaving students behind from the starting gate. Another option is that B.Ed students are actually improving more than is suggested, but in overestimating the baseline, the assessment items do not capture the growth accurately. The second explanation of these results is that, as the community of mathematics lectures in initial teacher education, across institutions, we do not yet know how to best leverage mathematical meaning making of our current students. We recognise that we do not yet have enough empirical experience to fully answer the question.

A New Research Focus: Theory, Practice and Impact of Mathematics in ITE

The PrimTEd mathematics results emphasised that the ‘problem’ of teaching mathematics to student teachers was not about individual lecturers failing to teach well, but rather that as a collective of lecturers in mathematics education, we needed to think differently about coursework serving mathematics initial teacher education. The generative question became less about describing the lack of progress of student teachers and more about exploring what actually works in mathematics initial teacher education in the South African context.

In 2019, Roberts and Mostert developed a Maths Intensive course for first year students, consisting of a full week of work focused on developing mathematics fluency and an online practice system to consolidate knowledge learned. A version of this course was undertaken across three years (2019–2021) at the University of Johannesburg and Cape Peninsular University of Technology. The first design trial resulted in learning gains (measured in shifts from a pre-test to post-test) of 14 percentage points (Roberts, 2020). The gains in a second design cycle (with a revised method owing to Covid-19) demonstrated similar gains (Roberts & Maseko, 2022). In the same period, a long-term design hub working with rural teachers in the Eastern Cape, known as the Magic Classroom Collective, shared a number of insights about the nature of current teachers’ relationship with knowledge (see Porteus, 2022) They had developed a number of locally generative heuristic tools (Porteus & Mostert, 2022), through which they demonstrated a significant increase in learners’ performance in foundation phase mathematics (Porteus et al., 2021).

The Project: ‘Maths4Primary Teachers’

In response to this confluence of challenges, a collective of mathematics lecturers across six institutions of higher education (Maths4Primary Teachers) has come together to discuss how to better collaborate in mathematics education for primary teachers in the context of initial teacher education. The shared goal was to design and develop coursework informed by both theory and empirical experience, trial the coursework in different institutional settings and measure the impact through structured pre and post-tests. The goals combined building a community of practice amongst lecturers, improving both the mathematics relationships and the competency of student teachers, and deepening our understanding of how to teach mathematics in this moment in South African history.

The Maths4Primary teachers collective placed ‘mainstream primary schools’ at the centre (Ramadiro & Porteus, 2017). We focus on building a knowledge project, re-centring the schooling system serving the majority of South African children. The collective purposefully sought to establish a dialectal engagement between the following influences: (1) mathematics education theory; (2) the lecturer collective; and (3) the PrimTEd standards. The goal was to make explicit the mathematics education theory upon which the course materials were based. Acknowledging that a great deal of mathematics education theory emerges from more resourced contexts with substantively different starting points, we aimed to critically engage with theory, translating it for the given context. Much of the experience of initial teacher education in mathematics resides in the experience of lecturers, much of it not written up. This experience is approached as a valuable asset. The experience feeds into the design continually, through processes of design and reflection. When experiences differ, the design team privileges the experience emerging at ‘previously disadvantaged institutions’ (in this case University of Fort Hare, Walter Sisulu University and Tshwane University of Technology)—universities that often serve students with an especially fragile claim on mathematical meaning making. The coursework makes explicit the PrimTEd standards it seeks to address, and measures impact through pre- and post-tests drawn from the PrimTEd assessment items. The work seeks to extend the work on standards to demonstrate how these standards can be achieved, and dialectically engage with the standards to ensure they are held accountable to instructional practice.

