Search

In this paper we present analyses of three textbooks currently used in the teaching of mathematics to year-one children in England. One is an established English-authored textbook, while the others are Singaporean-authored imports promoted by government as solutions to perceptions of systemic failure. Every task in each textbook was coded against a set of eight number-related competences known to support children’s learning in both short and long terms. Such a framework, which is literature-derived and curriculum-independent, enables meaningful comparison of materials deriving from different cultural contexts. Analyses of the proportions of all tasks coded for the different competences showed that none of the three books adequately addresses all eight competences, although the English-authored comes closest. Moving averages, undertaken to show the temporal location of the opportunities presented for children to acquire the eight competences, showed them distributed throughout the school year in the English-authored textbook but only during the first half of the school year in the two Singaporean-authored textbooks. Some implications for the importation of such materials are discussed., FoNS

Motivated by earlier research highlighting Swedish teachers’ beliefs that the setting of homework compromises deep-seated principles of educational equity, this paper presents an exploratory study of Swedish parents’ perspectives on homework in their year-one children’s learning. Twenty-five parents, drawn from three demographically different schools in the Stockholm region, participated in semi-structured interviews. The interviews, broadly focused on how parents support their children’s learning and including questions about homework in general and mathematics homework in particular, were transcribed and data subjected to a constant comparison analytical process. This yielded four broad themes, highlighting considerable variation in how parents perceive the relationship between homework and educational equity. First, all parents spoke appreciatively of their children receiving reading homework and, in so doing, indicated a collective construal that reading homework is neither homework nor a threat to equity. Second, four parents, despite their enthusiasm for reading homework, opposed the setting of any homework due to its potential compromise of family life. Third, seven parents indicated that they would appreciate mathematics homework where it were not a threat to equity. Finally, fourteen parents, despite acknowledging homework’s potential compromise to equity, were unequivocally in favour of mathematics homework being set to their children., FoNS

In this paper, motivated by the desire to understand which forms of parent-initiated activity are productively implicated in young children’s mathematics learning, we present a methodological critique of recent research. Many such studies, based on assumptions that parent-initiated activities can be categorised as formal or informal, direct or indirect, or advanced or basic, exploit surveys to elicit how frequently parents engage their children in various predetermined activities. While such survey data have the potential to yield important insights, the analytical procedures typically employed prevent them. Studies involving factor analyses yield uninterpretable factors, which are then used to create summative variables based on the scores of individual activities. Other studies, drawing on untested preconceptions, simply create summative variables. In all cases, these summative variables are based on such a wide range of qualitatively different activities that labels like formal or informal become arbitrary and the potential of individual activities to support learning gets lost beneath colleagues’ desires for statistical significance. In closing, we ask colleagues, albeit somewhat rhetorically, what is the purpose of such research? Is it to identify those activities that support learning or to offer statistically robust factors, which, due to the diversity of activities embedded within them, offer few useful insights?

used in the teaching of mathematics to year-one children inEngland. One is an established English-authored textbook, whilethe others are Singaporean-authored imports promoted by governmentas solutions to perceptions of systemic failure. Every taskin each textbook was coded against a set of eight number-relatedcompetences known to support children’s learning in both shortand long terms. Such a framework, which is literature-derived andcurriculum-independent, enables meaningful comparison ofmaterials deriving from different cultural contexts. Analyses ofthe proportions of all tasks coded for the different competencesshowed that none of the three books adequately addresses alleight competences, although the English-authored comes closest.Moving averages, undertaken to show the temporal location ofthe opportunities presented for children to acquire the eightcompetences, showed them distributed throughout theschool year in the English-authored textbook but only duringthe first half of the school year in the two Singaporean-authoredtextbooks. Some implications for the importation of such materialsare discussed.

