Mathematical fluency as a function of conservation ability in young children
Highlights
► First grade children were differentiated by level of cognitive development. ► Individual differences affect the ability to perform mathematics. ► Conservation ability resulted in greater mathematical fluency. ► Fluency was significantly greater in both addition and subtraction. ► Learning differences require developmentally appropriate instruction and materials.
Introduction
There has been general agreement that by the end of first grade children must have mastered single-digit addition and subtraction operations (Kilpatrick, Swafford, & Findell, 2001). Although untimed arithmetic tasks measuring accuracy tend to overestimate children's arithmetic skills (Baroody, Bajwa, & Eiland, 2009), timed fluency tasks measuring accuracy and speed are quite indicative of skill mastery (Binder, 1996, Duncan et al., 2007, Sarama and Clements, 2009). However, some children struggle more than others to achieve fluency in first grade due to differences in ability related to their level of cognitive development (Wübbena, 1977); indeed, these differences affect children's performance on basic math tasks (Ginsburg and Baroody, 1990, Ojose, 2008, Piaget, 1965). Educators must understand individual differences in mathematical fluency related to the level of cognitive development to provide appropriate instruction and adopt suitable expectations for student achievement. The present study examined the addition and subtraction fluencies and levels of cognitive development of first grade children.
Children's understanding of the relationship between addition and subtraction problems has been investigated extensively (Baroody, 1999, Canobi, 2004, Canobi et al., 1998, Canobi et al., 2002, Canobi et al., 2003, Siegler, 1987, Vilette, 2002), although these findings appear mixed. Siegler (1987) assumed that children automatically associate subtraction with addition, whereas Canobi (2004) suggested that an understanding of addition concepts precede that of subtraction and the addition–subtraction relationship. Baroody (1999) indicated that the inverse relationship between addition and subtraction is difficult for children to understand, thereby adversely affecting children's fluency in simple math-related tasks (Bryant et al., 1999, Fischer, 1990). Fluency is important; ultimately, when viewed as a combination of speed and accuracy, it indicates a level of skill mastery that allows unoccupied cognitive resources to be applied toward new and more advanced mathematical tasks (Binder, 1996, Skinner, 1998).
Seminal cognitive developmental theory may reveal why children differ in their abilities to understand the inverse relationship between addition and subtraction. Piaget (1952) theorized four levels of cognitive development: sensorimotor, preoperational, concrete operational, and formal operational. Children in first grade, at 6 or 7 years old, may be in the preoperational (non-conserving) or concrete operational (conserving) level of development (Ojose, 2008, Wübbena, 1977), as demonstrated by the conservation-of-liquid task (Piaget, 1967). In observing changes in the amount of liquid among different-sized cups, non-conserving children demonstrated centration on perceptual changes about the liquid—that is, focusing on one part with the exclusion of other equally important parts. Whereas, conserving children overcame perceptual influences using logic to judge that reversible changes in liquid remained constant (Copeland, 1974, Ojose, 2008). This allows conserving children to evaluate several parts of actions simultaneously, rather than focusing on a single aspect. While conserving children use logical reversible reasoning to understand the relationship of interdependent actions based on a classic conservation task, there has been less success in identifying links to school-based tasks involving fluency of addition and subtraction operations.
Cooper and Schleser (2006) and Ramos-Christian, Schleser, and Varn (2008) investigated the mathematical fluency of conserving and non-conserving children from a high-socioeconomic-level background. In using an instrument that combined addition and subtraction problems, conservation ability resulted in a greater mathematical fluency (i.e., combined accuracy and speed), although accuracy was comparable among children at both levels of cognitive development (Ramos-Christian et al., 2008). However, a gap in this literature is that fluency was not examined separately for addition and subtraction problems. Researchers have argued that solving inverse operations (i.e., addition and subtraction problems) successively requires increased cognitive demand beyond that required for separate operations (Baroody, 1999, Canobi, 2004, Vilette, 2002). It is possible, therefore, that non-conserving children are as fluent as conserving children when addition and subtraction problems have been separated. This study builds on previous research by investigating whether significant differences in fluency exist when addition and subtraction problems are separately administered to conserving and non-conserving children in first grade from a low-socioeconomic-level background.
Section snippets
Participants
The participants in this study were 97 children in the first grade from two low-socioeconomic-level elementary schools in Central Texas. The sample included 51 boys and 46 girls both aged 6 years and 8 months to 7 years and 9 months (Mage = 7.16; SD = 3.65). The ethnic composition was mixed (35% White, 65% Hispanic). Informed consent was obtained from all parents and guardians prior to children's participation in this study.
Materials
Mathematical fluency in addition and subtraction was assessed using two
Results
Two one-way analyses of covariance (ANCOVAs) were conducted. The independent variable was conservation ability and included two levels: conserving and non-conserving. The two dependent variables were addition fluency and subtraction fluency; age was the covariate. Familywise error was controlled using a Bonferroni correction (αFamilywise = .05/2 = .025).
Addition fluency scores ranged from 3 to 53, and subtraction fluency scores ranged from 0 to 52. The mean addition fluency score was 23.06 (SD =
Discussion
The present study investigated the relationship among levels of cognitive development and addition and subtraction fluencies of first grade children. The results suggest that conserving children have significantly greater fluency in addition and subtraction than non-conserving children. These results are consistent with findings in the literature indicating that children who have reached the concrete operational level have significantly greater mathematical fluency than children in the
Acknowledgments
Special appreciation is provided to the children, parents, teachers, administration, and others who supported this study. Further appreciation is provided to the assistance of Mark L. Garibaldi and the four peer-reviewers for providing feedback on earlier versions of this manuscript.
References (31)
- et al.
Children's understanding of the relation between addition and subtraction: Inversion, identity, and decomposition
Journal of Experimental Child Psychology
(1999) Individual differences in children's addition and subtraction knowledge
Cognitive Development
(2004)- et al.
Functional magnetic resonance imaging study of Piaget's conservation-of number task in preschool and school-age children: A neo-Piagetian approach
Journal of Experimental Child Psychology
(2011) Do young children grasp the inverse relationship between addition and subtraction? Evidence against early arithmetic
Cognitive Development
(2002)Progress monitoring and RTI system
Children's relational knowledge of addition and subtraction
Cognition and Instruction
(1999)- et al.
Why can't Johnny remember the basic facts?
Developmental Disabilities Research Reviews
(2009) Behavioral fluency: Evolution of a new paradigm
The Behavioral Analyst
(1996)- et al.
The role of conceptual understanding in children's addition problem-solving
Developmental Psychology
(1998) - et al.
Young children's understanding of addition concepts
Educational Psychology
(2002)
Patterns of knowledge in children's addition
Developmental Psychology
Closing the achievement gap: Examining the role of cognitive development level in academic achievement
Early Childhood Education Journal
How children learn mathematics: Teaching implications of Piaget's research
School readiness and later achievement
Developmental Psychology
A part–part–whole curriculum for teaching number to kindergarten
Journal for Research in Mathematics Education
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