In this paper, the multidigit arithmetic-related strategy adaptivity, strategy flexibility and solution accuracy of Danish compulsory school students is examined. Participants, 749 grade three, 731 grade six and 818 grade eight, were drawn from twenty demographically different schools. Drawing on a tri-phase assessment tool, each student completed a series of tasks designed to elicit shortcut strategies. First, students solved each task by means of their preferred strategy; those using shortcut strategies were construed as adaptive for that task. Second, students solved the same tasks by means of whatever alternative strategies they had available; those offering at least two strategies were construed as flexible for that task. Third, for each task, students were asked to indicate which of their strategies they believed was optimal. Across all grades, students were more flexible than adaptive. Overall, sixth graders exhibited higher levels of flexibility than third graders and marginally lower levels than eighth graders. Sixth graders exhibited higher levels of adaptivity than those in either grade three or grade eight. Students’ accuracy, which improved with maturation, was influenced positively by both adaptivity and flexibility, with flexibility having the greatest influence in grade three and adaptivity in grade six. The findings raise further questions concerning, inter alia, culture’s influence on students’ strategy choices and the interaction of adaptivity, flexibility and maturity on accuracy.

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.

In this paper we present a novel adaptation of a tri-phase assessment tool, originally devised to investigate students’ linear equations-related strategy flexibility, to evaluate Danish grade-six students’ multidigit arithmetic-related strategy adaptivity and flexibility. Participants, 731 students, median age 12 years and drawn from 20 demographically different schools, completed a series of positive integer addition, subtraction and multiplication tasks designed to elicit shortcut strategies. First, students solved each task by means of their preferred strategy, with those using a shortcut strategy being defined as showing adaptivity for that task. Second, students solved the same tasks by means of whatever alternative strategies they could devise; those that offered at least two strategies were defined as showing flexibility for that task. Third, for each task, students were asked to indicate which of their strategies they believed was optimal. Statistical analyses, based on the total number of tasks on which students demonstrated adaptivity or flexibility, found students to be more flexible than adaptive, with 55% of students showing moderate or high levels of flexibility but only 21% for adaptivity. Both strategy adaptivity and flexibility, which differed according to operation, were implicated in students’ solution accuracy, with adaptivity being a stronger predictor than flexibility.

Motivated by a curriculum privileging number-based strategies but national tests highlighting students’ reliance on standard algorithms, this study analyses 2,216 Danish Grade 3, 6 and 8 students’ solutions to various multidigit arithmetic tasks, each designed to elicit shortcut strategies, against background variables including sex, ethnicity and familial socio-economic status (SES), and outcomes including strategy flexibility, and national tests for both mathematics and reading. Students offering multiple solutions to a task were defined as flexible, while arithmetic experts (defined by accuracy) were distinguished by their use of shortcut strategies; routine experts never used them, while adaptive experts used them in at least one third of all tasks. With respect to mathematics achievement, experts scored 0.86 SD-units higher than non-experts, and within the former, adaptive experts scored 0.49 SD-units higher than routine experts. With respect to reading, experts achieved 0.57 SD-units higher than non-experts, while adaptive experts achieved 0.19 SD-units higher than routine experts. Boys were significantly more adaptive and flexible than girls. The proportion of experts increased from Grade 3 to Grade 8, whereas the proportion of adaptive experts increased from Grade 3 to 6 but then remained constant. Familial SES was significantly higher for experts than for non-experts but not for adaptive experts in relation to routine. Neither quarter of birth nor the existence of older siblings influenced any outcomes, although the proportion of experts was higher for children with Western backgrounds than for children with non-western background. The results suggest a relationship between adaptive expertise, strategy flexibility, and achievement.

In this paper, by means of a mixed methods approach to data collection and analysis, we examine the impact of linear-equations-related competence on Swedish upper secondary students’ later mathematics achievement. Quantitative data were 307 student solutions to an equations-related word problem posed at the start of their programme and the same students’ terminal grades to the first compulsory general mathematics course. Qualitative data were the written solutions to the word problem and interviews undertaken with purposively selected students. The results showed that students who solved the problem algebraically were not only confident in their solutions but significantly more successful on the terminal assessment than those who solved it by other means. Moreover, students who approached the word problem algebraically but solved it incorrectly, were no less successful than those who solved it successfully by other means.

Estimation is an essential competence with a developmental role in the learning of various mathematical topics. Yet, as previous studies highlight, this competence is either excluded or ambivalently included in intended curricula around the world. The current study investigates the estimation-related opportunities in the primary curricula of three Eastern Mediterranean countries (Cyprus, Greece, and Turkey). Our analyses are framed by four forms of estimation (computational, measurement, quantity, number line). As with previous studies in other contexts, computational estimation and measurement estimation are extensively addressed in the curricula of Cyprus and Turkey, yet without any meaningful justification for their inclusion. All three curricula fail to recognise the importance of number line estimation and quantity estimation, the two forms with the most significant developmental implications for the later learning of other mathematical concepts and areas of mathematics. Among the three curricula under scrutiny, the Greek is the one with the fewest and most superficial references to estimation. In closing, we discuss the implications of this study and suggestions for future research.

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. 

It is broadly accepted that teachers’ professional identities influence how they teach and what their pupils learn. In this paper, drawing on semi-structured interviews with 22 experienced primary teachers from the Republic of Cyprus, we explore the construction of informants’ professional identities with a specific focus on mathematics teaching. Analyses, undertaken according to the constant comparison method, yielded three broad themes, implicated in differing ways in the construction and manifestation of informants’ identities: prior experiences of mathematics, mathematical competence, and images of the self-as-teacher. Overall, teachers fell into two groups, which analyses led us to construe as either mathematical victors or mathematical victims. Mathematical victors had experienced success as learners of school mathematics, from which pleasure, pride, and confidence in their mathematical knowledge for teaching emerged. Their teaching, which emphasized pupils’ attainment of similar enjoyment and success, focused on abstraction and mathematical reasoning. Mathematical victims had experienced failure as learners of school mathematics, from which anxiety and a restricted mathematical knowledge for teaching emerged. Their teaching, which emphasized positive pupil experiences, focused on affect rather than cognition and an avoidance of “traditional” teaching. However, both groups, despite their confident assertions, appeared unaware of the potential of their actions for creating new victims. The findings, which are discussed in relation to existing literature, confirm the complex nature of mathematics teachers’ identities and highlight, in particular, the need for further research into the formative role of teachers’ prior experiences of mathematics, whether positive or negative.

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 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?