Exploring Conceptions of ‘Number Sense’ Evident in Pre-service Programmes for Primary School Teachers: A Review of Texts Used Across 11 University Programmes

Abstract

This paper examines the conceptions of ‘number sense’ as promoted in pre-service primary mathematics education courses at 11 South African Higher Education Institutions through the texts used by academics or prescribed for students. While all the participating institutions agree that the development of primary school learners’ ‘number sense’ is central to their mathematics methodology courses and that there is an overwhelming amount of research and literature on ‘number sense’ nationally and internationally, their conceptualisations of the nature of ‘number sense’ vary. Teacher educators, who develop pre-service teacher education courses, were asked to provide the texts, used to underpin the 11 universities’ mathematics education modules in the Bachelor of Education (Foundation and Intermediate Phases) and Post Graduate Certificate in Education (Foundation and Intermediate Phases) programmes. These texts were analysed drawing on Whitacre et al.’s emphasis on three ‘number sense constructs’ identified as Innate Number Sense, Early Number Sense and Mature Number Sense. The results show that there is no common language of description for ‘number sense’ across the 11 universities. This research implies that there is a need to develop a consistent understanding of ‘number sense’ and how it is developed across institutions.

Keywords:

• Primary pre-service teacher education
• conceptions of ‘number sense’
• innate number sense
• early number sense
• mature number sense

Introduction

The following review has emerged from the Primary Teacher Education: Number and Algebra (PrimTEd: NA) working group, which is one of three Primary Teacher Education (PrimTEd) mathematics working groups that formed part of the Teaching and Learning Capacity Improvement Project (t&lcip), funded through the Department of Higher Education by the European Union.

PrimTEd seeks to respond to the well-documented crisis of learner underperformance in mathematics and language in the schooling sector. Various international (Trends in Mathematics and Science Study & Southern African Consortium for Education Quality) and national (National Senior Certificate and Annual National Assessments) benchmarking tests suggest, firstly, that South African learners are underperforming in mathematics when compared with their international counterparts and, secondly, that the trajectory of underperformance starts in the Foundation Phase. The project aims to respond to this situation by focusing on two key domains within the Foundation and Intermediate Phases, namely, Number and Algebra, and precisely the way in which universities prepare pre-service teachers to conceptualise and develop primary school learners’ understanding of Number and Algebra.

In South Africa, much research (e.g. Graven & Stott, 2012) suggests that primary school learners are not able to work flexibly with numbers. Many learners in the Foundation Phase (Grades R–3) and Intermediate Phase (Grades 4–6) seem to rely on using tally marks to solve computations, suggesting a dependence on concrete counting strategies (Ensor et al., 2009). Consequently, the development of learners’ capacity to work flexibly with numbers, across different contexts, is severely limited (Graven et al., 2013).

Aunio et al. (2016) claim that the development of solid early numeracy skills is important for learners’ mathematics learning as primary school is a predictor of success in the higher grades. Their research suggests that one of the main challenges facing Foundation and Intermediate Phase teachers is progressing learners from concrete counting strategies through various informal strategies to the standard algorithms of elementary arithmetic. While concrete counting strategies draw on learners’ core knowledge (i.e. of collections), formal algorithms focus on non-core knowledge developed in interaction with the teacher and peers (Andrews & Sayers, 2015). This requires teachers to be able to assess their learners’ ‘number sense’ and plan learning pathways that enable them to progress. However, the Southern African Consortium for Mathematics Education Quality revealed that most Grade 6 teachers had lower levels of mathematics proficiency than the curriculum expectations (Taylor & Taylor, 2013). Despite the national government funding to increase teachers’ qualifications (Venkat & Osman, 2013), teachers’ content and pedagogical content knowledge still need to be addressed. This has given rise to numerous critiques of the teacher education system. Green (2011) argued that learners’ underperformance would continue if teacher education institutions do not produce well-qualified teachers. Research conducted by Fonseca et al. (2018) found little difference between the mathematics content knowledge of first- and fourth-year Foundation and Intermediate Phase Bachelor of Education students. As a result, ‘newly qualified teachers, through no fault of their own, are not competent to teach the school curriculum’ (Taylor & Mawoyo, 2022, p. 164). The t&lcip sought to strengthen the capacity of teacher educators to develop quality programmes and well-qualified primary school teachers. PrimTEd working groups were established to develop standards in Number and Algebra, Mathematical Thinking and Geometry. The focus of this paper is the PrimTEd: NA working group, which sought to develop standards that enable teacher educators to develop pre-service teachers ‘number sense’ with a view to them developing their learners’ ‘number sense’.

