English and Swedish year-one teachers’ number-related learning goals: the influence of intended and received curricula

Abstract

In this paper, drawing on semi-structured interviews with generalist teachers of year-one children in England and Sweden, we examine comparatively the influence of the intended curriculum (teachers in both countries work within mandated national curricula) and the received curriculum (the collectively assumed efficacious practices and goals handed down from one generation of teachers to the next) on teachers’ expressed number-related learning goals. Analyses, framed by a literature-derived and curriculum-independent set of eight forms of number-related competence each implicated in later mathematical learning, identified both similarities and differences in the two groups’ expressed goals. Key similarities concerned expectations that all children should become additively competent, supported by supplementary goals concerning systematic counting, number bonds, the number line and an appropriate mathematical terminology. Key differences concerned English teachers’ strongly-expressed emphasis on place value and a desire for children to learn to multiply. Overall, the strongly-framed English curriculum appears to influence teachers’ goals more than the weakly-framed Swedish, while Swedish teachers seem to draw on a received curriculum more closely aligned with the literature-derived developmental goals than the English. Finally, when set against the literature-derived and curriculum-independent developmental goals, the English curriculum, unlike the Swedish, expects year-one children to learn much age-inappropriate material.

KEYWORDS:

• Year-one children
• number-related learning
• teacher goals
• England
• Sweden
• intended curriculum
• received curriculum
• semi-structured interview

MSC:

• 97E99

1. Introduction

In many countries, including England and Sweden the systems here under scrutiny, teachers work within government-produced national curricula that specify in culturally unique ways what children are expected to learn during their years of compulsory schooling. However, despite the curricula expectations within which they work, teachers frequently follow didactical folkways, which are ‘warranted by their existence and taken-for-granted effectiveness’ (Buchmann, 1987, p. 154) and ‘uncritically accepted from our ancestors’ (Lauwerys, 1959, p. 294). In other words, it would seem that teachers operate within an intended curriculum, represented by the government-mandated goals that children are expected to attain, and a received curriculum, represented by the culturally-situated beliefs and practices that all teachers acquire by dint of their membership of a particular cultural group (Andrews, 2011). In this paper, by means of an interview study, we investigate the influence of differently framed curricula (Bernstein, 1975) on the number-related learning goals espoused by English and Swedish teachers of year-one children.

Historically, while many scholars have framed their analyses against different conceptions of curricula, none have adequately represented the didactical folkways discussed above. For example, one of the best-known models is that employed by the Second International Mathematics Study, which exploited a three-level model comprising the intended curriculum, the implemented curriculum and the attained curriculum (Eggen et al., 1987; Robitaille & Garden, 1989). Others, albeit in slightly different ways, have expressed similar conceptions. For example, Robitaille (1980), in an article prescient of the second international study, writes of the intended, implemented and realised curriculum. Other conceptualisations, typically offered in relation to particular analytical purposes, concede the role of the intended curriculum, but refer to the enacted rather than implemented curriculum, as with Boesen et al.’s (2014) intended and enacted curricula, Kurz et al.’s (2010) intended, planned and enacted curricula, and Seitz’s (2017) intended, enacted and assessed curricula. Overall, curriculum analysts seem to agree on both the meaning and use of the term ‘intended curriculum’, although Remillard and Heck (2014), by way of an alternative, discuss the ‘official curriculum’; those ‘curricular elements that are officially sanctioned’ (p. 708) and which are typically presented as ‘statements of what students should ideally attain as a result of instruction’ (p. 709). The important debate, it seems to us, lies in what follows logically from the intended or official curriculum. For example, enacted or implemented curricula, which Remillard and Heck (2014) construe as the operational curriculum, imply actions on the part of teacher that reify the intended. Such a conceptualisation may be problematic if, as with this paper, the goal is to analyse teachers’ curriculum-related beliefs rather than actions and, importantly, distinguish between didactical practices located in curricular observance and those located in didactical folkways.

Curriculum-related research has, until relatively recently, received little attention (Li & Lappan, 2014), which may be a consequence of an inconsistently defined construct, whereby the word curriculum has different meanings in different contexts, typically located in the distinction between educational systems that impose national curricula and those that do not. With respect to the former, the context of this paper, the content of the intended curriculum, and in some cases even the manner of the implemented curriculum, is typically determined by the ministry responsible for education. Thus, systemic expectations with respect to what is taught, when it is taught and how it is taught vary according to context, as found with recent analyses of the curricula of Denmark, Norway and Sweden (Sunde et al., 2022), the four constituent nations of the United Kingdom (Andrews et al., 2022), Cyprus, Greece and Turkey (Xenofontos et al., 2023).

With respect to the latter, particularly the United States, curriculum-related research has attended not to curricula, as Europeans would understand them, but curriculum materials. Typically, these would be textbook series, along with any supplementary materials, developed to support teachers in their work with children. In such a context, the work of Janine Remillard, which has been at the forefront of US research for a quarter of a century and offered many important insights into their development and implementation (Remillard, 1999, 2005; Remillard et al., 2014, 2021; Remillard & Bryans, 2004; Remillard & Heck, 2014), can more closely be construed as addressing the implemented rather than the intended curriculum, although the distinction may become becomes blurred, particularly when scholars from Europe and the US collaborate. For example, in reporting their comparative study of Finnish, Flemish, Swedish and US teachers’ use of instructional materials, Remillard et al. (2021, p. 1333) assert that in ‘both Finland and Sweden, teachers have a high degree of autonomy over curriculum selection, teaching strategies, and student assessment’. In fact, highlighting the need for terminological clarity, the first part of the assertion is factually incorrect; Finnish and Swedish teachers may have autonomy over the selection of curriculum materials but not the curriculum itself.

In this paper, we attend not to curriculum materials, however they are defined and exploited, but the curricula within which teachers are compelled to work. In an ideal world, it would be reasonable to expect the intended curriculum and the received curriculum to align; the intended would present developmentally appropriate goals in didactically coherent ways, while the latter, despite its historical roots, would resonate closely with the intended. Admittedly, while a few mathematics education studies have highlighted an ambivalent alignment between intended and enacted curricula (Boesen et al., 2014; Choppin et al., 2022; Kurz et al., 2010; Toh, 2022), little research has been undertaken with respect to how the intended curriculum and the received curriculum inform teachers’ articulated goals for their children’s learning. To address this research paucity, it would seem sensible to focus on a particular aspect of mathematics learning in relation to particular groups of students. To this end, acknowledging its importance in children’s ongoing mathematical development, we focus on how teachers of year-one children construe the teaching of number, a core underpinning of children’s later experiences of mathematics. In so doing, in order to provide a culture-independent framework, we draw on the eight categories of number-related knowledge that comprise foundational number sense (FoNS), each of which requires instruction and represents a literature-derived and curriculum-independent competence that all year-one children child should acquire if they are to become successful learners of mathematics (Andrews & Sayers, 2015).