Measures of association, which typically require pairwise data, are widespread in many aspects of educational research. However, due to the need to reduce their content to equal numbers of units of analysis, they are rarely found in the analysis of textbooks. In this paper, we present two methods for overcoming this limitation, one through the use of disjoint sections and the other through the use of overlapping moving averages. Both methods preserve the temporal structure of data and enable researchers to calculate a measure of association which, in this case, is the complementary Euclidean average distance, as an indicator of the books' similarity. We illustrate these approaches by means of a comparative analysis of three commonly-used English and Swedish mathematics textbooks. Analyses were focused on individual tasks, which had all been coded according to the presence or absence of particular characteristics. Both methods produce nearly identical results and are robust with respect to both densely and sparsely occurring characteristics. For both methods, widening the aggregation window results in a slightly increased level of quantified similarity, which is the result of the 'smoothing effect'. We discuss the relation between the window width and the choice of research question.

In this paper, motivated in part by evidence that Swedish teachers are sceptical of parents’ abilities to offer appropriate support, we present an exploratory investigation of the activities Swedish parents initiate to facilitate their year-one (first grade) children’s learning of mathematics. Data, derived from 25 semi-structured interviews conducted with parents from three demographically different schools, were subjected to constant comparison analyses and yielded three broad categories of activity. These concerned the use of games in the learning of mathematics, contextualised mathematics activities like cooking and shopping, and decontextualised mathematics activities like systematic counting. Collectively, the results indicate that while parents of year-one children are confident supporting their children’s learning of mathematics, they are also conscious of the need to avoid both undermining schools’ efforts and exacerbating educational inequity. With few exceptions, the activities parents described were age-appropriate and more likely to complement teachers’ actions than not., FoNS

This paper presents an exploratory study of English and Swedish teachers' perspectives on the role of homework in year-one children's learning of number. In order to ensure cultural integrity, data were analysed independently by two colleagues in each context. Analyses yielded three broad but cross-culturally common themes reflecting culturally situated notions of common sense. These concerned the existence of homework, the purpose of homework and the role of parents in homework's completion. While homework was unproblematic for all English teachers, half the Swedish cohort spoke against it, arguing that variation in home background would compromise principles of equity. All teachers who set homework, whether English or Swedish, spoke of homework as a means of supporting children at risk of falling behind their peers, a process by which children practice routine skills. English teachers' homework-related justifications were located in a discourse of target setting that was invisible in the Swedish.

In this paper, we examine the national curricula for primary mathematics for each of the four constituent nations of the United Kingdom (England, Northern Ireland, Scotland and Wales) for the estimation-related opportunities they offer children. Framed against four conceptually and procedurally different forms of estimation (computational, measurement, quantity and number line), the analyses indicate that computational estimation and measurement estimation were addressed in all four curricula, albeit from a skills-acquisition perspective, with only the Scottish offering any meaningful justification for their inclusion. The process of rounding, absent in the Northern Ireland curriculum, was presented as an explicit learning objective in the English, Scottish and Welsh curricula, although it was only the Scottish that made explicit the connections between rounding and computational estimation. In all curricula, both quantity estimation and number line estimation were effectively absent, as was any explicit acknowledgement that learning to estimate, irrespective of its form, has a developmental role in the learning of other mathematical topics.

The analysis of the content of school textbooks, particularly in a time of cross-cultural borrowing, is a growing field restricted by the tools currently available. In this paper, drawing on the analyses of three English year-one mathematics textbooks, we show how two approaches to the analysis of sequential data not only supplement conventional frequency analyses but highlight trends in the content of such textbooks hidden from frequency analyses alone. The first, moving averages, is conventionally used in science to eliminate noise and demonstrate trends in data. The second, Lorenz curves, is typically found in the social sciences to compare different forms of social phenomena. Both, as we show, extend the range of questions that can be meaningfully asked of textbooks. Finally, we speculate as to how both approaches can be used with other forms of ordered classroom data., FoNS

In this paper we present statistical analyses of three textbooks used by Swedish teachers to support year one children's learning of mathematics. One, Eldorado, is authored by Swedish teachers, another, Favorit, is a Swedish adaptation of a popular Finnish series and the third, Singma, is a Swedish adaptation of a Singapore series. Data were coded against the eight categories of foundational number sense, which are the number-related competences literature has shown to be essential for the later mathematical success of year one learners. Two analyses were undertaken; the first was a frequency analysis of the tasks coded for a particular category, the second was a time-series analysis highlighting the temporal location of such opportunities. The frequency analyses identified statistically significant differences with respect to children's opportunities to acquire foundational number sense. Additionally, the time series showed substantial differences in the ways in which such tasks were located in the structure of the textbooks. Such differences, we argue, offer substantial didactical challenges to teachers trying to adapt their practices to the expectations of such imports.