The concept of ‘number sense’ was delineated by Dantzig 69 years ago. He wrote,

Man (sic!), even in the lower stages of development, possesses a faculty which, for want of a better name, I shall call Number Sense. This faculty permits him to recognize that something has changed in a small collection when, without his direct knowledge, an object has been removed or added to the collection. (Dantzig, 1954, p. 1)

Since then, literature has been abundant on ‘number sense’, the consequences of poor ‘number sense’, and how to develop learners’ ‘number sense’ in primary schools. As it currently stands, the term is contested, and a precise and elaborate description thereof remains elusive (Berch, 2005). Griffin (2004) suggests that while ‘number sense’ is difficult to define, a mastery of ‘number sense’ is easily recognisable. Many researchers have attempted to develop frameworks (e.g. McIntosh et al., 1992) or characteristics (e.g. Berch, 2005) of ‘number sense’ to operationalise the term. However, Berch (2005) argues that the need for more precision in explaining the term ‘number sense’ is compounded by the different descriptions of ‘number sense’ provided by cognitive scientists and mathematics educators. This complexity has led researchers, such as Berch (2005), to suggest that ‘number sense’ is not only developed with experience but is also part of our biological makeup. Andrews and Sayers (2015) refer to this as preverbal and verbal ‘number sense’. Preverbal number sense is present in infancy and develops without instruction (Dehaene, 1997/2011). Verbal number sense consists of the ‘number sense’ developed through language before and during schooling and beyond. As a critique of the views of McIntosh et al. (1992), Berch (2005), Andrews and Sayers (2015) and others, and the challenge of developing a workable definition of ‘number sense’ Whitacre et al. (2017, 2020), raise concerns about attempts to treat ‘number sense’ as a single construct.

In this article, we will ascertain how ‘number sense’ is conceptualised by drawing on texts from 11 higher education institutions (HEIs) in their primary teacher education programmes. The research questions driving this review are: what conceptions of ‘number sense’ are used in the main texts used to teach number in pre-service primary education in this cohort of 11 South African universities; and to what extent is an understanding of ‘number sense’ consistent or divergent across these texts?

Theoretical Framework

Although ‘number sense’ is easy to recognise (Griffin, 2004), the concept is multifaceted and complex (e.g. van de Walle et al., 2016). In his efforts to develop an explanation of ‘number sense’, Berch (2005) drew on a wide variety of research from both the cognitive sciences and the mathematics education literature. Like McIntosh et al. (1992) and Andrews and Sayers (2015), Berch (2005) lists several features of ‘number sense’. Whitacre et al. (2017) argue that this assumes that there is ‘a single number sense construct’ (p. 97). In their review of the literature on ‘number sense’, Whitacre et al. (2020) argue that attempts to define ‘number sense’ assume that the term is used synonymously across different constructs. Their critique of the ‘number sense’ research and the inconsistency in what is regarded as ‘number sense’ results from different ‘perspectives’. They argue that the inconsistency of using the term ‘number sense’ across the research literature results from polysemy (i.e. using one term to describe different concepts) and that there are three distinct ‘number sense’ constructs. Whitacre et al. (2020) maintain that ‘number sense’ research will only be able to advance with recognition that there are three different constructs, each with their own assumptions that reflect different orientations and interests. They label these constructs as innate number sense, early number sense and mature number sense (Whitacre et al., 2017, 2020).

Innate number sense (INS) is based on the work of cognitive scientists (e.g. Dehaene, 2011) who analyse the functioning of the brain when primates engage in various tasks. Dehaene (2011) and Spelke (2000) maintain that all human and non-human primates are biologically endowed with ‘number sense’. This ‘number sense’ is perceptual and relates primarily to ‘the description of magnitudes rather than explicit knowledge of number words or symbols’ (Whitacre et al., 2020, p. 104). In other words, INS is preverbal (Andrews & Sayers, 2015). Dehaene (2011) maintains that INS is not predictive of future mathematics success in school as all humans are equally endowed with ‘number sense’ at birth.

Early number sense (ENS) develops as one engages in society and is regarded as verbal ‘number sense’ (Andrews & Sayers, 2015). It refers to the number skills learned during early childhood (up to Grade 2) that form part of the curriculum in the early years. Such learned skills include number recognition, systematic counting, awareness of the relationship between number and quantity, quantity discrimination, understanding different representations of number, estimation and simple arithmetic competence (Andrews & Sayers, 2015). Unlike INS, ENS is regarded as a predictor of school mathematics success, as ENS is based on the experiences of young children and the curriculum from pre-school to Grade 2 (Andrews & Sayers, 2015). Whitacre et al. (2020) maintain that the key researchers in the field of ENS are Andrews and Sayers.