1.1. Characterising foundational number sense

Widely acknowledged as an important outcome of learning, number sense has been broadly described as the ability to operate flexibly with number and quantity (Aunio et al., 2006; Clarke & Shinn, 2004; Gersten & Chard, 1999). Such conceptions, however, have been construed as problematic (Andrews & Sayers, 2015; Whitacre et al., 2020), not least because they are too vague to support teachers in their day-to-day activities. Indeed, while Berch (2005, p. 133) asserts that number sense can be expressed as attributes like ‘awareness, intuition, recognition, knowledge, skill, ability, desire, feel, expectation, process, conceptual structure, or mental number line’, he offers little practical advice for teachers struggling to support their learners. Consequently, a number of scholars, including Berch himself, have attempted to categorise more precisely its characteristic properties. In this respect, Berch (2005) identified thirty components, Howell and Kemp (2005, 2006) almost thirty and Purpura and Lonigan (2013) twenty-five. Such large numbers are also problematic, for at least two reasons, one of particular relevance for teachers and the other for researchers. First, from the perspective of teachers, and we illustrate this point by reference to Howell and Kemp’s (2006) distinction between rote counting to five and rote counting to ten, such large numbers allude to the creation of what might be called micro-goals that may prompt teachers to pay too much attention to the small picture and too little to the big. In this respect, it is interesting to note that while English teachers of mathematics compare favourably with their European colleagues with respect to the development of students’ conceptual knowledge and procedural knowledge, they differ significantly in almost never connecting this knowledge (Andrews, 2009). Moreover, the creation of large numbers of micro-goals may lead to the over-complication of activities focused on the professional development of teachers, whether preservice of inservice. Second, from the perspective of researchers and acknowledging the cultural uniqueness of school mathematics, such large numbers are unlikely to support analyses of curricula and the opportunities to learn embedded in them. In similar vein, they are unlikely to support analyses of the content of textbooks, classroom activity or teachers’ beliefs. Put simply, these concerns warrant the need for simpler ways of characterising number sense.

In their recent review paper, Whitacre et al. (2020) identified three distinct forms of number sense; namely, approximate number sense, early number sense and mature number sense. Importantly, from the perspective of this paper, ‘early number sense includes learned skills that involve explicit number knowledge, such as counting items using number words and comparing numbers represented symbolically as numerals’ (Whitacre et al., 2020, p. 104). Also, from the perspective of this paper, Whitacre et al. identify two articles as being of seminal importance in characterising early number sense, one of which, Andrews and Sayers (2015), was the outcome of an attempt to develop ‘a simple to operationalise framework … with the potential to inform teacher education, facilitate classroom evaluations and provide a warranted tool for use in cross-cultural studies’ (Andrews & Sayers, 2015, p. 258). Interestingly, and something of an aside, Whitacre et al. (2020) assert that Andrews and Sayers (2015) ‘describe number sense as a single construct’ (Whitacre et al., p. 97), which is incorrect, as Andrews and Sayers expressly write of preverbal number sense, foundational number sense and applied number sense, each of which resonates closely with each of the three forms identified by Whitacre et al. Moreover, Andrews and Sayers (2015) construe foundational number sense (FoNS) as a multidimensional construct comprising eight categories of competence. In our view, the eight categories of FoNS, which require instruction (Andrews & Sayers, 2015), provide the cognitive and didactical link between Whitacre et al.’s (2020) approximate number sense and mature number sense. Therefore, it is these eight categories of competence, which are summarised in Table 1, that frame this study. In the following, we discuss in more detail the characteristics of the eight FoNS categories of competence, outline their implications for learning and, where possible, augment Andrews and Sayers’ (2015) original conception with more recent literature.

Table 1. Summaries of the eight categories of foundational number sense (After Andrews & Sayers, 2015).

FoNS category	In relation to the numbers 0–20, year-one children should be able to …
Number recognition	identify, name and write number symbols;
Systematic counting	count systematically, forwards and backwards, from arbitrary starting points;
Number and quantity	understand the one-to-one correspondence between number and quantity;
Quantity discrimination	compare magnitudes and deploy language like ‘bigger than’ or ‘smaller than’;
Different representations	recognise and make connections between different representations of number;
Estimation	estimate in a range of contexts;
Additive operations	undertake simple addition and subtraction operations;
Number patterns	recognise and extend number patterns, identify a missing number;

1.1.1 Number recognition

The first of Andrews and Sayers’ essential competences is number recognition, which, they argue, entails recognising number symbols, knowing associated vocabulary and meaning (Malofeeva et al., 2004), alongside being able to identify particular symbols from a collection of number symbols and naming a number when shown its symbol (Gersten et al., 2005; Yang & Li, 2008). Recognising number symbols, irrespective of the manner of their presentation, predicts being able to order digits (Sasanguie & Vos, 2018), while those who experience difficulty with number recognition may experience later mathematical problems (Lembke & Foegen, 2009), particularly with subitising (Koontz & Berch, 1996; Stock et al., 2010). Young children who recognise numbers presented symbolically are more able to undertake arithmetic successfully than those who cannot (Desoete et al., 2012; Koponen et al., 2016; Krajewski & Schneider, 2009), a competence stretching into adulthood (Sasanguie et al., 2017). Similarly, being able to articulate numbers represented symbolically is implicated in arithmetical competence in both children (Lyons et al., 2014) and adults (Sasanguie & Reynvoet, 2013). Finally, the connected skills of number recognition in young children predict mathematics achievement in general (Geary et al., 2009; Martin et al., 2014) as late as adolescence (Geary, 2013; Pollack et al., 2022).

1.1.2 Systematic counting

The second of Andrews and Sayers’ eight competences highlights the importance of counting systematically (Berch, 2005; Clarke & Shinn, 2004; Gersten et al., 2005; Van de Rijt et al., 1999). Systematic counting, which includes an understanding of ordinality (Ivrendi, 2011; Jordan et al., 2006; LeFevre et al., 2006; Malofeeva et al., 2004), entails children being able to count upwards to twenty and back from arbitrary starting points, knowing that every number occupies a unique position in the list of all numbers (Jordan & Levine, 2009; Lipton & Spelke, 2005). Being able to count systematically, a competence that supports children’s understanding of concepts like the successor principle (Hartnett & Gelman, 1998), underpins arithmetical competence in general (Gersten et al., 2005; Hannula-Sormunen et al., 2015; Koponen et al., 2016; Martin et al., 2014; Passolunghi et al., 2007; Stock et al., 2010) and mental arithmetic competence in particular (Lyons & Beilock, 2011). Alternatively, children who experience difficulties with counting are more likely to experience later arithmetical difficulties (Desoete et al., 2009). Importantly, competent finger counting, a skill that can be taught effectively (Gracia-Bafalluy & Noël, 2008) and which is known to correlate positively with increasing socio-economic status (Jordan et al., 1992; Levine et al., 1992), has also been implicated in the arithmetical competence of both children (Fayol et al., 1998; Jordan et al., 1992; Noël et al., 2005) and adults (Domahs et al., 2010; Morrissey et al., 2016, 2020). Finally, while finger counting typically impacts positively on arithmetical competence, different approaches to finger counting impact differently (Neveu et al., 2023).