In addition the collective has deliberately drawn on interventions trialled in universities, as well as in mainstream schools. In the initial phase, the coursework took forward much of the design work initially developed through the Maths Intensive described above. The design team incorporates innovations from other universities, and especially the applied work of the Mathematics Chairs at both University of the Witwatersand (Venkat, 2022) and University of Rhodes (Graven, 2022). The coursework is informed by a deepening understanding of teacher and learner meaning making in mainstream primary schools. The design team makes extensive use of material from the Magic Classroom Collective, a long-term design hub embedded in a cohort of schools in the rural Eastern Cape (Ramadiro & Porteus, 2017; Porteus & Mostert, 2022; Porteus, 2022). We also draw from other South African mathematics interventions in the early grades, such as R Maths (Spencer-Smith et al., 2022), Jump Start (Moloi et al., 2022) and Number Sense (Brombacher & Roberts, 2022).

The first module developed by Maths4Primary teachers was entitled Emergent Number Sense. This 8-week module was designed as the first module for first year initial education in mathematics; it establishes our estimation of a ‘starting line’. The lecturer collective made numerous choices relating to the foci of Emergent Number Sense. Key amongst these were to:

1. establish the basis of two pathways into number knowledge (a counting and a measurement pathway), each framing a different representational trajectory to build early number sense;

2. be explicit that underlying both pathways is a structured approach to number, which draws on early algebra;

3. pay deliberate attention to language use in mathematics, attending to both English and isiXhosa for the bilingual rural context;

4. develop the students’ mental mathematics fluencies with particular focus on 2s, 5s and 10s (in both additive and multiplicative relations);

5. slowly induct students into engaging with mathematics education literature; and

6. attend to students’ wounded personal stories of their relations with mathematics and of learning mathematics (see Roberts, 2017), so that they re-experience mathematics with a growth rather than a fixed mindset, and build their ‘mathematical acting and thinking’ capacities.

This theoretical paper establishes the theoretical argument motivating the first decision. The two pathways into number knowledge were defined as: (1) a counting pathway, making use of discrete objects or events which can be quantified by counting in ones; and (2) a measurement pathway, making use of continuous extents which can be quantified by introducing a unit of measurement. Each has a distinct metaphor, strategy for the reification of the 10 and supporting representation.

Research Aim

In this paper we set out to theorise our choice of establishing a counting pathway and measurement pathway as a conceptual starting point for initial teacher education in mathematics for foundation and intermediate phase teachers.

Theoretical Discussion

Emergent Number Sense consists of six imithamo (bite sized pieces or sub-modules), each organised over roughly a week with at least two lectures (Roberts et al., 2021). When including assessment requirements, the module spans 8 weeks. The module is based on theories pertaining to mathematical representations and developmental pathways into number knowledge, in conjunction with our hypotheses about an appropriate ‘starting point’ for initial teacher education in South Africa.

Mathematical Representations

The theoretical choices guiding the introductory module emphasise the use of mathematically meaningful representations. There is a wide body of mathematics education literature focusing on the importance of representations in supporting children’s problem-solving processes in mathematics (Askew, 2012; Barmby et al., 2011). The notion of ‘representation’ signifies a reference to meaning or conceptual significance. ‘External representations’ (hereafter ‘representations’) denote teachers’ and children’s talk, markings, drawings and writings in mathematics. Representations are externalised tools, available for observation, discussion and manipulation making internal modelling (thinking) visible to self and others (Goldin, 2020). In their summary of research exploring classroom representations and children’s mental imagery, Askew and Brown (2003) suggest that children’s mental imagery of mathematics strongly reflects the mathematical representations used by their teachers (whether objects, verbal written, or pictorial). They suggest that learners who do well in mathematics build an early mental imagery beyond counting in ones.

