Number words in young children’s conceptual and procedural knowledge of addition, subtraction and inversion

Abstract

Three studies addressed children’s arithmetic. First, 50 3- to 5-year-olds judged physical demonstrations of addition, subtraction and inversion, with and without number words. Second, 20 3- to 4-year-olds made equivalence judgments of additions and subtractions. Third, 60 4- to 6-year-olds solved addition, subtraction and inversion problems that varied according to the inclusion of concrete referents and number words. The results indicate that number words play a different role in conceptual and procedural development. Children have strong addition and subtraction concepts before they can translate the physical effects of these operations into number words. However, using number words does not detract from their calculation procedures. Moreover, consistent with iterative relations between conceptual and procedural development, the results suggest that inversion acquisition depends on children’s calculation procedures and that inversion understanding influences these procedures.

Introduction

Conceptual knowledge of addition and subtraction involves understanding that addition increases the number of objects present and subtraction has the opposite effect. However, there are conflicting views on the role of number words in this understanding. Indeed, number words may play a different role in children’s conceptual understanding and their procedural skills. One interpretive difficulty with previous addition and subtraction research is separating the effects of number words and concrete referents on young children’s problem-solving skills. Accordingly, the first goal of the research was to explore the role of number words in children’s addition, subtraction and inversion concepts. The second goal was to investigate the role of number words and concrete referents in children’s procedures for solving addition, subtraction and inversion problems.

The nature of arithmetic development is subject to intense debate. Some research suggests that even infants can predict the outcomes of additions and subtractions of 1–3 objects (e.g., Wynn, 1992). However, there is evidence that infants are influenced by non-numerical features that co-vary with number, such as total surface area or contour length (Clearfield and Mix, 1999, Mix et al., 2002). Infants’ success on addition and subtraction tasks has also been attributed to an underlying system for reasoning about individual objects via “object files” (Feigenson and Carey, 2003, Feigenson et al., 2002, Uller et al., 1999). Some proponents of object-file explanations have also theorised that preschoolers’ knowledge of count words enables a restructuring of initial, object-based representations, leading to mathematical reasoning (for a review and critique, see Gelman & Butterworth, 2005). In contrast, others have claimed that key aspects of children’s addition and subtraction understanding are not dependent on their knowledge of number words (Dehaene, 1992, Gallistel and Gelman, 1992, Gelman and Butterworth, 2005, Starkey et al., 1990). The finding that infants appear to recognise additions and subtractions for numbers of objects that exceed working memory limits (e.g., 5 + 5 = 10), while continuous-extent variables are controlled, addresses some of the criticisms of earlier work (McCrink & Wynn, 2004). However, even if such competences are present in infancy and might therefore be expected among preschoolers, the ways in which these competences develop and relate to number-word knowledge require further research attention.

Even before attending school, children recognise the effects of additions and subtractions on number words. In particular, by demonstrating that children’s judgments are above chance level, researchers have shown that 5-year-olds recognise that additions and subtractions alter number-word labels for large groups of objects (Lipton & Spelke, 2006). Similarly, Sarnecka and Gelman (2004) found that 3-year-olds recognize that adding or subtracting one changes the number-word labels of two to six objects, although Condry and Spelke (2008) found that 3-year-olds did not recognize the effects of additions and subtractions involving number words ranging from five to fourteen. Preschoolers can use number words to predict the direction of additions and subtractions (Zur & Gelman, 2004). They can also recognise addition principles (e.g., 3 + 2 = 2 + 3) in the context of objects with number-word labels (Canobi, Reeve, & Pattison, 2002). However, it is unclear whether these accomplishments in the context of number words are based on an abstract, conceptual understanding of addition and subtraction. Children’s early understanding might be abstract in the sense that they can predict the effects of additions and subtractions without reference to specific numbers.