1.1.3 Awareness of the relationship between number and quantity

Andrews and Sayers’ (2015) third category of competence concerns the need for children to understand the relationship between number and quantity. In particular, underpinning notions of cardinality, children need to understand the unique correspondence between the name given a number and the quantity represented by that number, as well as recognising that the last number counted represents the total of objects being counted (Jordan & Levine, 2009; Malofeeva et al., 2004; Van Luit & Schopman, 2000). This correspondence between number, whether name or symbol, and its represented quantity is a human construction requiring instruction (Geary, 2013). Children who for whom this mapping process is difficult tend to experience later mathematical difficulties (Kroesbergen et al., 2009; Mazzocco et al., 2011). For year-one children, the ability to undertake digit comparison tasks successfully – deciding quickly and accurately which of two numbers presented symbolically represents the larger quantity – predicts arithmetical competence (Bartelet et al., 2014; Sasanguie & Vos, 2018). In particular, understanding the relationship between number and quantity alludes to subitising – the visual enumeration of a small number of objects, a competence known to be one of the most significant predictors of both mathematical competence (Butterworth, 2005; Groffman, 2009; Penner-Wilger et al., 2007) and arithmetical difficulties (Desoete et al., 2009).

1.1.4 Quantity discrimination

The fourth category of competence, quantity discrimination, alludes to an awareness of magnitude and the ability to compare between different magnitudes (Clarke & Shinn, 2004; Ivrendi, 2011; Jordan et al., 2006; Jordan & Levine, 2009; Yang & Li, 2008). Learners are encouraged to use terms like bigger than or smaller than (Gersten et al., 2005) and understand that nine represents a bigger quantity than five but is smaller than twelve (Lembke & Foegen, 2009). Children who are magnitude aware have ceased to view counting as ‘a memorized list’ detached from the words’ numerical magnitudes (Lipton & Spelke, 2005, p. 979). Understanding magnitude in this way, irrespective of ability or age, is a predictor of mathematical achievement (Aunio & Niemivirta, 2010; De Smedt et al., 2013; Desoete et al., 2012; Holloway & Ansari, 2009; Nan et al., 2006; Stock et al., 2010). Moreover, comparing two numbers presented symbolically and then asserting which represents the greater quantity, has been shown to underpin all manner of mathematics achievement (Bugden & Ansari, 2011; Lyons et al., 2014), an outcome that can persist for many years (Pollack et al., 2022; Sasanguie et al., 2012).

1.1.5 Understanding different representations of number

Andrews and Sayers’ fifth category of competence lies in recognising that numbers can be represented in a variety of ways (Ivrendi, 2011; Jordan et al., 2007; Yang & Li, 2008) and that these representations ‘act as different points of reference’ (Van Nes & Van Eerde, 2010, p. 146). Students who understand the number line, for example, tend to be arithmetically more competent than those who do not (Booth & Siegler, 2008; Siegler & Booth, 2004). Children who understand partitions as specific representations of a number are more likely to understand numerical structures (Thomas et al., 2002) and acquire arithmetical skills (Hunting, 2003). The use of manipulatives, particularly linking cubes, facilitates counting and the identification of errors (Van Nes & Van Eerde, 2010), while the use of fingers to represent numbers underpins a range of learning outcomes (Crollen & Noël, 2015; Domahs et al., 2010; Fayol et al., 1998; Gracia-Bafalluy & Noël, 2008; Jordan et al., 1992; Morrissey et al., 2016; 2020; Noël et al., 2005). In other words, children with a repertoire of connections between different representations of number are more likely to become arithmetically competent (Mundy & Gilmore, 2009; Richardson, 2004; Van Nes & De Lange, 2007; Van Nes & Van Eerde, 2010).

1.1.5 Estimation

The sixth category of competence concerns the ability to estimate, whether the size of an object (Ivrendi, 2011), the number of objects in a set of objects (Berch, 2005; Jordan et al., 2006, 2007; Malofeeva et al., 2004; Van de Rijt et al., 1999) or the position of a number on an empty number line (Booth & Siegler, 2006). In particular, integer number line estimation competence is a predictor of mathematical learning difficulties (Andersson & Östergren, 2012; Bull et al., 2021; Siegler & Opfer, 2003; Van der Weijden et al., 2018; Wong et al., 2017), mathematical competence generally (Maertens et al., 2016; Sasanguie et al., 2012; Schneider et al., 2018; Simms et al., 2016), and, irrespective of age, arithmetical competence (Dietrich et al., 2016; Friso-van den Bos et al., 2015; Lyons et al., 2014; Wong et al., 2016; Zhu et al., 2017), particularly with respect to fractions (Bailey et al., 2014; Fazio et al., 2014; Hansen et al., 2015; Van Hoof et al., 2017). By way of contrast, quantity estimation, or the ability to identify the number of objects in a set without recourse to counting (Crites, 1992; Siegler & Booth, 2005), has been implicated in the development of children’s arithmetical competence (Bartelet et al., 2014; Wong et al., 2016).

1.1.6 Simple additive competence

Andrews and Sayers’ (2015) seventh competence concerns the ability to perform simple arithmetical operations (Ivrendi, 2011; Jordan & Levine, 2009; Malofeeva et al., 2004; Yang & Li, 2008), here construed as the transformation of small sets through addition and subtraction (Jordan & Levine, 2009). Such skills are known to underpin arithmetical fluency (Berch, 2005; Jordan et al., 2007) and are stronger predictors of later mathematical achievement than measures of general intelligence (Geary et al., 2009; Krajewski & Schneider, 2009). Furthermore, advanced counting strategies, such as counting on rather than counting all, have been implicated in mathematics achievement and the identification of potential learning difficulties (Geary, 2011; Geary et al., 2004; Nguyen et al., 2016).

1.1.7 Awareness of number patterns

The final competence of Andrews and Sayers’ framework concerns the role of number patterns in children’s learning, and, in particular, the ability to identify missing numbers (Berch, 2005; Clarke & Shinn, 2004; Gersten et al., 2005; Jordan et al., 2010). Number patterns-related skills, particularly skip counting (DuVall et al., 2003), reinforce counting skills, facilitate later arithmetical operations (Gibbs et al., 2018; Grünke, 2016; Van Luit & Schopman, 2000) and underpin an understanding of place value (Nurnberger-Haag et al., 2021). Importantly, failing to identify missing numbers in a sequence is a strong indicator of later mathematical difficulties (Chard et al., 2005; Clarke & Shinn, 2004; Gersten et al., 2005; Lembke & Foegen, 2009). By way of contrast, the ability to recognise mathematical patterns, which lies at the very heart of mathematics (Zazkis & Liljedahl, 2002), has been identified as a key precursor of general mathematics achievement (Mulligan & Mitchelmore, 2009; Mulligan et al., 2013; Papic et al., 2011).

Finally, with respect to warranting its analytical relevance, the eight categories of foundational number sense have exposed significant differences in the ways in which opportunities for year-one children to learn are structure in Swedish-, Finnish- and Singaporean-authored textbooks used in Sweden (Sayers et al., 2021) and English- and Singaporean-authored textbooks used in England (Petersson et al., 2023). In short, and acknowledging that few aspects of mathematics education are not informed by cultural norms and unarticulated traditions, the FoNS framework should be an appropriate tool for understanding how curricular expectations, mathematics education research and teachers’ expressed goals for year-one children’s learning of mathematics align.