The wide international research relating children’s thinking to representational tools (Sasman & Linchevski, 1999; Sfard, 2008; Tall, 2004; Wheatly, 1992) emphasises the importance of a child’s ability to move between the multiple representations, however these are defined. While the mathematics education community in South Africa has historically accepted this premise, providing multiple forms of flexible representation, we have been re-thinking the issue in the South African context. The use of multiple representations assumes the consolidation of a conceptual map through which to mediate diverse representations. As emphasised by the work of Askew (2012), in a system with weak mental models of mathematics for many teachers, introducing representational models requires two steps of internalisation. First children must learn to use the model effectively. Then, over time, this must translate into the internalisation of the tool in a way that a child reproduces the tool (without signalling the use of the tool) for her own productive processes. His work emphasises that this takes working with common representations over long periods of time (Askew, 2012).

Most of the literature focuses on representations used by children to both make meaning and externalise their thinking. In our case we are focusing on the meaning making of teachers, and the capacity of this meaning making to translate into meaning making of children into the future. We posed the following question: what starting points build representations that are capable of disrupting and re-structuring teacher’s mental maps of mathematics, can travel with teachers and children to anchor fundamental concepts over time, scaffold more accurate and productive instructional talk between teachers and children and are easily carried into classrooms with few resources beyond a chalkboard? We draw upon sociocultural theorists in recognition that tools are inherently ‘cultural’, in that they may resonate in one context and not in another, reflecting a range of historical and cultural influences.

The Developmental Pathways into Number Concept

The Emergent Number Sense module asserts that there are two main routes or pathways into quantification: a counting pathway (with an emphasis on discrete/cardinal quantities) and a measurement pathway. (with an emphasis on continuous/ordinal quantities). Accepting that the suggestion that these two ways are distinct is an oversimplification; we nevertheless believe that explicitly introducing number through these two pathways (and developing representations to build these imageries over time) provides a productive basis for students to re-claim mathematical meaning in the early grades.

Dehaene (1992) referred to a triple-code model of number where number comprises a number symbol (a visual Arabic code in which numbers are represented as sequence of digits), a quantity (an analogical quantity or magnitude code) and a number name (a verbal code in which numbers are represented as sequences of words). This conceptualisation links the number word to a symbol and to a quantity. Quantification is at the heart of number concept, to which both cardinality and ordinality are considered to be key foundations. The notion of quantity itself is only efficient for large numbers when it is structured—for example, into a base-10 system of place value.

There has been debate in the literature about what constitutes a children’s ‘starter pack’ (Butterworth, 2005). The dominant view of early number learning asserts that learning begins with the cardinal concept of number, involving enumerating and grouping collections (the numerosity of a set of objects). Butterworth (2005) proposes that humans are born with a capacity specialised for recognising and mentally manipulating numerosities (cardinal values) and that this capacity is likely to be embodied in specialised neural circuits. Sinclair and Coles (2015) suggest that the current dominant emphasis placed on cardinality was highly influenced by Butterworth’s (2005) emphasis on children working with objects, whether through subitising or adding on. They also point out that the influential work of Gelman and Gallistel (1978), identifying five counting principles, further reinforced a dominant focus on cardinality (Sinclair & Coles, 2015: 253).

However, the dominant view privileging following a counting pathway into number is challenged in several ways. First, researchers have suggested that the dominance of the counting pathway may reflect circular arguments in the literature. Sinclair and Coles (2015) challenge recent development in neuroscience. They argue that cognitive scientists have designed tasks from an assumed starting point of cardinality, and their task design reflects their founding assumption. They theorise that if tasks were designed with an emphasis on an ordinal conception of number, ordinality may also be seen as foundational and therefore part of the ‘starter kit’. There is emerging evidence that suggests that the capacity to articulate relations between numerals flexibly, such as their relative quantity and order, is crucial for further mathematical success (Sinclair & Coles, 2015). Emerging neuroscience research suggests that children’s awareness of ordinality may be distinct from their awareness of cardinality, requiring the development of distinct neural pathways (Sinclair & Coles, 2015).