This paper presents an exploratory study of English and Swedish teachers' perspectives on the role of parents in year one children's learning of number. Drawing on the results of semi-structured interviews, data from each cohort were analysed independently to ensure the cultural integrity of any response categories and the results of this process compared. Two broad themes were identified concerning implicit and explicit forms of parental involvement. The former, manifested similarly across the two cohorts, concerned the importance of parents presenting children with positive attitudes towards mathematics. The latter, incorporating three comparable subthemes, focused on the creation of number-rich home environments, home–school communication and parents' role in the completion of homework. All three subthemes differentiated the cohorts in ways that highlighted teachers' culturally situated perspective on teaching and learning. Some implications are discussed, particularly with respect to the challenge this study poses for developers of cross-cultural survey instruments.

Computational estimation, typically construed as an approximate mental calculation of an arithmetical problem, is an important skill in everyday life and a wide range of professional contexts. Despite its importance, textbooks and curricula address it inadequately, with the consequence that many teachers are uncertain as to why and how they should teach it. In this paper, we present a narrative literature review that brings together the extensive research of the cognitive psychologists and the limited research of the mathematics educators to clarify the nature of computational estimation and its development. Focused initially on the strategies used in computational estimation before turning to children’s and adults’ computational estimation competence, the review shows that computational estimation, which develops over time, draws on a wide range of strategies reciprocally dependent on a secure understanding of numbers and arithmetic. It shows that the poor estimation competence of children and adults’ is susceptible to interventions, particularly with respect to addressing a common misconception that the purpose of computational estimation is the mental calculation of exact solutions.

In this paper, we examine the national curricula for primary mathematics for each of the four constituent nations of the United Kingdom (England, Northern Ireland, Scotland and Wales) for the estimation-related opportunities they offer children. Framed against four conceptually and procedurally different forms of estimation (computational, measurement, quantity and number line), the analyses indicate that computational estimation and measurement estimation were addressed in all four curricula, albeit from a skills-acquisition perspective, with only the Scottish offering any meaningful justification for their inclusion. The process of rounding, absent in the Northern Ireland curriculum, was presented as an explicit learning objective in the English, Scottish and Welsh curricula, although it was only the Scottish that made explicit the connections between rounding and computational estimation. In all curricula, both quantity estimation and number line estimation were effectively absent, as was any explicit acknowledgement that learning to estimate, irrespective of its form, has a developmental role in the learning of other mathematical topics.

In this paper, we introduce an adaption of the Lorenz curve as a powerful tool for graphicallypresenting content analyses of mathematics textbooks. Three textbooks currently used withyear-one children in Sweden, one Swedish-, one Finnish and one Singaporean-authored, wereanalysed against the eight categories of Foundational Number Sense (FoNS), which is a set ofliterature-derived and instruction-dependent competences that all children need to acquire ifthey are to become successful learners of mathematics. The adapted Lorenz curvehighlighted, inter alia, major differences in the distribution of FoNS-related tasks across thethree textbooks. In general, the Swedish-authored textbook offers repeated cycles of FoNSrelated opportunities, the Finnish-authored textbook offers such opportunities continuously,and the Singaporean-authored textbook typically offers FoNS-related opportunities onlywithin its earlier pages., FoNS - Foundational Number Sense

