Working memory and domain-specific precursors predicting success in learning written subtraction problems
Introduction
Acquiring written calculation skills is a fundamental goal of the primary school arithmetical curriculum, and also represents a difficult point of transition essential to learning more complex mathematical skills at higher education levels (Dowker, 2003).
The goal of the present study was to analyze the factors that can predict the successful acquisition of a specific written arithmetic algorithm, i.e. multi-digit subtraction with borrowing, that is typically taught in second grade. Learning this algorithm represents not only one of the main goals of the arithmetic classes for this age group, but also one of the crucial difficulties encountered by primary school children (Selter, 2001, Venneri et al., 2003). To date, subtraction has not been thoroughly studied. Most research in the field of arithmetic learning has focused on addition or multiplication. Even less has been done to examine the domain-general and domain-specific competences associated with the ability to solve subtraction problems in primary school (Barrouillet et al., 2008, Lemaire and Calliès, 2009, Robinson, 2001, Torbeyns et al., 2004).
A sizable body of research (see e.g. Fuchs, Geary, Compton, Fuchs, Hamlett, et al., 2010a, Fuchs, Geary, Compton, Fuchs, Hamlett, et al., 2010b) has shown that a number of domain-general competences may predict success in learning at school, particularly in arithmetic. In the present study, we focused on the case of working memory (WM), which has already emerged as an essential aspect of numerical cognition (DeStefano and LeFevre, 2004, LeFevre et al., 2005, Noël et al., 2001, Passolunghi and Lanfranchi, 2012, Raghubar et al., 2010). The concept of WM refers to a set of processes or structures that are intimately associated with many arithmetical processes. In the present research, we distinguished between simple-storage and complex WM span tasks, which are considered as distinct constructs (Kintsch et al., 1999, Miyake et al., 2001). Simple storage tasks only involve storing information (e.g. remembering a series of numbers or letters), while complex WM span tasks entail both storing and processing information (Kail and Hall, 2001, Miyake and Shah, 1999). We also distinguished between verbal and visuospatial WM processes because an increasing amount of research has shown that different WM components are associated with arithmetic at different ages (Krajewski and Schneider, 2009, McKenzie et al., 2003). Visuospatial WM appears to be crucially implicated in the mathematical performance of younger children who are still learning basic arithmetical skills (see also Bull et al., 2008, Holmes et al., 2008, Maybery and Do, 2003, McKenzie et al., 2003, Rasmussen and Bisanz, 2005).
In the present study, we concentrated on the period when children learn multi-digit written subtraction with borrowing. This stage can be clearly identified in the life of a primary school class and it requires explicit teaching (VanLehn, 1990). It is a stage that may be associated with potentially different predictors from those relating to other aspects of arithmetic. In fact, a number of experimental studies have shown that the role of WM components in calculation depends on several factors, such as the type of algorithm required, the presentation format, or the involvement of carrying/borrowing procedures (Ashcraft and Kirk, 2001, DeStefano and LeFevre, 2004, Trbovich and LeFevre, 2003). The verbal WM component seems to be involved in addition and multiplication problems (Seitz & Schumann-Hengsteler, 2002), while the visuospatial one seems to relate to subtractions (Lee & Kang, 2002), though this may be mediated by cultural effects (Imbo & LeFevre, 2010). Other studies found that operations with carrying or borrowing procedures increased the demands on WM, prompting a selective involvement of WM components (Caviola et al., 2012, Imbo et al., 2007, Mammarella et al., 2010).
During their formal education, children gradually learn procedures for using multi-digit algorithms. The mathematics curricula in many countries conventionally distinguish between three main arithmetical approaches: written standard algorithms (procedural knowledge, also called the routine approach), written informal algorithms (making notes or using equations), and mental arithmetic, or the strategic approach (that involves applying strategies drawn from a given individual's strategy repertoire) (Heinze et al., 2009, Selter, 2001). Multi-digit arithmetic is still taught mainly with paper-and-pencil written standard algorithms, however (Selter, Prediger, Nührenbörger, & Hussmann, 2012). A mathematics curriculum for teaching arithmetic that is based on the written standard approach requires that children in grade 1 (typically 6 years old) consolidate their counting skills and start learning the principles of adding and subtracting, while the procedures for solving written additions and subtractions are taught (in that order) in grade 2 (Cornoldi & Lucangeli, 2004).
