This is an open access article distributed under the terms of the Creative Commons Attribution License (Although few would disagree that acquiring subject-related expertise is an important goal of mathematics teaching, it is so rooted in cultural expectations that some have argued that “there can be no fixed conception of what constitutes expertise in mathematics teaching” (Brown & Coles, 2011, p. 862). Moreover, further highlighting the problem, few studies of students’ mathematics-related expertise have offered anything other than a token acknowledgement of the cultural context in which they were undertaken, typically assuming that expertise requires no definition (see Jurow et al.’s, 2008 study of US eighth grade and Eronen & Kärnä’s, 2018 study of Finnish ninth grade students). Even international tests of mathematics achievement operationalise mathematics in ways that privilege different forms of expertise, whether it is the mathematics of a hypothetical curriculum assessed by TIMSS or the everyday real-world applications evaluated by PISA. In this paper, acknowledging such definitional inconsistency, we aim to contribute to the literature on student expertise in multidigit arithmetic by drawing on the context-independent distinction between adaptive expertise and routine expertise (Hatano & Inagaki, 1984) to examine Danish students’ arithmetical solution strategies.
Hatano and Inagaki’s (1984) context-independent conceptualisation presented routine experts as having acquired “a body of procedural knowledge… along with the skills necessary for applying that knowledge” (p. 28). Such people, when operating in familiar environments, and despite limited understanding of the principles that underpin their skills, are able to function well with just procedural knowledge. Alternatively, adaptive experts not only have well-developed procedural skills, but “understand the meaning of the skills” (ibid, p. 28) and are able, as a consequence, to manage with confidence “unfamiliar situations and to invent new strategies” (ibid, p. 30). However, highlighting the role of culture in the development of an individual’s adaptive expertise, the extent to which members of particular communities become adaptive experts is dependent on whether routine expertise is the privileged form of knowledge and people are discouraged from investigating variations in those skills (ibid).
Adaptive Expertise and Mathematics Education
The development and manifestation of adaptive expertise in the context of mathematics learning has been increasingly researched (Hickendorff, 2018; McMullen et al., 2016; Nemeth et al., 2019; Verschaffel et al., 2009), particularly from the perspective of strategy adaptivity, where evidence suggests that following the introduction of standard algorithms even students with access to multiple strategies typically resort to just one (Sievert et al., 2019). In short, teaching that promotes algorithms tends also to encourage routine experts (Selter, 2009), who, with limited understanding, can solve school exercises efficiently (Baroody, 2003; Hatano & Oura, 2003). Such students acquire a “static, sparsely connected knowledge that can only be applied to typical tasks”, often with little transferability to new situations (McMullen et al., 2020, p. 1). By way of contrast, adaptive experts are not only able to integrate both procedural knowledge and conceptual knowledge (Hatano & Oura, 2003) but do so in ways that allow them to go beyond algorithms and manage unfamiliar situations (Hatano & Inagaki, 1984). By drawing on such connections, adaptive experts' understanding of each step of a chosen strategy or procedure, may offer alternative steps within that strategy or procedure, or even invent new ones (Hatano & Inagaki, 1984).
From the particular perspective of arithmetic, the focus of this paper, it is known that adaptive experts are able to identify optimal strategies for a given problem (McMullen et al., 2016), and even young children, when given the opportunity, are able to switch strategies thoughtfully (Lemaire & Callies, 2009; Torbeyns et al., 2018). Developing a strategy repertoire for any given problem, alongside the competence to select appropriately from that repertoire, underpins not only mathematical reasoning and computational accuracy (Sievert et al., 2019; Threlfall, 2009; Torbeyns et al., 2017) but also the goals of reform-oriented curricula. In this respect, the teaching of arithmetic in Denmark, the site of the research reported below, is expected to focus on number understanding and number-based strategies rather than standard algorithms (Ministry of Children and Education, 2019). However, despite such goals, when presented with tasks designed to elicit shortcut strategies (that is, strategies that involve the flexible adaptation of numbers and operations, as with the compensation-related transformation of 199 + 323 to 200 + 323 – 1 or the indirect addition transformation of 703 – 686 to 14 + 3, whereas a non-shortcut strategy or routine strategy could be an algorithmic approach, see for example Linsen et al., 2015), students rarely apply such strategies (Hickendorff, 2018, 2022; Jóelsdóttir & Andrews, 2024b). This lack of shortcut strategy use highlights the extent to which instructional practices may influence which strategies are privileged in mathematics classrooms, particularly as “adaptive expertise involves habits of mind, attitudes, and ways of thinking and organising one’s knowledge that are different from routine expertise and that take time to develop” (Verschaffel et al., 2007, pp. 32-33).
In recent years, research into the development of students’ adaptive expertise has been augmented by an emphasis on flexibility. Indeed, Nemeth et al. (2019) write of a “consensus among mathematics researchers and educators that the abilities to use various strategies for solving a problem (flexibility) as well as to use efficient strategies (adaptivity) are important mathematical competencies students should gain” (p. 2). However, the distinction between the two constructs has not always been transparent and may be due to Hatano and Inagaki’s (1984) earlier observation that adaptive experts possess “flexible and adaptive conceptual knowledge” (p. 30). In this paper, we make a distinction between the concepts of flexibility and adaptivity; with flexibility referring to the existence of a strategy repertoire for solving a particular problem and the ability to switch between those strategies and adaptivity referring to the ability to select the optimal strategy from that repertoire (Hickendorff, 2022; Verschaffel et al., 2009). When viewed in this way, adaptive experts are both flexible and adaptive, as knowing and being able to switch between various strategies are preconditions for adaptivity (Verschaffel et al., 2007).