Recent research into specific language impairment (SLI) suggests that number-word knowledge plays a different role in the development of arithmetic concepts and procedural skills (Arvedson, 2002, Donlan et al., 2007, Fazio, 1996). For example, Donlan and colleagues found that 8-year-olds with SLI had profound deficits in producing the count-word sequence and basic calculation procedures. In contrast, the 8-year-olds had unimpaired conceptual understanding, as measured by their knowledge of additive commutativity (e.g., 2 + 3 = 3 + 2). These results suggest that conceptual development is not constrained by number-word knowledge in the way that procedural development is. The results also suggest that conceptual achievements can be made independently of procedural skills. (For reviews of the concept-first vs. procedures-first debate in arithmetic development, see Cowan, 2003, Rittle Johnson and Siegler, 1998.) However, the commutativity principle (the basis for Donlan and colleagues’ conceptual task) is learned before other important arithmetic concepts such as associativity (e.g., [2 + 3] + 4 = 2 + [3 + 4]), subtraction-complement (e.g., if 5−2 = 3 then 5−3 = 2) and inversion (e.g., 2 + 3− 3 = 2) principles (Baroody et al., 1983, Canobi, 2004, Canobi, 2005, Canobi et al., 1998, Canobi et al., 2003, Canobi et al., 2002, Langford, 1981). In addition, children understand abstract relations based on commutativity more readily than those based on additive composition and associativity (Canobi et al., 2003). Therefore, it is important to explore the role of number words in children’s acquisition of concepts other than commutativity.

In particular, the role of number-word knowledge in the development of children’s inversion requires research attention. Appreciating the inverse relation between addition and subtraction is fundamental to a conceptual understanding of arithmetic (Bryant et al., 1999, Piaget, 1952). In an early study, Starkey and Gelman (1982) found that 3-year-olds solved easy inversion problems such as 2 + 1−1, without overtly counting. However, Vilette (2002) found that only 4- to 5-year-olds performed above chance-level in rejecting impossible answers to such problems and accepting the correct answers. Unfortunately, children’s use of object-file representations or covert counting procedures in these two studies cannot be ruled out. Nonetheless, with the inclusion of control problems, Bisanz and colleagues demonstrated that 4- to 6-year-olds can use inversion-based shortcuts (Klein & Bisanz, 2000) even when cues based on the identity of objects and length of rows are removed (Rasmussen, Ho, & Bisanz, 2003). However, it is still unclear whether inversion develops from computational experience with specific numbers or through reasoning about the concepts of addition and subtraction more generally (Bisanz, Sherman, Rasmussen, & Ho, 2005).

Children’s early addition and subtraction skills could be based, at least in part, on their knowledge of the physical actions of adding and subtracting. Despite this, researchers often use word problems without concrete referents as the context for assessing the role of number words in children’s problem-solving procedures (e.g., Barth, La Mont, Lipton, Dehaene, Kanwisher & Spelke, 2006; Jordan et al., 1995, Levine et al., 1992, Rasmussen and Bisanz, 2005). For example, Levine and colleagues found that 4-year-olds could match the (concealed) final set after watching objects being added or subtracted but were unable to solve equivalent word problems. Similarly, Barth and colleagues found that children could judge animated dot array problems but could not solve or judge verbal word problems. However, the participants in these studies were asked to solve verbal word problems without concrete referents. In contrast, the non-verbal problems were based on a physical demonstration. Therefore, it is unclear whether the presence of concrete referents or the absence of number words helped the children to succeed. Jordan, Huttenlocher, and Levine (1994) also found that low-income children were disadvantaged by being asked to give verbal responses to a non-verbal calculation task. However, in this study, it is possible that the source of children’s poor performance was difficulties in overcoming a mismatch between presentation-type and response-type rather than using number words per se.

Accordingly, the current research was designed to explore the role of number words and physical objects in the development of young children’s concepts and procedures for addition, subtraction and inversion. Study 1 addressed children’s conceptual knowledge that addition and subtraction change the number of objects present and are inverse operations. Study 2 was a follow-up study, designed to clarify the nature of children’s difficulties in applying number words to addition and subtraction concepts. Study 3 addressed the role of number words and concrete referents in children’s problem-solving procedures.