Mature number sense (MNS) also refers to the numerical skills learned in school. The focus is on ‘number sense’ related to multi-digit whole numbers and rational numbers (Whitacre et al., 2020). Typically, MNS is developed in the Intermediate Phase. Unlike ENS, MNS is typically described as lists of components (e.g. McIntosh et al., 1992) that refer to ‘conceptual structures and habits of mind, rather than skills’ (Whitacre et al., 2020, p. 101). These components include flexibility, efficiency and accuracy in calculating mentally and in writing, and being able to judge the reasonableness of the results (McIntosh et al., 1992). According to Whitacre et al. (2020), the seminal text on MNS is that of McIntosh et al. (1992).

We draw on the theoretical framework provided by Whitacre to analyse and explain the conceptions of ‘number sense’ evident in the research and professional texts used by teacher education departments in 11 HEIs.

Methodology

Each participating institution in the PrimTEd: NA working group, had a designated lecturer assigned to the group. Many of these lecturers are responsible for coordinating and/or teaching primary mathematics education courses in collaboration with their colleagues. These institutional coordinators (for the project) were asked to send five key ‘number sense’ articles that they used in preparing or teaching the ‘number sense’ modules of their mathematics education courses. Two institutions sent more than five texts.

The 11 universities in the study offer Post Graduate Certificate in Education and/or Bachelor of Education degree programmes for Foundation and/or Intermediate Phase.

A total of 54 texts were obtained from the 11 participating institutions after removing duplicate texts. These texts were uploaded onto a database and categorised firstly, according to the phase (Foundation or Intermediate Phase) for which they were intended, and secondly according to the type of text (research-based or professional texts). Research-based texts were explicitly about research, whereas the professional texts were guides for pre- and in-service teachers. Most of these professional texts drew explicitly on research. As noted in Table 1, 24 were categorised as professional texts and 30 as research-based texts.

Table 1. Types of texts

	Journal articles	Book	Chapter	Conference paper	Research report	Study guide	Website	Excerpt
Research texts	17	6	4	2	1
Professional texts	2	17	1			2	1	1

Of the initial number of texts, nine research texts were used by two institutions and three professional texts were used by two or more institutions.

Each research text was presented on an Excel spreadsheet using the following headings: aims, assumptions, conceptualisation of ‘number sense’, explanation of how ‘number sense’ is developed, methodology, empirical setting and participants, findings and implications. The professional texts were also presented on an Excel spreadsheet. Of interest in these texts was how ‘number sense’ was conceptualised. The spreadsheets were populated by five of the institutions’ colleagues before a writing retreat. To identify the conception of ‘number sense’ in each of the texts, we scanned each text for the term ‘number sense’. The explanations of ‘number sense’ provided by each text were entered onto the Excel spreadsheet. During the writing retreat, we analysed the ‘conceptions of number sense’ by looking for themes across all the texts. This we did individually and then as a writing group.

During the writing retreat, the group was confronted with a methodological dilemma. Some texts were either not specific to ‘number sense’ or did not use the term ‘number sense’. The group examined each of the texts to ascertain if there were terms used that were consistent with the ‘number sense’ constructs that had been identified. These included number concept, mathematical thinking and computational fluency. The texts that were not directly related to ‘number sense’ were read to understand the focus of each text and its possible inclusion.

Having identified key themes emergent from the literature inductively, we then drew on the three constructs (INS, ENS and MNS) to identify the assumptions underpinning each of the texts used.

Themes Emerging from the Literature

The main themes that emerged from the texts dealing with the conceptualisation and teaching of ‘number sense’ were:

Theme 1: The variability of the types of chosen texts across institutions.

Theme 2: Not all texts focused explicitly on ‘number sense’.

Theme 3: The relationship between the ‘constructs of number sense’ in the different texts.

Theme 4: Not all texts used the term ‘number sense’.

Theme 5: What does ‘number sense’ constitute?

Interrogating the five themes that emerged from the selected readings from the different institutions enabled the authors to answer the above-mentioned research questions.

Theme 1: Variability in Choice of Texts

There is significant variability in the choice of texts that the teacher educators in the PrimTEd: NA working group draw on in developing their courses and promoting the pre-service teachers’ understanding of how to develop learners’ ‘number sense’. Drawing on Berch (2005), these were initially broadly categorised by the authors of this article as ‘cognitive science texts’ and ‘mathematics education texts’. The ‘cognitive science texts’ are based on the INS construct, while the ‘mathematics education texts’ are informed by the ENS and MNS constructs.

Eight of the texts are based on research within the field of cognitive science. These texts reported on research predominantly conducted by the authors (e.g. Feigenson, 2011) or drew explicitly on research conducted in the field (e.g. Butterworth, 2005). The ENS and MNS texts generally had a professional orientation. The aim of these texts is to assist pre- and in-service teachers in developing learners’ ‘number sense’. While some texts are theoretically informed, there is not necessarily any explicit reference to research in the text (e.g. Drews & Hansen, 2006). The McIntosh et al. (1992) article was used by two institutions that differed from the two who drew on the field of cognitive science. The McIntosh et al. (1992) text focuses predominantly on MNS (Whitacre et al., 2020).