This paper presents an exploratory study of English and Swedish teachers’ perspectives on the role of homework in year-one children’s learning of number. In order to ensure cultural integrity, data were analysed independently by two colleagues in each context. Analyses yielded three broad but cross-culturally common themes reflecting culturally situated notions of common sense. These concerned the existence of homework, the purpose of homework and the role of parents in homework’s completion. While homework was unproblematic for all English teachers, half the Swedish cohort spoke against it, arguing that variation in home background would compromise principles of equity. All teachers who set homework, whether English or Swedish, spoke of homework as a means of supporting children at risk of falling behind their peers, a process by which children practice routine skills. English teachers’ homework-related justifications were located in a discourse of target setting that was invisible in the Swedish., FoNS - Foundational Number Sense

This paper draws on semi-structured interviews undertaken with twenty teachers of year one children in Sweden. Interviews focused on teachers’ construal of their own and their pupils’ parents’ roles in supporting year one children’s learning of early number. Data, which were analysed by means of a constant comparison process, yielded homework as a theme that dichotomised teachers between those who set homework for learning number and those who do not. Of those who set homework, the majority construed it as a means of facilitating number-related fluency, particularly for children in danger of falling behind their peers. Of those who do not, the majority argued that differences in family backgrounds would compromise societal principles of equality of opportunity., FoNS

In this paper we present analyses of popular grade one textbooks, one from each of England and Sweden. Focused on Foundational Number Sense (FoNS), we examine how each book’s tasks facilitate children’s learning of those number-related competences that require instruction and which underpin later mathematical learning. Analyses identified both similarities and differences. Similarities lay in books’ extensive opportunities for children to recognise and write numbers and undertake simple arithmetical operations. However, neither offered more than a few tasks related to estimation or simple number patterns. Differences lay in the Swedish book’s greater emphases on different representations of number, quantity discrimination and relating numbers to quantity, highlighting conceptual emphases on number. The English book offers substantially more opportunity for students to count systematically, highlighting procedural emphases.

Evidence indicates that children with a well-developed number sense are more likely to experience long-term mathematical success than children without. However, number sense has remained an elusive construct. In this chapter, we summarise the development of an eight-dimensional framework categorising what we have come to call foundational number sense or those non-innate number-related competences typically taught during the first years of schooling. We also show, drawing on grade one lessons from Hungary and Sweden, how focused instruction on conceptual subitising, the teaching of children to identify and use easily recognisable groups of objects to structure children’s understanding of number, facilitates children’s acquisition of a range of foundational number sense-related competences.

Acknowledging that what teachers believe informs how they teach we argue the importance of understanding, at both entry and exit, teacher education students’ beliefs about mathematics, mathematics teaching and themselves as learners of mathematics. In this paper we report on the development and trial of a simple to use online survey instrument focused on uncovering teacher education students’ mathematics-related beliefs. Twenty items, targeted on a range of constructs reported in the literature, were set against five point Likert scales. The instrument was found to be reliable and an exploratory factor analysis yielded seven interpretable belief dimensions. The interactions of these dimensions allude to groups of student likely to prove problematic during their programme.

It is known that an appropriately developedfoundational number sense (FONS), or the ability tooperate flexibly with number and quantity, is a powerfulpredictor of young children’s later mathematical achievement.However, until now not only has FONS been definitionallyelusive but instruments for identifyingopportunities for children to acquire its various componentshave been missing from the classroom observationtools available. In this paper, drawing on a constant comparisonanalysis of appropriate literature, we outline thedevelopment of an eight dimensional FONS framework.We then show, by applying this framework to three culturallydiverse European grade one lessons, one English,one Hungarian and one Swedish, that it is both straightforwardlyoperationalised and amenable to cross culturalanalyses of classroom practice. Some implications arediscussed.

For a variety of reasons children start school with differing number-related skills, leading to differences in later mathematics achievement. Such differences prompt the question, what number-related experiences are necessary if the first year of school is to prepare children appropriately for their learning of mathematics? In this paper we discuss the development of an eight dimensional framework, foundational number sense (FoNS), that characterises those learning experiences. We then demonstrate the framework's analytical efficacy by evaluating episodes from two sequences of lessons, one Polish and one Russian, focused on the use of the number line. The results show that the FoNS framework is cross-culturally sensitive, simply operationalised and analytically powerful.