Adaptive Expertise and Individual Differences
It is generally agreed that students’ arithmetic-related strategy choice draws on their prior knowledge of arithmetic (Schulz, 2018; Threlfall, 2009), including both their conceptual and procedural knowledge (Schneider et al., 2011; Star et al., 2015), and the learning opportunities they receive (Sievert et al., 2019). Furthermore, as highlighted in a recent review paper, and of particular relevance to this study, individual and familial variables like, for example, student sex and grade level, alongside socio-economic status (hereafter SES) and other family characteristics have been found to influence strategy flexibility and adaptivity (Verschaffel, 2024).
Sex differences in strategy choice for single-digit arithmetic have been found from the early school years (Carr & Jessup, 1997; Sunde et al., 2020), with girls showing a preference for counting strategies and boys a preference for retrieval strategies. Such differences, in respect of multi-digit arithmetic, resonate with Dutch studies highlighting boys’ preferences for mental or short-cut methods and girls’ preferences for written methods (Hickendorff, 2018). Elsewhere, at least in the context of eighth and ninth graders’ learning of algebra, girls have been found to be more adaptive and flexible than boys (Star et al., 2015). However, research on the influence of sex on strategy choice remains limited.
From the perspective of maturation, instruction, particularly textbooks’ timing of the introduction of standard algorithms, may limit not only students’ strategy repertoires but their preferred strategies (Hickendorff et al., 2019; Sievert et al., 2019; Torbeyns et al., 2017). Indeed, notwithstanding evidence that students’ adaptive strategy choices tend to become more likely with age (Lemaire & Callies, 2009; Torbeyns et al., 2018), in Denmark, the context of this study, both strategy adaptivity and flexibility seem to peak in Grade 6, with students in Grade 6 showing higher levels of both adaptivity and flexibility than those in Grades 3 and 8 (Jóelsdóttir & Andrews, 2024b). Also, children born early in the school year typically achieve higher than those born later and, while some scholars have argued that these differences disappear with age (Milling-Kinard & Reinherz, 1986), the consensus seems to be that they persist (Bell & Daniels, 1990). For example, a recent reanalysis of PISA data found, across mathematics, science and literacy, that at age 15, children born at the start of the school year benefit both cognitively and affectively in comparison with their younger peers (Givord, 2020). Indeed, Smith (2009) has estimated, with respect to Canadian children, that the test scores on core knowledge domains like mathematics and literacy of “older children exceed those of younger children by 0.259 – 0.400 standard deviations (σ) in Grade 4, declining to between 0.104 – 0.242 by Grade 10. Such results complement a Chilean study that found delaying the start of school for the youngest children increases the achievement scores of fourth and eighth grade students by more than 0.3 standard deviations (McEwan & Shapiro, 2008). However, despite such differences, little is known about the impact of birth month on students’ multidigit arithmetic strategy choices. In conclusion, while maturity and a child’s age relative to his or her cohort are known to impact achievement, little is known about their influence on arithmetic-related strategy choices.
Evidence with respect to the influence of siblings on educational achievement seems ambiguous. For example, a recent Chinese study found, at both grades four and eight, that children raised with no siblings outperformed those with siblings on mathematics, Chinese, science and English (Li et al., 2021). Elsewhere, a study undertaken in Florida, found, with respect to the influence of older siblings, positive influences in low-income households but negative in high income households (Karbownik & Özek, 2023). However, acknowledging our interest in mathematics, Gabay-Egozi et al. (2022) found that women were more likely to study STEM subjects if they had been raised in homes with either small sibling groups, male sibling group dominance, or if they had an older sister with high mathematics achievement. Also, acknowledging the influence of socio-economic status (SES) on mathematics achievement generally (Kalaycioğlu, 2015), it would be reasonable to assume that students from high SES backgrounds would exhibit higher levels of adaptive behaviour than their peers from low SES backgrounds, an assumption that has not, to the best of our knowledge, been investigated. Finally, in addition to highlighting achievement differences located in students’ sex and familial SES, international studies like PISA remind us of the achievement gap between students with an immigrant background and their non-immigrant peers, even when controlling for SES (OECD, 2023). Overall, while the above highlights the influence of a range of background characteristics, both individual and familial, on general mathematics achievement, little is known about their influence on the different forms of expertise and students’ strategy choices.
The Current Study
Acknowledging such matters, the research reported in this paper examines the influence of a range of individual and familial characteristics on the development of routine expertise and adaptive expertise with respect to Danish Grade 3, 6 and 8 students’ approaches to multidigit arithmetic. The study is guided by the question:
How, in respect of multidigit arithmetic, do adaptive experts and routine experts differ with respect to achievement and both individual and family characteristics?
Specifically, the study examines how adaptive and routine experts differ with respect to age, ethnicity, sex, older sibling, and their parents’ socio-economic status (SES). In addition, the study examines the relationships between mathematics achievement and reading, as measured by Danish national tests, and students’ arithmetic-related strategy adaptivity and flexibility. We use reading as a proxy for general achievement, to find if the differences between adaptive and routine experts are similar for achievement in other school subjects, compared to mathematics.
Operationalising Expertise
In the following, an expert is defined as a student who is able to calculate accurately across different operations. Experts defined in this way are then distinguished according to whether they are routine experts – students who never use shortcut strategies – or adaptive experts – students who regularly use shortcut strategies. Such an approach should provide new insights into how various individual characteristics impact the two forms of arithmetic-related expertise. It will also extend our understanding of routine and adaptive experts by going beyond an examination of the influence of family background and SES to consider the impact of student flexibility in arithmetic strategy choice and both mathematics and reading achievement.