The professional text by Naudé and Meier (2016) was used by five institutions participating in the PrimTEd: NA working group. This is unsurprising given that it is one of the few South African texts focusing on number development in the Foundation Phase. Two texts written by Van der Walle and colleagues (Van de Walle & Lovin, 2006; Van de Walle et al., 2016) also proved to be popular amongst the HEIs. Van de Walle et al. (2016) used by four HEIs and Van de Walle & Lovin (2006) was used by two HEIs. These professional texts emphasise how to teach ‘number sense’ in primary school.

The variability in the choice of texts indicates that there is no unified sense of what ‘number sense’ is across institutions and that the dominant constructs vary across the institutions. Two institutions foreground INS, while the rest focus primarily on ENS and MNS.

Theme 2: Not all Texts Focus Explicitly on ‘Number Sense’

Many of the texts did not focus explicitly on ‘number sense’. Such literature presented research on how the brain computes (Gallistel & King, 2010), learning and development (Gelman & Gallistel, 1978; Hughes, 1986; Piaget, 1964; Butterworth, 2005), commognition (Sfard, 2008), working memory (Feigenson, 2011), linguistics (Gopnik & Meltzoff, 1997; Negen & Sarnecka, 2012; Huang et al., 2013), what constitutes mathematics in pedagogical situations (Baker et al., 1971; Davis, 2013), numeracy pedagogical practices (Venkat & Naidoo, 2012; Ensor et al., 2009), algebraic pedagogical practices (Sfard, & Linchevski,1994), the gap in making the transition from unit counting to the abstract algorithm (Schollar, 2008; Ensor et al., 2009), numeracy performance relating to gender, language and school-type (Aunio et al., 2016); hypothetical learning trajectories (Ensor et al., 2009; Graven & Stott, 2012; Fritz et al., 2013) and assessments based on hypothetical learning trajectories (Wright et al., 2006, 2014; Fritz et al., 2014). While these texts are not the focus of this paper, they form part of the broader repertoire of texts used by the 11 institutions developing pre-service teachers’ understanding of how children develop an understanding of Number and Algebra.

Theme 3: The Relationship Between the ‘Constructs of Number Sense’ in the Different Texts

(a) Texts that focus on INS or that draw on INS and (b) texts that focus on ENS and/or MNS

Evidence from the texts suggests a significant variability in the precise conception of ‘number sense’ linked to the different ‘number sense constructs’.

Texts that focus on INS or draw on INS

As mentioned above, only two of the 11 HEIs participating in the PrimTEd: NA working group submitted all the texts that focused on INS. These texts are categorised as exclusively focusing on INS or both INS and ENS. Table 2 categorises these texts into the two afore-mentioned constructs.

Table 2. The relation of the HEIs texts to the three ‘number sense constructs’

INS	INS/ENS	ENS	ENS/MNS	MNS
Carey (2001)	Aunio et al. (2016)—use the term ‘early numeracy skills’ (acknowledge INS, but focus on ENS)	Confer (2005)	Anghileri (2006)	McIntosh et al. (1992)
Dehaene (2011)	Clements and Sarama (2009)	Griffin (2004)	Berch (2005)—also includes INS
Feigenson (2011)	Berch (2005)	Jordan et al. (2007)	Charlesworth and Lind (2010)
Feigenson et al. (2004)	Butterworth (2005)—use the terms ‘numerosity’ and ‘early arithmetic’	McDermott and Rakgokong (1996)	Drews and Hansen (2007)—use the term ‘mathematical thinking’
Izard et al. (2008)	Carey (2001)	Naudé and Meier (2016)	Hopkins et al. (2001)
Spelke (2000)	Fritz et al. (2014)—use the terms ‘key numerical concepts’ and ‘arithmetical calculation’ (acknowledge INS)	Van de Walle and Lovin (2006)	Reys et al. (2009)
	Fritz et al. (2013)—use the terms ‘key numerical concepts’ and ‘arithmetical calculation’ (acknowledge INS)	Wright et al. (2006)	Russell (2000)—focuses on ‘computational fluency’
	Gelman and Gallistel (1978)	Way (2011)	Siemon et al. (2014)
	Sarama and Clements (2009)	Wright et al. (2014)	Sonnabend (2009)
			Van de Walle et al. (2015)
			Van de Walle et al. (2016)
			Williams and Shuard (1982)
			Yessel and Dill (1987)

INS, Innate number sense; ENS, early number sense; MNS, mature number sense.