Beyond neuroscience, there are several educational arguments for an emphasis on the measurement pathway to number. In a measurement-based introduction of number, the measurement ‘unit’ is central to knowing how much (or how many) of a unit is needed to constitute a larger quantity (Bass, 2015). As such, it emphasises the notion of ‘chunking’ or structuring of number as an inherent characteristic of quantity. The number line, seen as an important representation into later mathematics, is in essence an ordinal representation. Research suggests that a measurement-based introduction to number may provide a more coherent development of the number line as a conceptual mathematical tool (Bass, 2015). Advocates of a measurement pathway also suggest that it may allow a smoother transition from whole to rational numbers, and facilitate algebraic thinking. As such there is a growing body of work that privileges a measurement-based introduction of number as more important than the counting pathway.

Reviewing the literature, Mulligan et al. (2018) suggest that it is important to balance ordinal and cardinal aspects of number sense development in the primary grades. They note the difficulties of this shift, arguing that ‘this will require some reflection on the ingrained ways in which cardinality is now privileged, as well as further creative explorations of how ordinality can be mobilised to promote the development of other number-related awareness such as place value’ (p. 149).

The Emergent Number Sense module asserts both pathways as fundamental to reconstructing mathematical meaning in initial teacher development in South Africa. The two pathways—counting and measurement—were used as an organising framework to cluster several big ideas of relevance to primary school teachers, presented in Figure 2. We continued to lean in to the counting pathway (four modules), but established the measurement pathway (two modules).

Figure 2. The counting pathway and the measurement pathway (simplified typology)

Source: Roberts et al. (2021: 12)

We are aware that both sets of representations can theoretically be applied to both kinds of quantities (discrete and continuous). We took a strong decision, in the first years of initial teacher development, to select a few productive mathematical representations and build them over time (as opposed to emphasis on the interpretation of multiple representations). Given student teachers’ already fragile relationship to mathematics, we opted to link each pathway to one representational system best giving form to that number imagery. We suggest that number picture representations lend themselves most effectively to a discrete imagery of number (linked in our typology to a counting pathway), while number lines lend themselves more easily to a continuous imagery of number (linked to the measurement pathway). We now turn our attention to each pathway, and the opportunities each pathway brings to meaning making.

The Counting Pathway

The student manual introduces the counting pathway as follows: ‘children are born with a mathematical starter kit, which is strengthened over time by interaction with caring adults’ (Roberts et al., 2021: 7). The introductory framing is presented in Figure 3.

Figure 3. Introduction to the counting pathway

Source: Roberts et al. (2021: 14)

In a simplification of the counting pathway, we adopt a set-based conception of number as collections of discrete objects, which can be grouped or enclosed in a container. The counting pathway can be linked, in theory, to a number of representations to establish mental imagery. Given the fragility of basic meaning making, we chose a representation that was familiar to teachers, but with the capacity, if transformed, to provide a strong heuristic tool for place value and additive relations. The representational system for teachers was initially put forward as a form of children’s representation for additive relations by Roberts (2019), and was developed more systematically for teachers by Porteus and Mostert (2022).

Number pictures start by depicting discrete objects as small circles (or tallies), and lend themselves to each marking becoming an element of a set. Numbers can be increasingly structured, such that 10 ones, for example, becomes 1 ten. The grounding metaphor used is the container (Lakoff & Núñez, 2000). The objects can be randomly arranged or structured. In this metaphor, ‘zero’ is the empty set. We use the heuristic tool of number pictures to externalise the early counting pathway, and then as a tool to bridge the vast chasm between number imageries of discrete ones, to a reified notion of a 10, developing the basis for place value. Through this progressive tool, we encourage movement away from unstructured number pictures towards structured number pictures which privilege twos, fives and 10s. We depicted this shift, as a developmental trajectory from early unstructured use of number pictures with counting units in ones, to increasing levels of structure, with an emphasis on 100s, 10s and ones (see figure 5).

In its most fundamental form, the container metaphor is a mental image of plates (sets) of tomatoes (discrete objects), whereby ultimately each set has the same number of objects (see Figure 4).