Method
Participants
Public schools in five demographically different municipalities in Jutland, the central region of Denmark, were invited to participate. This yielded 20 schools, large and small, urban and rural, and a total of 2298 pupils in 37 Grade 3, 39 Grade 6 and 45 Grade 8 classes. These grades were chosen as national tests for mathematics, undertaken in the same grades, and reading, undertaken in Grades 6 and 8, provide the achievement measures used below. In the Danish school system students start in Grade 0 in August of the year in which they turn 6 years, with the consequence that children born in the first quarter (Q1) are almost a year older than their peers born in the fourth. This paper is based on data from the 2,216 pupils (96.4% of the tested sample) for whom family background and other data were available (Table 1). Prior to children’s participation, parents/carers (hereafter parents) received two letters. The first, through their child’s school’s communication platform, outlined the project and confirmed parents’ right to withdraw their child for no reason and with no consequences. The second, by means of an email, informed parents about, for example, how personal data would be managed within the framework of the EU’s general data protection regulation (GDPR). With the exception of three students whose parent denied permission, all students present on the day of testing participated.
Table 1
Participant Totals, Median Age and Distribution on Grade and Sex
| Grade level | Classes (n) | Students (n) | Median age | Girls (n) | Girls (%) |
|---|---|---|---|---|---|
| Grade 3 | 37 | 722 | 9 | 353 | 49 |
| Grade 6 | 39 | 687 | 12 | 358 | 52 |
| Grade 8 | 45 | 786 | 14 | 380 | 48 |
| Total | 121 | 2216 | 1091 | 50 |
Operationalising Routine and Adaptive Expertise
All students undertook a Tri-phase Flexibility Assessment (hereafter TriFA) of their multidigit arithmetical competence. Originally designed and validated for evaluating students’ equations-related strategy flexibility (Xu et al., 2017), the TriFA has been successfully adapted for evaluating students’ multidigit arithmetic strategy choices (Jóelsdóttir & Andrews, 2024a). During Phase 1 students solved each task by whatever strategy they wish; during Phase 2 they solved the same tasks again but with as many different strategies as they can; during Phase 3 they identified the optimal strategy from those they offered. All responses were given in writing. As the aim was to assess strategy repertoire, solution speed was not assessed.
All tasks in this adaptation, which was completed in an hour of class time, were designed to elicit shortcut strategies. Details of each task can be found in Jóelsdóttir and Andrews (2024a). Briefly, acknowledging students’ likely experiences of arithmetic, Grade 3 students undertook four addition and four subtraction tasks. Those in Grade 6 undertook three additions, three subtractions and three multiplications, while those in Grade 8 undertook four each of addition, subtraction and multiplication. Where possible, the same tasks were used across the grades, with each designed, as indicated, to elicit different types of shortcut strategies (see Jóelsdóttir & Andrews, 2024a, 2024b). Importantly, division was excluded as the low levels of division-related competence yielded by the Danish national tests indicated that an analysis of division-related strategy selection would be unproductive.
All written responses were coded independently by at least two raters, the first author and research assistants trained to code the various multidigit arithmetic tasks. Students’ solutions to Phase 1 tasks were coded first for accuracy (correct, incorrect or no solution) and then for strategy (shortcut or not shortcut). In this context and irrespective of grade, an expert was defined as a student who made no more than one error across all test items and an adaptive expert was an expert who offered a shortcut strategy on the task in question. With respect to Phase 2, the total number of solutions was recorded and each additional solution was coded for strategy type as with Phase 1. Next, the solutions identified as optimal during Phase 3 were coded as either shortcut or not. Any disagreements among coders were discussed and a third colleague invoked when necessary.
As our interest lies in understanding the characteristics and impact of the different forms of expertise, where an expert was any students who submitted no more than one incorrect solution. The figures of Table 2 show the number of tasks experts should solve by means of shortcut strategies to be classed as either routine or adaptive. Here, experts who failed to solve any task by means of shortcut strategies were classified as routine experts. Second, due to the unexpectedly low number of students implementing shortcut strategies, the bar for defining adaptive experts was set lower than originally intended. Consequently, an adaptive expert was defined as one who uses shortcut strategies regularly, that is, on at least three tasks in Grades 3 and 6, and four in Grade 8. This pragmatic definition ensured that shortcut use was unlikely to be accidental. For example, any sixth grade student offering shortcuts on three different tasks would either be offering shortcut solutions to all three tasks addressing the same operation, each of which is designed to elicit a different shortcut strategy, or to tasks from different operations, indicating an ability to use shortcut strategies across the operations (Jóelsdóttir & Andrews, 2024a, 2024b). Experts using shortcuts on fewer tasks cannot with confidence be said to be adaptive and were categorised as mixed experts. Finally, Figure 1 shows the proportions of students at each grade level classified according to the above protocol.
Table 2
Defining Characteristics of Experts, Routine Experts, Mixed Experts and Adaptive Experts
| Group | |||
|---|---|---|---|
| Tasks solved accurately | |||
| Grade (Total number of tasks) | Grade 3 (8) | Grade 6 (9) | Grade 8 (12) |
| To be an expert (and then) | 7-8 | 8-9 | 11-12 |
| Tasks solved adaptively | |||
| Routine experts | 0 | 0 | 0 |
| Mixed experts | 1-2 | 1-2 | 1-3 |
| Adaptive Experts | 3-8 | 3-9 | 4-12 |
Figure 1
Distribution of Students by Levels of Expertise and Grade
Student Background Characteristics, Family Background and SES
To determine the influence of students’ sex (female or male as assigned by birth), ethnicity, family background and SES, register data were obtained from Statistics Denmark. Student ethnicity, following Statistic Denmark’s definition, was defined as Western or non-Western, an older sibling student was defined as having at least one older sibling in the student home, and a Q1 student was born in the first three month of the calendar year. Finally, acknowledging its widespread use elsewhere (Shavers, 2007), parents’ educational level was used as a proxy for SES. In this instance, a child where at least one parent had a university level qualification was scored 1 otherwise 0.