The texts that focus on INS draw on research with human and non-human primates. Spelke (2000), Carey (2001), Feigenson et al. (2004), Izard et al. (2008), Dehaene (2011) and Feigenson (2011) suggest that there are two non-verbal systems related to the representations of number that are innate and independent of instruction. These are the ‘object tracking system’ (OTS) and the ‘approximate number system’ (ANS). These systems are referred to by Spelke (2000) as core knowledge, that is, a set of ‘building blocks’ (p. 1233).

The system for the precise representation of a small number of objects, referred to as the OTS (Carey, 2001; Izard et al., 2008), is a ‘common system for quantification’ (Feigenson et al., 2004, p. 309). Carey (2001), Izard et al. (2008) and Feigenson et al. (2004) suggest that infants are responsive to exact numerosities; for example, they can identify two objects as distinct from three objects without knowing the number words. This precise representation of distinct entities enables children to ‘build representations of objects as complete, connected and solid bodies that persist over occlusion, and maintain their identity through time’ (Spelke, 2000, p. 1233). Results show that precise representation of numerosities is domain specific (i.e. it only applies to discrete objects as opposed to continuous objects) and is limited to a small number of objects (up to three) (Spelke, 2000). This core system becomes ‘integrated with the symbolic number system used by children and adults for enumeration and computation’ (Feigenson et al., 2004, p. 310). It forms the basis of quantification, underpins verbal counting skills (Gelman & Gallistel, 1978) and early arithmetic (Andrews & Sayers, 2015), which are central to ENS.

The system of approximate numerosities, referred to as ANS, suggests that numerical magnitude occurs only in an approximate manner (Carey, 2001; Feigenson et al., 2004). Xu and Spelke (cited in Feigenson et al., 2004) maintain that 6-month-old infants can already distinguish two quantities with a 1:2 ratio (by 9/10 months, they can distinguish the relation of a 2:3 ratio). Dehaene (2011) notes that these comparisons are always approximate. The ANS supports later subitising and estimation activities. Subitising is when children can ‘see’ a number of items up to five without counting them one by one (Wright et al., 2010).

Dehaene (2011) argues that language and symbol systems extend the core knowledge we are born with. Drawing on Dehaene, Berch (2005) suggests that a focus on only one of these conceptions of ‘number sense’ is limited and suggests that ‘core representations become connected to other cognitive systems as a consequence of both development and education’ (p. 334). Berch (2005) maintains that teachers should embrace a comprehensive view rather than the limited but entrenched view of curriculum policies and mathematics education perspectives. Whitacre et al. (2020) also argue that these two forms of representation, the OTS and ANS, form the basis of children’s early and school-related ‘number sense’. In other words, ENS develops from INS. As noted above, several researchers (e.g. Sarama & Clements, 2009) regard INS as the foundation for ENS. While INS is not a predictor of later mathematics achievement, many texts focusing on ENS suggest that it forms the basis for verbal ‘number sense’ development. In other words, ANS and OTS are the innate ‘building blocks’ for developing an understanding of quantification, subitising, estimation and simple arithmetic.

Texts that focus on ENS and/or MNS

Although the cognitive sciences perspective on ‘number sense’ is pertinent, it is the mathematics education perspective that most institutions seemingly regard as pivotal to pre-service teachers’ development of ‘number sense’ in the classroom. The research literature on mathematics education in schools can be divided into two of Whitacre et al.’s (2017, 2020) constructs: early number sense and mature number sense. Interestingly, in the research literature used by the 11 institutions, there is far more emphasis placed on ENS (or the link between ENS and INS) than MNS (Table 2). The only article that focuses explicitly on MNS solely is by McIntosh et al. (1992).

Research focusing on ENS tends to concentrate on curriculum topics (Jordan et al., 2007). This is consistent with Whitacre et al.’s (2017) findings. In their assessment of Grade 1 children’s ‘number sense’, Jordan et al. (2007) described ‘number sense’ as consisting of counting skills, enumeration, number recognition, comparison of quantities, word problems and context-free calculations. For Sarama and Clements (2009), simple arithmetic includes ‘counting, comparing, unitizing, grouping, partitioning, and composing’ numbers (p. 27). In contrast, the texts that included both ENS and MNS (e.g. Anghileri, 2006; Sonnabend, 2009) or MNS (McIntosh et al., 1992) only, moved beyond ‘number sense’ as a list of curriculum topics to include the ‘capacity for logical thought, reflection, explanation, and justification’ and problem-solving’ (Kilpatrick, 2001, p. 116). The authors of this article broadly describe these as process skills, that is, the knowledge that produces the discipline.

Most of the professional texts are textbooks for pre- and in-service teachers. These texts focus on pedagogical content knowledge, the amalgam of content knowledge and pedagogical knowledge (Shulman, 1986, 1987). Given the concerns raised earlier in the article about pre-service teachers’ content and pedagogical content knowledge, these texts seem to be key for preparing pre-service teachers.