1.2. Compulsory education in England and Sweden

Superficially, the two educational systems under scrutiny, England and Sweden, share a number of similarities. Both have government-mandated national curricula that, according to Bernstein’s (1975) notion of classification, clearly separate mathematics from other forms of disciplinary knowledge; both systems employ, at different stages of a child’s development, national tests of mathematics achievement, and both have school inspection regimes. That said, within these nominal similarities are substantial differences. All Swedish schools are comprehensive, with all children following the same curriculum in mixed-attainment classes while English education is typically stratified, both within and across schools, leading to variation in children’s access to the curriculum. In Sweden, the first year of compulsory school, known as the preschool class, is for children who turn seven during the school year, while in England the same year, known as the reception year, is for children who turn five during the school year. In both countries, year-one follows those introductory years, confirming that year-one children in Sweden are typically two years older than their English equivalents. In the English context, national tests are considered high stakes, while in the Swedish, where teacher assessment remains a key element of the process, national tests are low stakes. From the perspective of their respective inspection regimes, and indicative of a systemic trust of teachers in Sweden but not England (Ozga et al., 2015), the English reflects a ‘hard’ governance, whereby emphases on target-setting, performance management and benchmarking are used to foster competition, and the Swedish a ‘soft’ governance, whereby emphases on mediation and brokering are used to foster cooperation and collaboration (Ehren et al., 2015).

Within the curricula themselves there are differences, some trivial some not. For example, the English national curriculum includes number patterns within the domain of number, while the Swedish locates the same topic in algebra. More importantly, perhaps, are differences in the ways the two curricula frame (Bernstein, 1975) mathematical knowledge. The English curriculum, specifying in detail the outcomes expected by the end of each year, including year-one, presents a strong framing of number-related knowledge. By way of contrast, the Swedish national curriculum, which mandates a small set of broad outcomes to be achieved after three years, offers a weak framing of such knowledge. The former, which is accompanied by an extensive set of exemplifications, leaves little scope for teacher autonomy, while the Swedish, which offers little additional exemplification, implies considerable teacher autonomy and highlights the significant role of textbooks in structuring teachers’ planning.

1.3. The current study

In this paper, framed by the eight categories of foundational number sense (Andrews & Sayers, 2015), we investigate, by means of a comparative study of English and Swedish teachers of year-one children, the interaction of the intended curriculum and the received curriculum on teachers’ expressed goals for year-one children’s learning of number. In so doing, we address, the following questions

In what ways are teachers’ number-related goals for year-one children informed by the intended curriculum?

In what ways are teachers’ number-related goals for year-one children informed by the received curriculum?

How do the combined effects of the intended and received curricula play out in two different cultural contexts?

The data on which this paper is based derive from semi-structured interviews undertaken with 19 (20 were planned but one was cancelled by the prospective participant) teachers of year-one children in England and 20 in Sweden. Acknowledging the relative homogeneity of each group, 20 interviews were considered sufficient for achieving thematic saturation, the point after which no new categories of response were generated by the analyses (O’Reilly & Parker, 2013). In practice, teachers in a diverse range of schools in different geographical locations in each country were contacted and, after having had the nature of the project and what participation would entail explained to them, invited to participate. This approach continued until twenty volunteers in each country, representing a range of professional experiences, had been identified.

Despite being self-selected, participating teachers formed an appropriately representative sample in terms of gender, age and years of experience, while represented schools broadly reflected the diversity of size and location of primary schools in both countries. Details of the two cohorts and their schools are shown in Tables 2 and 3 for England and Sweden respectively. In both countries, teachers’ ages and lengths of teaching experience accorded with recent OECD figures showing that 31% of English teachers are under 30 years of age (OECD, 2019a), compared with only 8% in Sweden (OECD, 2019b). Teachers’ declared genders were slightly skewed towards female as the same OECD figures show that 85% and 76% of English and Swedish primary teachers respectively are female. From the English perspective, however, this may have been due to the fact that a disproportionately high number of male primary teachers are headteachers and removed from teaching responsibilities.¹ With respect to the socio-economic status of represented schools, and acknowledging that different measures were available, the range of English children whose mother tongue was not English ranged from 2% to 43%, while the proportion of Swedish children with a foreign background ranged from 4% to 66%. That said, six Swedish schools included fewer than ten such children with the consequence that no such figures are publicly available.

Table 2. Characteristics of English teachers and their schools.

Pseudonym	Gender	Age	Years of experience	Location of school	Roll	% Mother tongue English
Carol	F	51	30	Large town	160	96
Christina	F	49	20	Large town	276	98
Charlie	F	28	6	Small town	431	91
Jessica	F	39	17	"	"	"
Amanda	F	47	12	"	"	"
Rachel	F	36	13	Small town	178	99
Jo	F	41	19	"	"	"
Gemma	F	32	10	Village	204	96
Megan	F	28	2	Small town	457	96
Lola	F	44	16	"	"	"
Anna	F	27	2	Small town	418	84
Kate	F	40	2	"	"	"
Louise	F	30	9	Large town	367	88
Peter	M	38	10	Large town	402	63
Rowena	F	37	6	"	"	"
Mary	F	51	25	Large town	415	83
Jenny	F	49	23	Large town	188	57
Sarah	F	22	1	Village	150	98
Michael	M	24	2	Large town	405	78
	Mean	34.8	10.3		299	79

Table 3. Characteristics of Swedish teachers and their schools.

Pseudonym	Gender	Age	Years of experience	Location of school	Roll	% Foreign background
Julia	F	44	10	Village	140
Mona	F	51	8	Affluent suburb	165
Matilda	F	40	17	Poor suburb	370	66
Marita	F	63	20	"	"	"
Isabelle	F	30	8	"	"	"
Wilma	F	40	7	Affluent suburb	350	7
Susanne	F	38	11	Mixed suburb	350	24
Ellinor	F	41	16	Large town	200	18
Lena	F	61	40	Affluent suburb	110	–
Marianne	F	49	18	Mixed suburb	380	17
Kerstin	F	53	13	"	"	"
Irene	F	48	20	Affluent suburb	150
Claudia	F	57	20	Affluent suburb	180
Anders	M	46	16	Village	160
Hanna	F	32	7	Small town	120
Erika	F	38	13	Large town	700	5
Sofie	F	39	3	"	"	"
Pauline	F	24	1	"	"	"
Lovisa	F	41	15	Small town	360	4
Jenny	F	35	5	Small town	450	13
	Mean	41.75	13.35		279

Confidentiality, anonymity and the right to withdraw were assured and informants’ pseudonyms agreed. The interview schedule, broadly focused on the teaching of number to year-one children, comprised a number of open questions, including one focused explicitly on what teachers expected their children to learn by the end of year-one. Typically lasting around 50 minutes, the video-recorded interviews were conducted at places of interviewees’ choosing. On completion, interviews were transcribed and participant-verified in preparation for analysis. Because our aim was to elicit teachers’ ambitions for their year-one children’s learning of number, a bottom-up rather than a top-down analytical process was thought necessary (Harry et al., 2005). Consequently, a constant comparison analysis (Conlon et al., 2015; Hardman, 2013; Thornberg & Delby, 2019), involving ‘a process of coding data and then grouping those codes into concepts in an increasingly hierarchical fashion’ (Wasserman et al., 2009, p. 358), was undertaken in the manner outlined below.

In practice, in order to preserve any culturally unique perspectives, each data set was analysed independently by two members of the project team according to the following schedule. First, each transcript was read and re-read. Second, all excerpts involving homework were identified. Third, following the constant comparison traditions, categories of responses were identified and, with each new category, previously read excerpts were re-read to determine whether the new category also applied to them. Fourth, the two analysts for each data set met to agree their categories before arranging them into broader themes. Finally, the resultant broad themes from each country were compared and contrasted. This process, drawing effectively on four independent analyses, resulted in the set of themes used to frame this paper. Importantly, no Swedish data were translated until excerpts selected for inclusion in the report had been identified. At this point, they were translated into English, including transforming Swedish idioms into forms recognisable to English-speakers.