Student Achievement and Flexibility
Measures of mathematics achievement (Grades 3, 6 and 8) and reading (Grades 6 and 8) were obtained from the Danish National tests, which are mandatory in mathematics for students in Grades 3, 6 and 8 and for reading in Grades 2, 4, 6 and 8 (Ministry of Children and Education, 2019). Finally, based on their responses to the TriFA, three additional scores were calculated for each student. First, flexibility referred to the percentage of TriFA tasks solved by means of more than one strategy. Second, flexibility with shortcut referred to the percentage of TriFA tasks solved flexibly and by at least one shortcut strategy during either Phase 1 or Phase 2. Third, shortcut optimal referred to the percentage of TriFA tasks for which students identified a shortcut strategy as optimal during Phase 3. We did not distinguish between correct or incorrect answers.
Statistical Analysis
The analysis is in two parts. First, the characteristics of the different expert groups in respect of the different background variables were determined. That is, for each student group, pooled across Grade 3, 6 and 8, and classified as (i) either experts or non-experts and (ii) (within the expert group) as either routine, mixed or adaptive experts, we examined, with 95% confidence levels, the proportions that were (a) girls (versus boys), (b) western (versus non-western), (c) in families with older siblings, (d) born in Q1 or (e) in families with at least one parent with a higher education qualification. Next, t-scores derived from differences in these proportions relative to their SES enabled us to test (two-tailed hypothesis) whether (i) experts differed from non-experts and (ii) adaptive experts differed from routine experts with respect to these background variables. In addition, drawing on data from the mandatory Danish national tests, we examined the mean and spread of the different student groups’ achievement scores in mathematics (Grades 3, 6 and 8) and reading (Grade 6 and 8). All such scores, based on a national sample, are standardised with M = 0 and SD = 1. Finally, we quantified and tested mean scores for (a) flexibility, (b) flexibility with shortcut, and (c) shortcut optimal.
Second, by means of multiple logistic regressions that account for possible between-class variation as random effects, we quantified the total and partial effects of the above variables as predictors with regard to classifying (i) experts versus non-experts and (within the expert group) as (ii) routine versus adaptive experts. Seven fixed-effect predictor variables were entered in the analyses. Five of these were the same dichotomous variables examined in part one. In addition, two variables were entered as covariates: achievement score in national test in mathematics and flexibility, as determined by the percentage of tasks solved by means of more than one strategy.
National test scores in reading were not included in this initial analysis, as no test scores were available for Grade 3 students. Also, the variables labelled flexibility with shortcut strategies and shortcut optimal were not included as the first is a subgroup of flexibility and the second is partly included in the definition of adaptivity. Prior to the analysis, we created a correlation matrix for all predictor variables to reveal possible collinearity. A priori, we defined a numerical value of Pearson’s r of ≥ 0.7 (where 50% of the variation in one predictor variable would be explained by variation in another predictor variable) as a criterion for only including one of any inter-correlated predictors. As all |r| values between eligible predictors were ≤ 0.39 (see Appendix A in the Supplementary Materials), none of the aforementioned predictors were excluded.
For each set of analyses, (i) experts versus non-experts and (ii) adaptive experts versus routine experts, we first established univariate models for each of the aforementioned predictors (yielding total effects of each predictor). In a second round, we established a full model, comprising all seven fixed-effect predictors, yielding partial effects of each predictor alongside the effect of other variables. In a third round, we used a forward stepwise procedure to establish a model consisting of statistically significant terms only (least complex adequate model). As model selection procedure, we used forward stepwise selection with students clustered in classes and with significance level for addition to the model p < .05 (default option in Stata). For all models comprising multiple fixed-effect predictors (Round 2 and 3), as post hoc operations we tested for possible two-way interactions between statistically significant main effects.
We first analysed data for Grade 3, 6 and 8 separately. Then, to reveal any variation in fixed effect size between grades, we analysed data for all three grades (3, 6 and 8) in a single model, where we accounted for grade as main effect. We also included interaction terms between grade and the other predictors where such interactions were statistically significant. For an interaction term between grade and a given fixed effect variable (X) to be included, the variable X should be statistically significant (p < .05) as main effect in the first place, as should the interaction term grade-by-variable X, as well as the combined statistical significance of the main effect of variable X and its interaction term with grade. In so doing, we tested the extent to which effect sizes of significant terms in the full model (β2) deviated from the initial effect size (β1), using a z-score calculated as z = (β1 – β2)/(SE12 + SE22)0.5. Unless otherwise stated, all analyses were carried out in Stata/SE 15.1. Finally, for simplicity, we present only the results for the combined sample; results concerning analyses at the grade level, which are not explicitly focused on our research questions, can be seen in Appendix B (Tables B1 to B6) in the Supplementary Materials.
Results
Proportion of Non-Expert and Expert Groups Divided on Grade and Sex
As shown in Figure 1, the proportion of students categorized as experts increased from 20% in Grade 3, to 37% in Grade 6 and 52% in Grade 8 (Pearson’s χ22 = 34.4, p < .0001). In addition, Figure 2 shows that Grade 3 boys were classified as experts significantly more often than Grade 3 girls (two-tailed t-test: t720 = -2,80, p = .005).
Figure 2
The Percentage of Students Divided on Sex and Grade Categorized as Experts (Subdivided Into Routine, Mixed and Adaptive Experts)
Note. p-values from a two-tailed t-test indicate the probability that the proportions of students categorized as (i) experts (ii) routine experts (iii) adaptive experts are similar for boys and girls within a given grade.