Theme 4: Not all Texts Use the Term ‘Number Sense’

As explained earlier in the article, some texts were included by the 11 HEIs that did not use the term ‘number sense’ per se. The terms used in these texts were mathematical thinking, number concept, computational fluency and an intuitive awareness of number. Interestingly, these texts either focused on ‘number sense’ as curriculum topics (e.g. McDermott & Rakgokong, 1996) or process skills related to ‘number sense’ (e.g. Russell, 2000).

Drews and Hansen (2007) use the term ‘mathematical thinking’ instead of ‘number sense’. Mathematical thinking consists of components: problem-solving, thinking flexibly about numbers and number operations, identifying the tools best suited to a mathematical situation, identifying patterns and generalising findings, making sense of problems though systematic and logical processes, explaining and justifying one's solution strategies, working collaboratively, demonstrating a productive disposition to mathematics, making connections across mathematics topics, and constantly evaluating their mathematical competence with a view to improvement.

Russell’s (2000) article is about ‘computational fluency’. Computational fluency consists of three core ideas: efficiency, accuracy and flexibility. Flexibility is the ability to choose an appropriate calculation strategy. Efficiency refers to choosing a strategy that can be carried out quickly, and accuracy relates to using a chosen strategy with precision. While the above two texts draw on important components of number sense (e.g. number operations), they are consistent with what Whitacre et al. (2017) call Mature Number Sense as they privilege the importance of the process skills in developing learners’ understanding of number.

Texts focusing on ENS emphasise number components (e.g. systematic counting). This is emphasised in the texts by McDermott and Rakgoking (1996) and Bosman et al. (2004). For McDermott and Rakgokong (1996) ‘number sense’ is seen as ‘number concept’. Number concept is the feeling for and understanding of the numerosity or value of a number necessary for calculating. Bosman et al. (2004) elaborate on the explanation of ‘number concept’ to include counting, number operations and understanding the relationships between numbers.

A sense of a number’s numerosity is a link between the cognitive science and mathematics education perspectives of ‘number sense’ (e.g. Aunio et al., 2016). Butterworth (2005) describes numerosity as an abstract view or a cardinal awareness of a set of objects. Children who can instantly recognise the numerosity of a set of objects (usually up to five) without counting are able to subitise (Wright et al., 2006). ‘Number sense’ continues developing as learners engage with number operations. In this process, they begin to develop a sense of place value (Schollar, 2008; Clements & Sarama, 2009). Other topics that surface from the literature are quantity, number knowledge, calculations, counting principles, number combinations (Gelman & Gallistel, 1978; Jordan et al., 2007), magnitude comparison, counting skills, number line insight, ordering, estimating, equivalence, calculation strategies and operations (Aunio et al., 2016; Gelman & Gallistel, 1978).

The fact that numerous texts do not use the term ‘number sense’ may indicate no consistent conception of ‘number sense’ or that ‘number sense’ is not a universal term.

Theme 5: What Does ‘Number Sense’ Constitute?

In the texts that referred explicitly to ‘number sense’, ‘number sense’ is viewed differently. Way (2011), drawing on Bobis, identifies three different conceptions of ‘number sense’, that is, ‘number sense’ as a framework, ‘number sense’ as a series of curriculum topics and ‘number sense’ as a selection of process skills (e.g. mental calculation, computational estimation, problem-solving). In addition, the authors of this article found that ‘number sense’ is also conceptualised as a set of dispositions.

‘Number sense’ as a framework

The most cited text that focuses on ‘number sense’ as a framework is that of McIntosh et al. (1992). McIntosh et al. (1992) claim that while there are several lists of components of ‘number sense’ (e.g. Berch, 2005), how these components fit together needs to be added to the literature. The framework suggested by McIntosh et al. (1992) attempts to ‘articulate a structure which clarifies, organizes, and interrelates some of the generally agreed upon components of basic number sense, many of which have been conjectured by different people over many years’ (p. 5). The impetus for the generation of a framework is their view that ‘number sense’ continually grows and expands as learners progress through the schooling system and beyond. The components in the framework by McIntosh et al. (1992) include curriculum topics (e.g. place value) and process skills (e.g. the ability to ‘invent’ strategies).

‘Number sense’ as curriculum topics

To provide a more explicit conception of numbers sense, many researchers (Clements & Sarama, 2009; Gellman & Gallistel, 1978; Jordan et al., 2007; Schollar, 2008; van de Walle et al., 2013) describe it in terms of key concepts, topics or characteristics. Confer (2005) refers to ‘number sense’ as a collection of topics that learners need to understand, such as counting, number relationships, decomposing and composing numbers, benchmark numbers (e.g. 5 and 10) and calculation strategies. Likewise, Clements and Sarama (2009) linked ‘number sense’ to numerical cognition that includes ‘number knowledge, counting and arithmetic’ (p. 225), the ability to group and quantify sets quickly, estimation and the use of multiple strategies to calculate. The text is organised into topics (e.g. quantity, number, subitising, and counting). To assist teachers ‘with meeting the students “where they are” and helping them build on what they know’ (p. ix), they have developed hypothetical learning trajectories of each curriculum topic.