Figure 4. A container metaphor for number

Source: Roberts et al. (2021: 14)

Figure 5. A learning trajectory for number pictures

Source: Roberts et al. (2021: 98)

Based on this familiar starting point, we seek to bridge the chasm from a unitary imagery of discrete number (early counting) to a reified 10 (whereby a 10 becomes a strong mental image, divorced from its 10 discrete units) to more complex calculating strategies based on grouping and an strong mental map for place value. The simple starting point (tomatoes on a plate) is a useful mental image for the development of children’s experience of counting in ones (without grouping). It provides a useful heuristic tool to progress from counting all, to counting on from a first number, to counting on from the larger number, to counting the difference by reaching a target (Askew & Brown, 2003; Carpenter and Fennema, 1999). The imagery also scaffolds the movement beyond counting in ones, often referred to as moving from counting to calculating. Calculating refers to more sophisticated strategies, independent of unit counting, often combining a sense of grouping and building upon known facts. Early calculating often leverages off of a reified 10, and fluent bonds of five and 10.

Understanding the mental imagery of some students as a long line of discrete ones (a long ‘songololo’ (centipede) of ones), we ask students to structure number pictures so that we can quickly see (structured by five for conceptual subitising), leading to structuring 10 small circles as two groups of five. This creates the conditions for a reified 10, in which we can draw a 10 without drawing its discrete elements. We ultimately build up to drawing a reified 100. The instructional narrative focuses attention on building up and breaking down 10s and 100s.

We then adopt the calculation trajectory proposed by Porteus and Mostert (2022) to establish the method of breaking down the second number as the ‘go to’ strategy through Grade 2 (or until consolidated in Grade 3). See Porteus and Mostert (2022) for the representational framework that scaffolds the representational trajectory from random ones (in Grade R and into Grade 1) through to two- and three-digit addition in Grade 2 and 3.

Across this work, we build on Treffers and Buys (2008) notion of the counting a sequence in ones being the ‘small number song’, the counting sequence in 10s being the ‘medium number song’, and the counting sequence in 100s being the ‘large number song’. We emphasised fluency in reference to counting forwards and backwards, starting at any number, using the small, medium and large number songs.

The Measurement Pathway

It is common cause that teachers in South Africa resonate more easily with discrete imagery of number, and largely struggle to make mathematical meaning with number lines. This alone may argue against introducing the measurement pathway as an early entry into mathematical meaning making in South Africa. However, we believe the argument may be circular: the over-emphasis on discrete imagery of number in the South African context has not provided the conditions through which a number line becomes a useful mathematical imagery and tool. Our hypothesis is that the systematic introduction of the measurement pathway, tightly linked to systematic development of number line representations (from highly structured to unstructured), has the potential to unleash mathematical meaning making in the system in the future.

The instructional traditions in the Netherlands, known as Realistic Mathematics Education, offer insight into the mathematical value of number line work across time. With enough early exposure to number lines, children can more seamlessly shift from whole numbers to real numbers. The form and function of a number line (as a continuous length) allow for counting in fractions, as well as the introduction of negative numbers (Bass, 2015; Schmittau, 2003). Further, a number line establishes the starting point for working with the Cartesian plane. It is therefore considered to be a powerful representation which travels from primary school to secondary school and beyond.

The introductory framing for the measurement pathway is presented in Figures 6 and 7.

Figure 6. Introduction to the measurement pathway

Source: Roberts et al. (2021: 8)

Figure 7. The ‘point moving along a path’ metaphor for continuous extents (lengths)

Source: Roberts et al. (2021: 14)

We adapted Lakoff and Núñez’s (2000) grounding metaphor of movement along a trajectory (where they invoke a train moving along its tracks as a seemingly universal metaphor). As trains are not commonly encountered in rural South African contexts, we simplified this metaphor to movement of a person walking along a path, where their movement is measured in steps. We applied this to ordinality in the context of the length of a path (as a continuous extent), which can be quantified by introducing equal steps as a unit of measure. The equal steps are necessarily sequenced in order radiating away from the starting point (zero).