However, no such differences were found for Grade 6 (p = .59) or Grade 8 (p = .64). Figure 2 also shows that while the proportions of students classified as routine experts were similar for boys and girls in Grade 3 (13% and 12% respectively), the proportion of routine expert boys was significantly lower than girls in both Grade 6 (10% and 19% respectively) and Grade 8 (23% and 34% respectively). Hence, while a multilevel logistic regression found the proportion of girls categorised as routine experts to increase across the three grades (Grade 3 vs. Grade 6: β = 0.53, SE = 0.21, p = .012; Grade 6 vs. Grade 8: β = 0.80, SE = 0.18, p < .0001), the proportion of boys categorized as routine experts was constant between Grade 3 and 6 (β = -0.32, SE = 0.24, p = .17), but doubled between Grade 6 and 8 (β = 1.04, SE = 0.22, p < .0001). By way of contrast, Figure 2 also shows that the proportion of boys identified as adaptive experts was, irrespective of grade level, around three times that of girls: whether in Grade 3 (8% and 2% respectively), Grade 6 (16% and 6% respectively) or Grade 8 (14% and 5% respectively). Despite such differences, the proportions of adaptive experts, irrespective of sex, grew significantly from Grade 3 to Grade 6 – for boys the proportion doubled (from 8% to 16%) (β = 0.75, SE = 0.24, p = .002) and for girls it tripled (from 2 to 6%) (β = 1.15, SE = 0.45, p = .010) but remained stable from Grade 6 to Grade 8 (boys p = .29, girls p = .72).
Characteristics of Non-Expert and Expert Groups
Table 3 shows the percentage, by individual characteristics, for non-experts compared with experts, and routine experts compared with adaptive experts. Routine experts differed from adaptive experts in the use of standard algorithms, with means of 91% and 21% respectively of tasks solved in this manner. While the percentage of boys and girls were comparable for both the non-expert and the expert groups, differences within the expert group were evident; more than half of all routine experts were girls (59%) while only one quarter of adaptive experts were girls. Significantly more experts than non-experts had western backgrounds (91% vs. 86%), while the proportion of students with a western background was equal for routine experts and adaptive expert (Table 3). Neither being born in the first quarter of the year nor having an older sibling had any influence on group proportions (Table 3). However, SES, as measured by at least one parent having a university qualification, was significantly higher for experts (68%) than for non-experts (53%) and, albeit to a lesser extent, for adaptive experts (75%) than for routine experts (65%).
Table 3
Percentage of Students, With Standard Error in Parentheses, Classified as Either Non-Expert or Expert, and (Within the Expert Group) as Either Routine, Mixed or Adaptive by Individual Characteristics and Family Background
| Variable | Experts vs. non-experts | Routine experts vs. adaptive experts | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Non-expert N = 1,398 | Expert N = 818 | df | t | p | Routine Expert N = 425 | Adaptive Expert N = 188 | df | t | p | |
| Girls | 51% (1) | 49% (2) | 2193 | -0.70 | .485 | 59% (2) | 25% (3) | 604 | 8.24 | < .0001 |
| Western | 86% (1) | 91% (1) | 2194 | -3.02 | .003 | 90% (1) | 93% (2) | 604 | -1.24 | .215 |
| Older siblings | 59% (1) | 57% (2) | 2214 | 1.28 | .202 | 57% (2) | 60% (4) | 611 | -0.66 | .508 |
| Born Q1 | 23% (1) | 27% (2) | 2214 | -1.93 | .054 | 26% (2) | 27% (3) | 611 | -0.06 | .950 |
| High SES | 53% (1) | 68% (2) | 2214 | -7.35 | < .0001 | 65% (2) | 75% (3) | 611 | -2.36 | .019 |
Note. The p-values, derived from t-tests, indicate the statistical significance of pairwise differences.
The various charts of Figure 3 show, for the different student groups, the standardised results of national tests for mathematics, based on a mean of zero and a standard deviation of one. With means of, respectively, 0.70 and −0.16, students classified as experts, achieved 0.86 SD-units higher than non-experts (t2147 = −21.2, p < .0001). Within the expert group, adaptive experts scored 0.49 points higher than routine experts (t587 = −6.75, p < .0001). Also, further highlighting the potential impact of adaptivity, students in the mixed expert group who, by definition showed adaptivity on at least one task, scored higher than the routine experts but lower than the adaptive experts. With respect to reading, assessed only for Grades 6 and 8, the charts of Figure 3 show that students in the expert group scored 0.57 SD-units higher than non-experts (t1410 = −11.24, p < .0001). Also, within the expert group, adaptive experts scored 0.19 points higher than routine expert (t452 = −2.04, p = .04).
Figure 3
National Test Results for Mathematics and Reading for the Different Student Groups
Note. The results are standardised, based on a mean of zero and a standard deviation of one. For legal reasons concerning the remote possibility of identifying individual students, outliers are not shown.
Figure 4 shows the percentages of tasks solved flexibly by the different groups. Experts were significantly more flexible than non-experts (t2214 = −18.10, p < .0001), solving on average 46% of all tasks, compared with 23%, by means of more than one strategy. Much of this difference was a consequence of the proportion of students unable to solve any task flexibly – 16% of the expert group were unable to solve any task flexibly compared with 42% for the non-expert group. Within the expert group, adaptive experts solved a significantly higher proportion of tasks flexibly (54%) than did routine experts (42%) (t611 = −4.39, p < .0001).
Figure 4
The Percentages of Tasks Solved Flexibly by Each Expert Group, Task Solved Flexibly and by at Least One Shortcut Strategy by Each Expert Groups and Solved by Means of a Shortcut Strategy Identified as Optimal
Note. For legal reasons, outliers are not shown.
Figure 4 also shows, for each student group, the percentages of tasks solved flexibly with at least one shortcut strategy. Experts solved, on average, 24% of all tasks flexibly with at least one shortcut strategy, while the comparable figure for non-experts was 8% (t2214 = −16.96, p < .0001). Within the expert group, the figures were 43% and 15% for adaptive and routine experts, respectively (t611 = −12.85, p < .0001). Finally, Figure 4 shows the percentages of tasks, for each student group, for which a shortcut strategy was identified as optimal. On average, 17% of tasks fell into such a category for the experts, compared to 8% for the non-experts (t2214 = 10.03, p < .0001). Within the expert group, adaptive experts (49%) selected shortcuts as optimal seventeen times as often as routine experts (3%) (t611 = 30.20, p < .0001).