The text by Charlesworth and Lind (2010) defines ‘number sense’ as ‘the concept or understanding of number’ (p. 685), that is, the fundamental concepts of ‘one-to-one correspondence, number sense and counting, logic and classifying, comparing, shape, spatial sense, parts and wholes, language, and application of these concepts’ (p. viii).

The above texts focus on the concepts that learners should develop about number in most curricula. While it is important that learners develop an understanding of these concepts, this should be done to enable them to problem-solve, communicate their thinking and justify their solution strategies, that is, ‘number sense’ as process skills.

‘Number sense’ consists of process skills

Process skills are used rather broadly to categorise the knowledge and skills used to produce the discipline of mathematics in a school setting.

Confer (2005) focuses on ENS, particularly in Grade 1. She explains that her book ‘presents a picture of the kinds of investigations, activities, and games that promote the development of number sense’ (p. xvii). She concurs with McIntosh et al. (1992) and Reys et al. (2009) that ‘[c]hildren with well-developed “number sense” use numbers to solve problems. They make sense of numerical situations and use what they know to figure out what they don’t know’ (p. xiii).

Reys et al. (2009) draw on a list of components characteristic of ‘number sense’. Not surprisingly, these components are like those of McIntosh et al. (1992). They include understanding number concepts and operations, developing flexible strategies for operating with numbers, being accurate and efficient in computations and recognising the reasonableness of results, using flexible ways to make mathematical judgements and a realisation that mathematics is useful and makes sense. Like Clements and Sarama (2009), they maintain that ‘number sense’ is a lifelong process that develops in stages as learners progress up the schooling system.

Anghileri (2006) draws on the description of ‘number sense’ offered by McIntosh et al. (1992). ‘Number sense’ involves the ability to compute in flexible and efficient ways using various strategies, knowledge of how numbers relate to each other, the different representations and meanings of different number operations, making generalisations about patterns and linking new knowledge to prior knowledge. For Anghileri (2006), learners with ‘number sense’ can link their knowledge of number to real-life situations, for example, knowing that six is my age, but six is also half a dozen.

Van de Walle and Lovin (2006), like Reys et al. (2009), regard ‘[n]umber is a complex and multifaceted concept. A complete and rich understanding of number involves many different ideas, relationships and skills’ (p. 37). Both texts draw on the definition of Howden (1989) and maintain that ‘number sense’ refers to ‘a good intuition about numbers and their relationships which develops gradually as a result of exploring numbers, visualising them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms’ (p. 11).

McIntosh et al. (1992) describe ‘number sense’ as: ‘A propensity for and an ability to use numbers and quantitative methods as a means of communicating, processing and interpreting information. It results in an expectation that numbers are useful, and that mathematics has a certain regularity’ (p. 4). To explain this conception of ‘number sense’, McIntosh et al. (1992) developed a ‘number sense’ framework consisting of three key components: ‘knowledge of and facility of numbers’, ‘knowledge of facility with operations’ and ‘applying knowledge and facility with numbers and operations to computational settings’ (p. 4). The first two of these components focus on curriculum topics, and the last one is on problem-solving.

‘Number sense’ as disposition

Kilpatrick et al. (2001) describe a productive disposition as the ‘sense in mathematics, to perceive it as both useful and worthwhile’ (p. 131). Many of the texts highlighting process skills also emphasise the importance of a productive disposition in developing number sense (McIntosh et al., 1992; Anghileri, 2006; Confer, 2005; Reys et al., 2009; Sarama & Clements, 2009). These texts explain that learners with ‘number sense’ are able to link their knowledge of number to real-life situations.

Discussion and Conclusion

As noted in this paper, missing from the research is a common language of description with regard to ‘number sense’. Whitacre et al. (2017, 2020) argue that regarding ‘number sense’ as a single construct has led to much confusion. While it is clear how INS is conceptualised in the cognitive sciences, the distinction between ENS and MNS could be more explicit. Whitacre et al. (2020) draw on the work of Andrews and Sayers (2015) to explain ENS. However, Andrews and Sayers (2015) provide a restricted view of ‘number sense’. For them, ‘number sense’ is a list of curriculum topics (e.g. quantity discrimination). Missing from ENS are the important process skills that enable the development of an understanding of the curriculum topics. This Whitacre et al. (2020) refer to as MNS (the domain of number beyond Grade 2). Curriculum topics are a requisite for developing number sense, and thus the distinction between ENS and MNS might be redundant.