2. Results

The analytical process described above yielded five themes common to the two cohorts and one theme unique to the English. There were minor themes yielded by each cohort’s analysis but these typically involved too few teachers to warrant inclusion. Five of the themes, including the unique English theme, are construed as representing learning goals commensurate with supporting the sixth theme, which is arithmetical competence. We begin by reporting each of these five themes, concerning, respectively, the role in children’s learning of number of systematic counting, number bonds, number line, mathematical terminology and its conceptual implications, and place value. Finally, we discuss how the two cohorts construe year-one children’s learning of arithmetical operations.

2.1. Systematic counting

Across both cohorts, teachers spoke of the need for year-one children to be able to count systematically. From the perspective of similarities, Swedish teachers, typically spoke explicitly of wanting their pupils to understand and be able to count a specified range of integers. Of these, seven spoke of the number range 1–20, three of the number range 1–100 and one in general terms. From the perspective of 1–20, Julia commented that children ‘should consolidate the numbers one to twenty … it is really important to understand number, and their symbols … they should know also how to count up or down from a given starting point’. Her comments reflected those of her colleagues, as seen in Matilda’s assertion that children should ‘understand at least up to twenty, that you know what twenty means and that you do not have to count with your fingers every time’. Those who favoured the larger range offered similar comments, albeit in a little less detail, as with Wilma’s desire for her children to ‘count to one hundred’ and Marianne’s hope that they ‘can count up to a hundred’.

In similar vein, few of the English cohort did not mention counting as a desired learning goal. Two spoke of the number range 1–20, while the remainder spoke of the number range 1–100. From the perspective of the former, Louise asserted, with a caveat, that

we usually work on doing that up to twenty … being able to identify one more, one less. Being able to count on from any given number. Being able to count back. (However) in year one, they are supposed to be able to work with numbers up to a hundred.

Those who spoke of working with numbers up to 100 offered similar goals but with, naturally, an extended range. For example, Charlie, whose comments reflected those of her colleagues, said that it is important ‘that they can count … to hundred and back from hundred … That they are fluent in counting forwards and backwards, especially backwards’.

Despite such nominal similarities, there were differences; more than half the English cohort but none of the Swedish spoke of wanting children to skip-count. In particular, teachers spoke of the importance, as suggested by Lola, of the ‘need to be able to count in twos, fives, tens’. In similar vein, Jessica, acknowledging curricular mandates and reflecting others’ comments, asserted that ‘by the end of the year … they’re expected to be able to count in twos, fives and tens’.

In sum, the desire for children to learn to count systematically united both cohorts, although they were distinguished by the number range 1–20 dominating the Swedish discourse and the 1–100 range the English. Moreover, English teachers, conforming to curricular expectations, spoke frequently of skip-counting.

2.2. Number bonds

Although their vocabulary varied, few teachers did not mention the importance of children learning their number bonds. More than half the Swedish cohort spoke of the importance of number friends, number families or number twins. From the perspective of number friends, the dominant theme, Irene, whose comments reflected those of others, said that ‘I want them to just get around numbers, numbers, work with numbers, like 10-friends’, ‘number friends, I think, up to ten’ and ‘all the number friends up to ten, in addition and subtraction … basic mathematics that you do now in year-one. Others, like Lovisa, spoke of number families, saying that children need to

understand numbers and their relationships. Like, three plus five is eight, which is five plus three. That you have, that you have a good feeling for it … I also work a lot with number families. To … point them out. If two plus three equals five, then three plus two equals five, five minus three equals two and so on.

In related vein, Kerstin spoke uniquely about number twins, saying that ‘I call them twins … yes, twins. They learn that four plus four, for example, is a twin. A small number of teachers, such as Wilma, spoke of both number friends and number families. She asserted that

you should know the number friends, ten-friends, eight-friends … like up to ten … We also expect to work with number families, as we call them … if two plus three equals five, then five minus three equals two and so on.

In similar vein, more than half the English cohort spoke of wanting their children to understand number bonds. Typical of their comments were those of Charlie, who wanted children ‘to know their number bonds to ten and twenty, quite fluently’ and Lola, who said that,

we want them to be able to know their number bonds to ten, twenty and number bonds to other numbers. Be able to say them. Hopefully, sort of, recognise them when they see them written down … Number bonds, I think. Really important. Especially to ten.

In similar vein, tacitly acknowledging what his Swedish colleagues called number families, Peter spoke about why such goals mattered, saying

I can’t remember what the proper word is; that … seven and three make ten and ten take away three is seven, and ten take away seven is three … What’s that called, when you’ve noticed that the relationship between addition and subtraction? … Like, like, so if, if, if they know their bonds to ten and then they realise that that means that they can do eight plus two is ten and ten minus two is eight, and ten minus eight is two, and that, or that there is two plus eight is ten and they’re like ‘Oh, I’ve got four number sentences. They all look different; there isn’t anything else to do with these three numbers, I’ve got everything’.

In sum, both sets of teachers, despite different colloquialisms, spoke of the importance of number bonds in facilitating children’s recall of key number facts, distinguished only by the tendency for Swedish teachers to refer to the number range 1–10 and the English 1–20.

2.3. Number line

Almost three quarters of the Swedish cohort spoke of the importance of learning about the number line. For some, as seen in Matilda’s ‘there is the number line and related words and concepts’ or Wilma’s ‘it can be a simple thing like jumping on the number line’, teachers’ goals were simply expressed. Others related the number line to a range of related outcomes. For example, Ellinor spoke of the number line in relation to simple addition and subtraction, saying that it is important that ‘you can add, you can take away … you can see where it is on the number line, how it is related to other numbers’, while Anders commented that ‘they should understand … the number line and quantity and what is greater than and less than. It should be obvious (to them) that eight is greater than six’.

In related vein, only a quarter of the English teachers spoke of wanting their pupils to understand and exploit the number line. Sarah, speaking in general terms, said that ‘we do lots of, like, number line work, 100 squares … missing numbers in 100 squares, so filling them in and just counting all the time and things like that’. Others tied additive operations to the number line, as with Megan’s ‘I think they need to be able to add and take away along a number line … So, they can actually explain their learning to others as well’ and Lola’s ‘adding and taking away using number lines’.

Overall, a substantial number of teachers in both countries construe the number line as an important goal for year-one children, not least from the perspective of its supporting the completion of additive operations.

2.4. Mathematical terminology and its implied concepts

From the perspective of similarities, several teachers from both cohorts spoke of wanting children to understand the equals sign in ways that would underpin a relational rather than an instrumental understanding of fundamental concepts. For example, reflecting the views of her Swedish colleagues, Isabelle commented that children should understand ‘likewise, equal to … several or fewer’, while Marianne was even more direct, saying that ‘I think you should understand the equals sign’. Others focused explicitly on the importance of the equals sign, as seen in Matilda’s comment that ‘understanding the equals sign is something we work very much with. What does the equals sign mean? It’s not just writing answers, but what does it really mean?’. In similar vein, Julia added that

I’m thinking of the equals sign, which I think is important, that they understand not only that one plus one is two, but that we really consolidate that, so they can also say that two equals one plus one. That they really know that.