Multivariate Analysis
Differences Between Experts and Non-Experts
Table 4 shows the results of three multilevel logistic regression models developed to determine the predictive impact of the different variables under different conditions. The simple model, which adjusted for grade and class nested within school, shows that the probability that a student was classified as an expert was significantly higher for students with western background than for students with a non-western background, higher for students with a parent with higher education (SES) than for students without a parent with higher education. Being an expert also predicted higher achievement in the national test in mathematics and a higher proportion of tasks solved flexibly.
Table 4
Fixed Effects Coefficients of Multilevel Logistic Regression Models (Accounting for Class Nested Within School as Random Factors), Predicting the Conditional Probability That a Student Is Categorized as Expert as Opposed to Non-Expert
| Predictors | 1: Grade + single variablesa | 2: Full model (all variables) | 3: Significant effects only | ||||||
|---|---|---|---|---|---|---|---|---|---|
| β (SE) | z | p | β (SE) | z | p | β (SE) | z | p | |
| Intercept | -1.590 (0.185) | -2.520 (0.308) | -2.629 (0.221) | ||||||
| Grade 6 (vs. 3) | 0.963 (0.178) | 5.42 | < .0001 | 1.356 (0.263) | 5.16 | < .0001 | 1.327 (0.261) | 5.09 | < .0001 |
| Grade 8 (vs. 3) | 1.575 (0.179) | 8.79 | < .0001 | 2.111 (0.257) | 8.20 | < .0001 | 2.095 (0.255) | 8.21 | < .0001 |
| Type-3 test: Grade (3, 6 or 8); | = 77.48, p < .0001 | = 67.82, p < .0001 | = 67.96, p < .0001 | ||||||
| Girl (yes) | -0.059 (0.098) | -0.60 | .546 | -0.073 (0.113) | -0.65 | .517 | not included | ||
| Western (yes) | 0.506 (0.180) | 2.81 | .005 | -0.165 (0.214) | -0.77 | .441 | not included | ||
| Older sib (yes) | -0.186 (0.099 | -1.87 | .062 | -0.125 (0.114) | -1.10 | .273 | not included | ||
| Born in Q1 (yes) | 0.209 (0.114) | 1.83 | .067 | 0.291 (0.132) | 2.21 | .027 | not included | ||
| SES (yes) | 0.718 (0.109) | 6.57 | < .0001 | 0.111 (0.131) | 0.84 | .399 | not included | ||
| NT maths score | 1.199 (0.072) | 16.70 | < .0001 | 1.076 (0.078) | 13.71 | < .0001 | 1.068 (0.075) | 14.27 | < .0001 |
| Flexibilityb | 2.378 (0.180) | 13.23 | < .0001 | [1.418 (0.202)] | [7.01] | [< .0001] | [1.427 (0.228)] | [6.28] | [< .0001] |
| Test: Grade-by-flexibility | = 4.68, p = .097 | = 10.20, p = .006 | = 9.59, p = .008 | ||||||
| Flexibility in Grade 3 | 2.910 (0.348) | 8.37 | < .0001 | 2.421 (0.377) | 6.42 | < .0001 | 2.398 (0.374) | 6.42 | < .0001 |
| Flexibility in Grade 6 | 2.295 (0.304) | 7.54 | < .0001 | 1.018 (0.335) | 3.04 | .002 | 1.074 (0.332) | 3.23 | .001 |
| Flexibility in Grade 8 | 2.078 (0.282) | 7.36 | < .0001 | 1.006 (0.317) | 3.18 | .001 | 1.022 (0.312) | 3.27 | .001 |
| Significance of total model: | Wald = 350.99, p < .0001 | Wald = 349.19, p < .0001 | |||||||
aEffects shown for single variables in models with grade as mandatory fixed effect. bIn the “Full model” and “Significant effects only” model, ”[..]” indicates main effects of Flexibility if included in the model without interaction terms between Flexibility and Grade.
In the full model, the effects of the national mathematics test score and flexibility remained highly significant, while the significant effects of SES and ethnic background disappeared. Thus, western background and SES were excluded from the significant effects only model. The effect of being born in Q1 was borderline positive significant if entered in isolation and marginally significant in the full model, but not included in the significant effects only model. Furthermore, a statistically significant interaction term between grade and flexibility, suggested that the positive effect of flexibility was stronger in Grade 3 compared to Grade 6 and 8.
From Table 4 it is also clear that the effects of the national mathematics test score were of similar magnitude in the simple model and the full model (β = 1.20 vs. 1.08; z = 1.16, p = .24). The effect of flexibility was not significantly reduced from the simple to the full model in Grade 3 (β = 2.91 vs. 2.42; z = 0.95, p = .34), whereas this was the case for students in Grade 6 (β = 2.30 vs. 1.02; z = 2.64, p = .009) and Grade 8 (β = 2.08 vs. 1.01; z = 2.53, p = .012).
Differences Between Adaptive Experts and Routine Experts
The univariate as well as the complex models (adjusted for variation between grade and random variation between classes) all showed that the probability that an expert was adaptive rather than routine was substantially higher for boys than for girls (βgirl = -1.766, odds ratio = exp[βgirl] = 1: 4.8) and correlated positively with their national mathematics test and flexibility scores (Table 5). The effects of all three predictors appeared to be additive, as their partial effect sizes did not deviate significantly from the univariate effect sizes (Table 5; sex: z = 0.34, p = 0.73; NT math; z = 0.52, p = .60; Flexibility: z = 0.42, p = .75).