In an attempt to develop a conception of ‘number sense’ that would include the different ‘number sense constructs’, our joint description of ‘number sense’ across the 11 institutions incorporates INS and MNS. Developing learners’ ‘number sense’ enables them to make sense of the number-related curriculum topics. The curriculum topics are included in the PrimTEd standards for Number and Algebra. We suggest that more important in the development of ‘number sense’ however, are the process skills (e.g. using a variety of strategies for calculating). Process skills are often ignored in many classrooms and inhibit learners’ ability to work flexibly with numbers. The description of ‘number sense’ emergent from the PrimTEd: NA working group is as follows:

Number sense has core biologically endowed (innate) features and culturally acquired (learned, non-core) features. The innate features of number sense enable an early sensitising to quantity (‘muchness’) in human infants using the so-called Approximate Number System (ANS) and a system that enables humans to track a small number of objects (OTS). Both systems remain operative over the lifespan of humans. The non-core features of Number Sense develop off the core, biologically endowed system through focused learning and teaching that exploit incipient arithmetic affordances of ANS and OTS. Well-developed number sense exhibits increased facility and flexibility in working with numbers and number relations. (Number Sense Project Group, 2017)

The extent to which the non-core features of ‘number sense’ develop off the core features is currently limited to the courses of two institutions. Further deliberation is necessary on the relationship between these two different constructs and how what is taught in most institutions, ENS and MNS, connects with the features of INS. In so doing, we can consider how the INS enables and constrains the development of ENS and MNS.

While much research has been done on INS by cognitive neuroscientists and cognitive psychologists, it would be surprising if this focus gains much traction in mathematics education courses in teacher education programmes. A broader and dominant perspective on ‘number sense’ explicitly related to the development of ‘number sense’ in the classroom context is already entrenched (Berch, 2005). Such conceptions occur in curriculum documents, textbooks and national and international benchmarking tests.

Given the contested nature of ‘number sense’, clear frameworks like those developed by McIntosh et al. (1992) become appealing. These frameworks are useful for understanding ENS and MNS and provide guidelines for curriculum development, pedagogy and assessment. However, in the context of South Africa, as noted by Schollar (2008) and Ensor et al. (2009), they may not be sufficient. A more transparent guideline on how learners develop ‘number sense’ as they transition through the schooling system may be helpful in this regard.

Many texts focused on hypothetical learning trajectories (Clements & Sarama, 2009; Wright et al., 2006, 2014; Ensor et al., 2009; Graven & Stott, 2012; Fritz et al., 2013). It might be necessary that in preparing our pre-service teachers to develop learners’ ‘number sense’, and to bridge the transition from unit counting to abstract algorithms, teacher education institutions include this in their methodology courses and that further research is conducted in South Africa on the development of learners’ ‘number sense’ trajectories in primary schools. Such preparation will assist pre-service teachers in curriculum planning, pedagogy and assessment.

Once the duplicates were removed, there were 54 texts used across the 11 institutions to support teacher educators or pre-service teachers in developing learners’ ‘number sense’. Unsurprisingly, ‘number sense’ is foregrounded in all these institutions. This is partly a reaction to the move away from teaching formal procedures that learners learn through drill and practice to a view that mathematics is about sense-making. The variety of texts used across institutions show that ‘number sense’ is conceptualised in many ways and support the contested and nebulous nature of ‘number sense’. Furthermore, to draw on the work of Whitacre et al. (2017, 2020), it appears that ‘number sense’ is seen as a single construct. The question remains: how do we go forward as institutions in developing a conception of ‘number sense’ with some familiarity across the HEIs. Interventions such as the PrimTEd provide an opportunity to do this. However, of the 23 institutions that offer primary pre-service teacher education, only 11 participated in the PrimTEd: NA working group. While the PrimTEd: NA working group has developed a set of standards to create consistency concerning our conceptions of the development of ‘number sense’ and as a guide for institutions offering primary mathematics education courses, this is insufficient. The Number and Algebra standards were developed in a much-needed community of practice over four years of deliberation and debate. We propose that: (1) the project continues and that more institutions participate and learn from and with each other; (2) the standards be published for institutions to critique; (3) the participants participate in national and international conferences to ensure the dissemination of the PrimTEd outcomes; and (4) the standards be incorporated into the primary mathematics education curricula and trialed to ascertain their use in equipping pre-service teachers with the competence to develop learners’ ‘number sense’.

Disclosure Statement

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

Acknowledgement

We acknowledge Edward Domboka, Karin Hackmack, Mapula Ngoepe, Manono Poo and Marinda van Zyl for their participation in the writing retreat which provided the impetus for this article. The research is supported by the Department of Higher Education and Training in South Africa and funded by the European Union project ‘Teaching and Learning Capacity Improvement Project’.