From the English perspective, Rachel, who mentioned the equals sign six times, said that:

So, your equals sign and, it’s not … why it’s there? It means it’s got to be the same on each side and, you know, things like that. those, those core skills that are, are key for the children … I think in the past … the way the equals (sign) was taught, was that it’s, it’s a symbol on the end of a number sentence … we've worked a lot on what the equal sign actually means … We’re now turning those number sentences around.

In similar vein, Jo commented that

we spend a long time … talking about the equal sign and making sure that you know what that means. It's not just ‘We put an answer after it’. So, it's this skill of being able to read, you know, what's put in front of them in a mathematical sense.

From the perspective of difference, Swedish teachers, which is not what was found in the English utterances, also focused on comparative terms like bigger or smaller, more or less, or more and fewer. For example, Irene spoke of children needing to understand ‘bigger than, smaller than, more, fewer, double, half (…) biggest, smallest’, while Marita argued that children should ‘understand the signs: plus, minus, equal to, greater than, less than’. Moreover, while not mentioning the distinction between continuous and discrete quantities, several, as did Irene above, spoke of the distinction between less and fewer. Their comments were well-exemplified in Matilda’s view that children need

to understand that fewer and less are not the same thing … I think, as with so many concepts, you need to take a lot of time in grade one. Yes, you need a common language. Right now, we keep the word fewer. They want it to be less, but in mathematics fewer and less are not the same.

In sum, a substantial proportion of teachers from both countries argued for children to acquire a relational understanding of the equals sign. However, a dominant theme to emerge from Sweden but not England was the need to acquire a comparative vocabulary.

2.5. Place value

Thirteen English teachers, but not one Swedish, spoke about wanting children to understand place value. Reflecting the views of the majority, Lola commented on the need for children

to understand … why a two-digit number is a two-digit number; that it’s made up of something. So, understanding that twenty-six is actually two sets of tens and six, you know … It’s that sort of … trying to get them to represent those bigger numbers and understand.

A few, like Gemma, added details concerning the position of the digits of a number. She said that

to begin with, I think, understanding … place value … ensuring that they understand the position of numbers in terms of … which numbers come first, the value of each number … And then from that, being able to actually talk, later on in the year, about place value in terms of looking at tens and your ones … and the meaning behind that … So, it’s putting in place the conceptual understanding of the value … 

Overall, the comments of all thirteen teachers are summarised well by those of Amanda, who said,

I think they need to have a secure knowledge of place value. Certainly, in two-digit numbers, by the end of year one. They should be able to look at any two-digit number and say ‘this is how many tens, this is how many units’. If they can’t do that then we've got a problem.

Finally, alluding to a later section, two teachers spoke explicitly about the relation between addition and place value. In this respect, Jo’s comments linked arithmetical competence to an understanding of place value. She said that ‘this year … we started doing simple addition and subtraction … and we’re now realising, ‘Oh, we haven’t talked about place value’. So, now we’re going back to the place value’.

Overall, despite its absence in the Swedish discourse, place value seemed a core expectation of the English teachers, with a clear sense that children should recognise the number of tens and units in any two-digit number.

2.6. Arithmetical competence

Across both sets of interviews, the dominant theme concerned children’s acquisition of arithmetical competence. However, despite some similarities, there were substantial differences in how such competence was construed by the two sets of teachers.

With respect to similarities, the majority of teachers from both cohorts spoke of wanting their children to acquire basic additive competence. From the Swedish perspectives, several teachers offered general goals, as with those of Julia, who asserted that children ‘should understand that one can take away or add’, while both Isabelle and Anders simply suggested that pupils should ‘master addition and subtraction’. Others added flesh to the bones of such goals. Wilma, reflecting her own and others’ comments concerning number ranges, said that ‘you should know addition between zero and twenty, … and subtraction at least up to ten and preferably up to twenty’. In similar vein, all English teachers spoke of addition and its inverse. For several, their goals were expressed simply and with no elaboration, as with Amanda’s, ‘obviously, being able to add and subtract’ and Rowena’s ‘adding and subtracting’.

In addition, teachers from both cohorts offered conceptual supplements to such goals, albeit in different ways. From an English perspective, and speaking for several, Jessica asserted that children need to know that ‘when you're adding you're always getting more … and when you're subtracting, you're always going to get less because you're taking some away’. By way of contrast, several Swedish teachers spoke explicitly of wanting their children to acquire a transferable additive awareness based on an awareness of the properties of the base ten number system. In this respect Susanne’s view was typical. She said that it is important that children

understand the connection, that they can see the pattern, that if … I say seven plus three, they can see also seventeen plus three. And when it comes to subtraction … they will be able to see the relationship and use it.

In addition, Lena, speaking of the need for children to understand the inverse relationship between addition and subtraction, said that pupils

should be able to see … the relationship between addition and subtraction. I also want them to see that there is a connection there … if they have done subtraction tasks and I check them, I ask them, ‘do you know how I correct this? … I check it with addition’, I say.

Overall, teachers from both cohorts expressed largely similar goals, although when considering supplementary concepts, it could be argued that the Swedish perspective was more sophisticated than the English.

The key differences concerned multiplication and its inverse. Not one Swedish teacher expected year-one children to learn about multiplication, while more than half the English cohort spoke of multiplication and its inverse as important goals for year-one children. In this respect, Rachel, echoing the aims of several, spoke of the need for ‘early … multiplication and division, in terms of sharing, in terms of groups … and understanding the vocabulary around that early multiplication and division’. For others, this entailed preparatory learning, as seen Charlie’s assertion of the need ‘to understand that multiplication is repeated addition  …  and to know that division is sharing’.

Finally, tacitly critiquing curricular expectations, Anna, observed that

they find timesing hard, times and divide. Sometimes dividing is actually easier but we have to teach sharing and also grouping. So sharing is fine, they understand that. You know, cutting, splitting the cake or sharing sweets or something. And then we also have to teach grouping which does it the other way round. That confuses them. And it always confuses them. So, if you’re doing twelve divided by three, whether it’s into three groups of four or, whether you’re doing it into groups of three and there are four groups … It just baffles them.

In sum and in contrast with their Swedish colleagues, who did not mention multiplication, half the English cohort expressed ambitions for year-one children to learn multiplication and division, although there were some concerned that year-one children find it too difficult.

3. Discussion

In this paper, our aim was to investigate, by means of a comparative interview study conducted in England and Sweden, the interaction of the intended curriculum and the received curriculum on teachers’ expressed goals for year-one children’s learning of number. In so doing, three questions were addressed, namely;

In what ways are teachers’ number-related goals for year-one children informed by the intended curriculum?

In what ways are teachers’ number-related goals for year-one children informed by the received curriculum?

How do the combined effects of the intended and received curricula play out in two different cultural

Contexts?

The analyses, undertaken separately for each of the two data sets, yielded six themes, five of which were common to both sets of teachers and one unique to the English. In the following, we discuss these themes, summarised in Table 4, against various curriculum documents and the eight curriculum-independent categories of foundational number sense.

Table 4. Summaries of the six categories of response, highlighting key similarities and differences.