Table 5
Fixed Effect Coefficients of Multilevel Logistic Regression Models (Accounting for Class Nested Within School as Random Factors), Predicting the Conditional Likelihood That an Expert Student Is Categorized as Adaptive Expert as Opposed to Routine Expert
| Predictors | 1: Grade + single variablesa | 2: Full model (all variables) | 3: Significant effects only | ||||||
|---|---|---|---|---|---|---|---|---|---|
| β (SE) | z | p | β (SE) | z | p | β (SE) | z | p | |
| Intercept | -1.092 (0.422) | -2.174 (0.713) | -1.393 (0.252) | ||||||
| Grade 6 (vs. 3) | 0.792 (0.417) | 1.90 | .057 | 1.058 (0.480) | 2.20 | .028 | 1.049 (0.471) | 2.23 | .026 |
| Grade 8 (vs. 3) | 0.331 (0.407) | -0.81 | .417 | 0.023 (0.469) | 0.05 | .961 | 0.040 (0.457) | 0.09 | .931 |
| Type-3 test: Grade (3, 6 or 8) | = 10.02, p = .0067 | = 7.68, p = .021 | = 7.68 p = .022 | ||||||
| Girl (yes) | -1.640 (0.249) | -6.60 | < .0001 | -1.766 (0.276) | -6.39 | < .0001 | -1.537 (0.236) | -6.51 | < .0001 |
| Western (yes) | 0.577 (0.436) | 1.32 | .186 | 0.416 (0.526) | 0.79 | .428 | not included | ||
| Older sib (yes) | 0.062 (0.223) | 0.28 | .780 | 0.173 (0.259) | 0.67 | .506 | not included | ||
| Born in Q1 (yes) | 0.012 (0.245) | 0.05 | .961 | 0.306 (0.282) | 1.08 | .279 | not included | ||
| SES (yes) | 0.401 (0.261) | 1.53 | .125 | -0.023 (0.321) | -0.07 | .943 | not included | ||
| NT math score | 0.911 (0.156) | 5.84 | < .0001 | 0.788 (0.174) | 4.52 | < .0001 | 0.573 (0.125) | 4.57 | < .0001 |
| Flexibility | 1.045 (0.358) | 2.92 | .004 | 0.869 (0.422) | 2.06 | .039 | 1.071 (0.392) | 2.73 | .006 |
| Significance of total model: | Wald = 70.25, p < .0001 | Wald = 69.74, p < .0001 | |||||||
aEffects shown for single variables in models with grade as mandatory fixed effect.
Discussion
In this study, our goals were to determine, in relation to various background variables, the characteristics of student expertise in respect of multidigit arithmetic. In this context, having defined an expert as one who made no more than one error on a test of arithmetical competence, our principal focus was on the distinguishing characteristics of routine experts and adaptive experts, where a routine expert is one who never uses shortcut strategies and an adaptive expert is one who regularly uses shortcut strategies. The results, which we discuss in detail below, make a substantial contribution to the collective knowledge of not only the characteristics of routine and adaptive experts, but also why adaptivity should be a goal of mathematics teaching (Hatano & Oura, 2003). Furthermore, as a basis for the development of instructional practice, the study augments the current limited knowledge about the interplay of individual differences and the different forms of expertise. In the following, we briefly discuss the key differences between non-experts and experts, before discussing the more important differences between routine and adaptive experts.
Non-Experts Versus Experts
Not surprisingly, arithmetic experts achieved significantly higher on national tests of achievement than non-experts, a result that accords with earlier findings that basic arithmetic skills predict general mathematics achievement (e.g. Nunes et al., 2012). Far more interesting is the effect size of the difference, with the experts’ mean score in national test, on a normalised distribution of mean zero and standard deviation one, being 0.86 higher than that of the non-experts. It is also worth noting, that the partial and total effect size of mathematical achievement were of similar magnitude, demonstrating strong proximate effect of this predictor even when entered alongside other predictor variables. It is also worth noting that the national test assesses much more than just arithmetical competence, including evaluations of students’ algebraic, geometric, measurement and statistical competence.
The fact that the strong effect of mathematical achievement was not weakened when entered alongside other predictors in the full model is likely to be a consequence of SES and ethnicity, both highly significant univariate effects, disappearing in the multi-predictor models. Hence, being an expert in arithmetic and succeeding on the national mathematics test were two sides of the same coin. That said, the correlation between mathematical achievement and SES accords with large-scale international studies like TIMSS (Kjeldsen et al., 2020).
The positive effect of flexibility on the probability of being an expert was not straightforward, not least because the complex models indicated a stronger effect size for Grade 3 when compared to Grades 6 and 8, which the simple models did not. This could be related to the relationship between flexibility and conceptual and procedural understanding (Schneider et al., 2011), whereby students who develop strategies based on a knowledge of numbers and number relations become arithmetic experts earlier than those who develop an algorithmic expertise.
Routine and Adaptive Experts
The most pronounced differences between significant predictors of routine and adaptive experts concern (i) arithmetic flexibility, where adaptive experts have higher levels of flexibility than routine experts, (ii) mathematics achievement, with adaptive experts outperforming routine experts, and (iii) sex, with boys dominating the group of adaptive experts and girls dominating the group of routine experts.