It is worth noting here that, when children perform calculations, they allocate their cognitive abilities differently, not only according to the type of algorithm required, but also to the problem's complexity (Imbo et al., 2007), and they may use specific strategies (Robinson, 2001, Seyler et al., 2003). Some general assumptions can nonetheless be made on the main processes involved in learning subtraction. For instance, having learnt to deal with addition problems presumably helps children to learn subtractions, partly because the latter are explained in terms of the addition procedure in reverse (Selter et al., 2012, Torbeyns et al., 2009). Solving a multi-digit subtraction problem according to the written standard approach involves splitting the operands into digits that are then manipulated using explicitly prescribed procedural rules. In particular, the standard algorithm taught in the classroom requires that pupils know how to align the operands correctly and then process single columns, working from the unit (on the right) to the tens (on the left), and applying special rules to borrow ten when the bottom digit is larger than the top digit. This means that multi-digit subtraction demands a conceptual understanding of the meaning of symbols, place value and the base-ten number system, as well as the ability to identify quantities, encode and transcribe quantities in an internal representation code, and keep track of partial results while completing the next step. Other arithmetical abilities that are probably important for subtraction include an efficient fact retrieval and a good expertise in mental calculation (Butterworth, 2005, Dowker, 2005).
The present study was designed to shed further light on the precursors and mediators involved when children learn written subtractions by examining the role of individual differences in both the domain-specific and the domain-general precursors (i.e. numerical and procedural knowledge in the former case, simple and complex visuospatial and verbal WM components in the latter) that take effect when written subtractions are taught using the standard method. We followed up five classes of 2nd-grade pupils throughout a school year, testing them on three different occasions (at the start of the school year, in November, and in February). In the first session, we assessed variables that we assumed might predict the acquisition of subtraction with borrowing. Regarding the domain-specific abilities, we hypothesized that the acquisition of the subtraction algorithm is linked to competences in the calculation domain, such as arithmetical fact retrieval, and proficiency in solving written addition problems. We also hypothesized that more specific abilities in the number domain, e.g. the expertise needed to process judgments of magnitude or understand the value of each digit in a complex number, would affect a child's acquisition of the subtraction procedure. As for abilities in the general domain, we predicted a different influence of WM components. In particular, we assumed that performing subtractions would mainly engage visuospatial WM resources, while the influence of verbal WM would emerge as a mediator of the child's knowledge of the principles of addition.
In the second and third sessions we assessed arithmetical learning in terms of written arithmetic that we assumed might mediate the relationship between the predictors considered and the acquisition of subtraction with borrowing (which was also measured in the third step). For the mediators, we assumed that the acquisition of written additions, both simple and with carrying, and simple subtractions would be supported by the same predictors as those supporting complex subtractions (Dowker, 2005, Fuchs et al., 2006), and further involved in the acquisition of subtraction with borrowing. The acquisition of subtraction was assessed immediately after conducting two lessons on written subtraction procedures (one without and one with borrowing) to exclude any confounding influence of any other variables intervening between the acquisition of the subtraction algorithms and any later assessment. The teacher's role was controlled by asking the class teachers not to teach subtraction, and arranging for 2 h of structured lessons to be conducted by one and the same teacher in all the classes.
Section snippets
Participants
The study concerned 96 primary school pupils (60 M, 36 F) tested in 2nd grade. The children were attending 5 different classes at schools in north-eastern Italy. They were tested over the course of three sessions, starting from the beginning of the academic year. Parental consent was obtained beforehand for all the children. Children with certified special educational needs, intellectual disabilities or neurological/genetic disorders, or from families with a very low socio-economic status were
Results
Preliminary analyses showed that gender and class did not significantly affect performance in any of the arithmetical tasks, so these variables were not taken into account. Descriptive statistics for each test are given in Table 1, while the correlations between the measures are shown in Table 2.
Discussion
This study focused on the contributions of precursors – both domain-specific (numerical and procedural knowledge) and domain-general (WM components) – in children learning written subtractions, when the algorithm is taught according to the standard procedural methods. Children in 2nd grade were tested three times during the school year to obtain a sort of ongoing picture of their acquisition of written subtractions.
Our final path analysis model was able to explain a relevant proportion of the
Conclusion
The results of the present study have not only theoretical but also educational implications. They show that some children's difficulties in learning arithmetical operations can be foreseen, and possibly prevented, by assessing a number of crucial precursors. In particular, children's difficulties in solving written calculations may relate to both WM weaknesses and specific arithmetical mind bugs. Particular attention should therefore be paid to supporting children in difficulty – especially
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