	England (Age 5/6)	Sweden (age 7/8)
Systematic counting	Number range 1–100	Number range 1–20
Number bonds	Limited conceptual and colloquial variation on number range 1–20	Wide conceptual and colloquial variation on number range 1–10
Number line	Limited conceptual underpinning of addition and subtraction	Widespread conceptual underpinning of addition and subtraction
Mathematical terminology	Articulation of a relational understanding of the equals sign	Articulation of a relational understanding of the equals sign plus a robust comparative vocabulary
Place value	Understand the values of the digits in any two-digit number	No such emphasis
Arithmetical competence	Undertake addition, subtraction, multiplication and division	Undertake addition and subtraction

3.1. Systematic counting

Teachers from both countries acknowledged a need for children to be able to count both up and down from arbitrary starting points (Jordan & Levine, 2009; Lipton & Spelke, 2005). In so doing, they were clearly encouraging children to understand the successor principle (Hartnett & Gelman, 1998) and that each number occupies a fixed position in the sequence of all numbers (Jordan et al., 2006). In other words, teachers acknowledged, albeit tacitly, a need for children to understand ordinality (Ivrendi, 2011; LeFevre et al., 2006). Despite such similarities, there were key differences in the two cohorts’ perspectives on counting. The tendency among Swedish teachers, in accordance with the expectations of FoNS, was to restrict children’s counting to the number range 1–20, while the tendency among the English teachers was to promote the range 1–100. This difference is of particular interest when compared with the expectations of teachers’ respective national curricula. On the one hand, the strongly framed English national curriculum specifies the number range 1–100 for year-one children, indicating that English teachers’ perspectives on systematic counting align with the intended curriculum. On the other hand, the weakly framed Swedish national curriculum specifies no particular number range, indicating that Swedish teachers’ perspectives on systematic counting tend not only to align with a received curriculum but are more age-appropriately grounded (Andrews & Sayers, 2015). Despite such nominal similarities, there were differences; more than half the English cohort and none of the Swedish spoke of wanting children to skip-count. Such goals, which are clearly specified in the English national curriculum, represent a perspective informed by the intended curriculum. That being said, skip counting in the manner described by the English teachers, while not falling explicitly within the FoNS framework, could be construed as tacitly addressing number patterns (Andrews & Sayers, 2015), with its specific impact on later arithmetical operations (Gibbs et al., 2018; Grünke, 2016; Van Luit & Schopman, 2000) and understanding of place value (Nurnberger-Haag et al., 2021).

3.2. Number bonds

Both sets of teachers spoke of the importance of number bonds, or number friends as they are known in Sweden, in facilitating children’s recall of key number facts and later arithmetic. In addition, Swedish teachers also spoke of number families and number twins. Such variation in vocabulary seems unknown in the English context, as the English national curriculum not only expects children to ‘use number bonds and related subtraction facts within 20’ but, in so doing, offers a non-statutory support example entirely resonant with the Swedish definition of number families. In other words, Swedish teachers clearly distinguish a range of related concepts that the English construe as number bonds. In contrast with the English curriculum, no mention is made of number bonds, or any related manifestation, in the Swedish curriculum. From the perspective of FoNS, teachers’ emphases on number bonds and their related concepts resonate with different representations of number, particularly with respect to the role of partitions in the development of children’s understanding of numerical structures (Thomas et al., 2002) and arithmetical skills (Hunting, 2003). Moreover, such emphases accord well with the seventh FoNS category, simple additive competence, or the ability to perform simple arithmetical operations within the number range 1–20 (Ivrendi, 2011; Malofeeva et al., 2004; Yang & Li, 2008). Finally, English teachers’ number bonds-related goals resonate with the national curriculum expectation that children should ‘add and subtract one-digit and two-digit numbers to 20, including zero’. Overall, while both sets of teachers appreciate the developmental contribution of number bonds, the impression is that English teachers’ emphases are strongly located in an intended curriculum, while the Swedish are clearly located in a received curriculum.

3.3. The number line

Although not one of the dominant themes, the number line is included due to its being a strong discriminator between the two cohorts, being mentioned by three quarters of the Swedish cohort but only a quarter of the English. For Swedish teachers, the number line is an important support to children’s learning of various number-related concepts. Implicitly, these included various concepts relating to quantity and comparative mathematical vocabulary. In addition, both sets of teachers emphasised the role of the number line in supporting children’s additive competence. That said, the Swedish national curriculum makes no mention of the number line, indicating another example of the received curriculum within which Swedish teachers operate. By way of contrast, the English national curriculum mandates the number line as a particular representation of number, which differs from the English teachers’ emphasis on additive competence. In other words, despite explicit curricular expectations, English teachers’ beliefs also seem framed by a received rather than an intended curriculum. From the perspective of FoNS, the manner in which teachers connect the use of the number line with additive operations accords well with simple additive competence. Moreover, while the English curriculum implicitly addresses different representations of number, English teachers, as indicated above, do not share this goal. Disappointingly, much recent research has highlighted the role of number line estimation in support of various aspects of children’s later learning of mathematics, particularly arithmetic (Maertens et al., 2016; Schneider et al., 2018; Simms et al., 2016), place value (Dietrich et al., 2016) and fractions (Bailey et al., 2014). In other words, the curricula of both England and Sweden, along with the teachers of both countries, estimation and its implications seem to have gone unnoticed.

3.4. Mathematical terminology and its conceptual implications

Both sets of teachers emphasised different linguistic and related conceptual expectations. While there was commonality with respect to the introduction of the equals sign and the need to establish a relational rather than operational understanding. That is, teachers typically want their pupils to understand the equals sign as representing equality between two expressions rather than a command to operate (Alibali et al., 2007; McNeil et al., 2006). Interestingly, while the FoNS framework does not include reference to the equals sign, a relational understanding has been found to be positively related to students’ algebraic competence (Booth & Davenport, 2013; Fyfe et al., 2018), to the extent that a relational understanding in grade two impacts positively in grade four (Matthews & Fuchs, 2020). Such emphases, which can be found in both national curricula, show the impact of the intended curriculum on both sets of teachers’ ambitions.

The key difference between the two cohorts lay with the Swedish teachers collectively emphasising the importance of comparative vocabulary, particularly in relation to the distinction between discrete and continuous quantities. Interestingly, such vocabulary is not evident in the Swedish curriculum but explicit in the English, which expects children to ‘use the language of: equal to, more than, less than (fewer), most, least’. In other words, while Swedish teachers’ views seem located in a received curriculum, the lack of such emphases among their English colleagues seems at odds with the intended curriculum. Importantly, by way of justifying the need to understand comparative vocabulary and its associated concepts, there is evidence that primary-aged children who are comfortable with continuous quantities better understand proportionality than children who comfortable only with discrete quantities (Boyer et al., 2008; Jeong et al., 2007). From the perspective of FoNS, the fourth category of competence, quantity discrimination, expects children to deploy such comparative vocabulary (Andrews & Sayers, 2015), not least because children who are magnitude aware have moved beyond counting as ‘a memorized list and a mechanical routine, without attaching any sense of numerical magnitudes to the words’ (Lipton & Spelke, 2005, p. 979).