Flexibility
Adaptive experts not only scored higher on flexibility but also used shortcut strategies more frequently during the first two TriFA phases than routine experts. For example, acknowledging that routine experts did not offer any shortcut strategies during Phase 1 of the TriFA assessment, when given a second opportunity during Phase 2 to offer a shortcut strategy, only 15% of all tasks solved by routine experts involved shortcut strategies. By way of contrast, across Phases 1 and 2 adaptive experts solved 68% of all tasks by means of shortcut strategies. Such findings indicate, as with other studies (Hickendorff et al., 2019; Verschaffel et al., 2007), that students who have learned an algorithm tend to stick to that as their preferred strategy. Importantly, of the 68% of the tasks solved by means of shortcut strategies by adaptive experts, 72% were identified as optimal during Phase 3. In comparison, of the 15% of the tasks solved by means of shortcut strategies by routine experts, only 20% were identified as optimal during Phase 3. In other words, routine experts rarely recognize a shortcut strategy as optimal, even when they offer a shortcut strategy during Phase 2. That might be explained by individual skills (subject variable), or cultural context and what is appreciated in the classroom context (Verschaffel et al., 2009).
General Achievement (National Test in Mathematics and Reading)
Previous studies have shown that conceptual and procedural skills, along with their interconnections, are necessary prerequisites for the development of strategy flexibility and adaptivity (Baroody, 2003; Gilmore et al., 2017; McMullen et al., 2020; Schneider et al., 2011). Moreover, conceptual knowledge is an important predictor of mathematics achievement generally (Rittle-Johnson et al., 2001). In light of such matters, it is of little surprise to find that adaptive experts are generally high achieving in mathematics. However, two elements of our results are particularly striking. First, highlighting the likely importance of acquiring adaptive expertise, the mean achievement of adaptive experts was found to be almost a whole standard deviation higher than the mean for all students. While earlier research has shown a causality between measures of intelligence and strategy choice (Luwel et al., 2011), our data do not necessarily support such a conclusion, not least because it is not clear if adaptivity implies achievement or achievement implies adaptivity. Second, and even more striking, is a difference between adaptive and routine experts of almost 0.5 standard deviations in general mathematics achievement. Thus, by way of warranting the acquisition of adaptive expertise as a goal of instruction, while routine experts are mathematically high achievers, adaptive experts are exceptionally high achievers. Furthermore, highlighting the importance of domain specific aspects of achievement like number understanding (Shanley et al., 2017) and strategy choice (Linsen et al., 2015), differences in achievement between routine and adaptive experts are much greater in mathematics than in reading. This indicates that while general intelligence plays a partial role in explaining high achievement in mathematics, the type of expertise plays a more substantial role. Thus, while routine experts may draw on a static and sparsely connected procedural knowledge (Baroody, 2003), without the prerequisites of appropriate conceptual knowledge and number understanding, they will not have the basis for adaptivity (Baroody, 2003; Hatano & Oura, 2003; McMullen et al., 2020).
Sex, Siblings and Quarter of Birth
While no sex difference was found between non–experts and experts, there was a large difference between the two groups of experts, with boys being significantly more adaptive than girls. Previous studies (e.g. Hickendorff, 2018) have also found boys to be more adaptive than girls, but not as strongly as found in our study. For instance, the figures in Table 3 of Hickendorff’s (2018) study suggest an approximate odds ratio of 1.5:1, compared with the 4.8:1 indicated above. The underlying reasons for this substantial difference are not clear, although others have suggested a complex combination of biological, psychological and environmental variables (Zhu, 2007). The fact that the effect size of sex did not differ between the univariate and the complex models, suggest that the sex-effect was purely additive and hence not related to any of the other predictors in the analysis, mathematical achievement and flexibility included. Hence, even though male experts were substantially more likely to be adaptive experts than female, this sex-difference in the proportion of experts being adaptive did not predict any overall difference in mathematical achievement score between boys and girls within the expert group. The results indicates that further studies are needed to explain the sex differences. The lack of influence of early births and older siblings may be a consequence of the particular characteristics of Danish public preschools, whereby, before starting school, young children experience, in age-integrated groups, several years of informal non-directive learning opportunities driven by everyday experiences and free-play (Rothuizen & Harbo, 2017). In other words, the Danish preschool, with its explicit emphasis on socialisation rather than instruction, may ameliorate both age-related cognitive differences and the influence of older siblings.
Implications
The results of this study highlight the importance of adaptive expertise, as measured by students’ use of shortcut strategies, in underpinning mathematics achievement (Hatano & Oura, 2003). In the particular context of arithmetic, this implies, we posit, a need to encourage the development of students’ use of number-based strategies, particularly shortcut, as opposed to the privileging of standard algorithms. Our results further show that although flexibility, defined as knowledge of several strategies, is related to high achievement (Hästö et al., 2019), flexibility in itself is an insufficient predictor. Students who approach tasks adaptively, particularly recognising shortcut strategies as optimal, perform significantly better on tests of general achievement than routine experts. Thus, acknowledging that the existence of adaptivity is dependent on the existence of flexibility, it is imperative that educators encourage both flexibility and adaptivity. Finally, it is important to note that the correlational nature of what is effectively a cross-sectional study precludes any indication of causality, highlighting a need for longitudinal studies of strategy development and deployment.
Conclusion
Having distinguished between expert and non-expert solvers of arithmetical tasks, this study has examined the characteristics of two groups of experts: routine experts who never use shortcut strategies and adaptive experts who use them regularly. Although parental education and ethnicity initially distinguished between experts and non-experts, their effects disappeared when controlling for national mathematics test score and flexibility. Family variables did not vary at all between adaptive and non-adaptive experts. Flexibility, defined as having access to multiple strategies for a given task, appears a strong predictor of both expertise and adaptivity. Routine experts achieve significantly higher on national tests of reading and mathematics than non-experts, and adaptive experts achieve significantly higher than routine experts. However, in comparison with adaptive experts, routine experts rarely recognise a shortcut strategy as optimal, and boys are significantly more likely to be adaptive experts than girls. Moreover, while students in Grade 6 were more likely to be adaptive than students in Grade 3, indicative of a natural developmental progression, they were also more likely to be adaptive than students in Grade 8, indicative of a didactical privileging of standard algorithms.