The simple starting point (a person walking along a path) is a useful mental image for the development of children’s experience of ordinal counting, with an emphasis on number order. Given that most students have not had a productive relationship with ordinality, measurement or number lines, we first make explicit the early assumptions that establish a mental map for a number line (see Figure 8).

Figure 8. The measurement learning trajectory.

Source: Roberts et al. (2021: 188)

Measurement requires the introduction of a unit to quantify a continuous extent (such as a path). The attribute (trait or property) in focus for the ‘walking along a path’ metaphor is ‘length’. Learners have to define the starting point (zero) and then measure the length of path by counting their equal steps. Each step is a movement along the path.

The conceptual challenge of early number line work for students with little confidence in working with number lines is the notions that zero represents the starting point (as opposed to an empty set) and that the number is the interval—the movement along the pathway (the step). At first we used a fence as a mental image emphasising the distinction between the ‘poles’ (dash marking) and the ‘fences’ (the interval), recognising the fences as the number quantity. We have recently refined this to be more closely linked to the movement along a pathway metaphor: zero is the start of the pathway, and the action or motion of stepping is the interval. Each foot print is the dash marking.

The learning trajectory for number lines is presented in Figure 9. We begin from a highly structured number line from zero to 10, whereby all ones are numbered. Children order number cards along a line, or identify a missing number. We build to a number line from zero to 20, where all numbers are numbered, which can be extended to any number range. We build toward a semi-structured number line where marks of ‘poles’ or foot prints for each number are explicit, but not numbered. We progress to semi-structured number lines where some numbers have neither a mark nor a number. The work includes placing numbers in their proper position, without counting in ones (‘I know this is 17 because it is 2 more than 15; it is closer to 15 than 20’). We progress to working with an empty number line, again leveraging off of the small number (ones), medium number (10s) and large number counting songs (100s) (Treffers & Buys, 2008).

Figure 9. Learning trajectory for number lines

Source: Roberts et al. (2021: 165)

Conclusion

In this paper, we first speak to a number of confluences facing initial teacher education in mathematics. Taken together, these confluences suggest that we require a deeper understanding of mathematics coursework in initial teacher education for primary school teachers. University lecturers are mostly left on their own to teach based on our own instincts and understandings. We suggest that the moment demands developing a more collective understanding of the challenge. As a step in this direction, mathematics education lecturers from across six universities have come together to more collectively design, trial and assess BEd coursework in Maths4Primary teachers. The goal is not only to design and trial new coursework, but also to make explicit the theoretical basis of our starting points, and to assess the impact of these starting points across time.

This paper presents the theoretical starting points for our first major decision: that a starting line for initial teacher development in primary mathematics is a careful introduction of both the counting pathway and measurement pathway into number, tightly linked to a few select systems of representation. We propose that the introduction of the counting pathway be linked tightly to a system of number picture representations, and the introduction of the measurement pathway be linked tightly to the systematic development of number lines. We think that taken together, these starting points: (1) provide the basis for ‘re-learning’ mathematics flowing from productive imagery; (2) can travel with students to build the conceptual basis for mathematics well past the intermediate phase; and (3) provide productive heuristic tools for the reification of 10, and a succession of abstractions required to ‘re-tool’ mathematics as meaningful into the senior primary phase.

By making explicit the conceptual framework and theoretical starting points underpinning the Emergent Number Sense module, we hope not only that a wider group of primary mathematics lectures can learn from our experience, but also that it invites colleagues who may motivate for different starting points to share their perspectives. This would contribute to a more robust research culture focused on initial teacher development in mathematics, and one what works to develop effective mathematics teachers in South African primary schools.

Disclosure statement

No potential conflict of interest was reported by the author(s).