3.5. Place value

Of all the themes identified by the analyses, place value was the greatest discriminator, discussed by two-thirds of the English cohort but none of the Swedish. Specifically, teachers’ emphases were on the need for children to understand the structure of a two-digit number and, for some, its supporting role for arithmetic. In this respect, the English national curriculum expects year-one children to engage with place value in order that they can compare numbers up to 100. According to Howe (2014, p. 187), place value ‘is the central concept of arithmetic computation. It is not simply a vocabulary issue, of knowing the ones place, the tens place, and so on; it is the key organizing principle by which we deal with numbers’. However, despite its importance, there is a consensus among scholars that, despite Howe’s enthusiasm for introducing place value to first-grade children (year two in England), too few children are cognitively ready for such a complex concept. For example, a study of US grade two students (year three in England) found that while most children, with or without mathematics difficulties, were able to count to 16 and read the number 16, few knew what the 1 in 16 actually meant (Jordan & Hanich, 2000). Similarly, a study of Turkish second-grade students found that while students were competent with counting and, to a lesser extent, addition, their understanding of place value was limited (Birgin et al., 2022). Moreover, a German study found that by the end of grade two, typically functioning children were able to identify accurately the larger of two two-digit numbers, indicating that they only then had they begun to develop an intuitive understanding of place value. Effective teaching of place value requires several years of instruction (Rojo et al., 2021), and even well-designed interventions have achieved limited success with grade two students’ understanding of place value (Barner et al., 2018). Thus, available evidence suggests that English teachers of year-one children tend to operate within an age-inappropriate intended curriculum. Indeed, according to Van de Walle et al. (2018), place value daws on five overarching principles: understanding sets of 10; understanding the position of digits in numbers, patterns in numbers; composing and decomposing numbers in flexible ways; conceptualisation of larger numbers. Such principles accord well with the eight categories of FoNS, which does not acknowledge place value as an essential goal for all year-one children’s learning.

3.6. Arithmetical operations

All teachers in both countries wanted children to acquire arithmetical competence. In particular, in accordance with international expectations (Ivrendi, 2011; Jordan & Levine, 2009; Malofeeva et al., 2004; Yang & Li, 2008), they wanted their children to acquire additive competence and an awareness of the inverse relationship between addition and subtraction. Interestingly, while the English curriculum expects year-one children to engage with addition and subtraction within the integer range 1–20, which is precisely what the sixth category of FoNS recommends (Andrews & Sayers, 2015), English teachers did not specify a number range, while their Swedish colleagues, whose curriculum specifies no range, were confident that year-one children should be restricted to the range 1–20. Thus, at least from the perspective of additive competence, Swedish teachers seem to be working within a received curriculum clearly resonant with research, while the English seem to be located in an intended curriculum, albeit misrepresented by them, also resonant with research.

However, there was a significant distinction between the two cohorts. More than half the English cohort, with utterances located in a curriculum expectation that year-one children should be able to ‘solve one-step problems involving multiplication and division’, spoke of multiplication and its inverse as important goals for year-one children. As with additive competence, no number range was specified. Such ambitions, which were shared with not one of their Swedish colleagues, fall outside the FoNS framework. In this respect, despite evidence that grade one (year two in England) children can engage successfully with the principles of repeated addition (Hendrickson, 1979), the consensus with respect to age-appropriateness is that additive operations are suitable for grade one and multiplication for grades two or three (Burns et al., 2015; Henry & Brown, 2008). Consequently, it is probably unsurprising that internationally there is a tendency for textbooks to introduce multiplication in the second or third grades (Ashcraft & Christy, 1995; Boonlerts & Inprasitha, 2013). In sum, while English teachers’ enthusiasm for multiplication may be located within an intended curriculum, the literature suggests that such intentions may not be age-appropriate.

3.7. Teachers’ goals and the eight categories of foundational number sense

With respect to the eight categories of foundational number sense, there is clear resonance between the ambitions of both cohorts with systematic counting and additive operations. However, other categories are present by implication and others are missing entirely. With respect to those whose presence can be inferred, number recognition is implicitly present in teachers’ utterances about number bonds, the number line and arithmetical operations. Similarly, the relationship between number and quantity is implicit in, say, Swedish teachers’ ambitions concerning comparative vocabulary and both cohorts’ arithmetic-related goals. In similar vein, it could be argued that quantity discrimination was implicit in Swedish teachers’ comparative vocabulary ambitions. Different representations of number were tacitly present in teachers’ utterances about the number line and number bonds, while number patterns were implicit in English teachers’ desires concerning skip-counting. However, estimation was absent from the discourse of both sets of teachers and their respective national curricula. Indeed, recent studies have shown that estimation, despite much recent work highlighting its developmental significance, is exceptionally poorly conceptualised in the curricula of both England (Andrews et al., 2022) and Sweden (Sunde et al., 2022). Overall, the resonance between teachers’ espoused number-related goals and those competences necessary for their pupils’ successful learning of mathematics is disappointingly low. It could be argued, due to their tending to stay within the number range 0–20, that Swedish teachers were more sensitive to their pupils’ developmental needs than their English colleagues. But the former, working within a weakly-framed curriculum, are considerably less constrained than the latter, who are expected to adhere to goals – for example, place value, multiplication and division, and extending the number range from 1–20 to 1–100 – that are neither age- or developmentally-appropriate.

3.8. Closing thoughts

In this paper, we set out to examine how the intended and the received curricula inform year-one teachers’ privileged number-related learning goals in England and Sweden. In so doing, we have framed our analyses and discussion against the eight research-derived and curriculum-independent categories of foundational number sense, which represent a core set of competences necessary for the successful learning of mathematics (Andrews & Sayers, 2015) a decision that proved appropriate for evaluating the appropriateness of both teacher and curriculum goals. While both sets of teachers work within mandated national curricula, the extent to which their views about children’s learning are informed by either intended or received curricula seems to vary considerably. The English curriculum, with its strong framing of content, seems to have a greater influence on teachers’ beliefs about desirable learning outcomes than the Swedish with its weak framing of content. In many ways, this is probably not surprising, although the fact that Swedish children are typically two years older than their English counterparts adds an interesting dimension. For example, Swedish teachers seem to draw on a received curriculum more closely aligned with the developmental goals of foundational numbers sense than their English colleagues, who, influenced by the expectations of the English curriculum, seem to expect year-one children to learn much age-inappropriate material.

Overall, and acknowledging that teachers’ views were elicited in ways that allowed no time for prior preparation, our view is that their utterances drew extensively on their respective intended curricula. However, there was little evidence of any didactical folkways (Buchmann, 1987) ‘uncritically accepted from our ancestors’ (Lauwerys, 1959, p. 294). Indeed, particularly in the Swedish context and despite the limited awareness of some of the categories of foundational number sense, many teachers’ privileged goals appear informed by an awareness of not only what is age-appropriate but of a range of number-related competences that facilitate age-appropriate goals. Importantly, a key implication of this study, particularly for curriculum developers, is that where curricula are tightly framed teachers’ perspectives on learning are more likely to be driven by the intended curriculum than in those systems where the curriculum is weakly framed. Moreover, acknowledging the limitations of a study of this nature, the evidence indicates that a weakly framed curriculum may lead to better-informed and developmentally aware teachers than a strongly framed curriculum.

Acknowledgements

The authors acknowledge the contributions of Gosia Marschall and Anna Löwenhielm for undertaking and transcribing the interviews in England and Sweden respectively.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Swedish Research Council (Vetenskapsrådet) under project grant 2015-01066.

Notes

1 https://commonslibrary.parliament.uk/research-briefings/cdp-2022-0197/ Accessed March 5, 2024.