Multiplication facts and number sense in children with mathematics learning disabilities and typical achievers
Introduction
In modern society, efficiency in mathematics is essential for everyday life, for professional qualification, and for higher education. Mathematics education today emphasizes developing a number sense, without ignoring simple arithmetic (e.g., The Common Core State Standards (Mathematics); National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). About six percent of the population has math learning deficiencies (MLD; Shalev, Auerbach, Manor, & Gross-Tsur, 2000). The origins of MLD are considered to be neurological (Kucian, 2016), but its diagnosis criteria are behavioral. The diagnosis criteria include persistent difficulties in number sense, in arithmetic facts (i.e., single-digit problems) and procedures, and in mathematics reasoning (American Psychiatric Association, 2013).
Four- to five-month-old infants are already able to pre-verbally compute the arithmetical consequences of adding and subtracting small sets of objects (Wynn, 1995). Two- to three-year-old toddlers learn to count and use counting, together with countable objects (especially their fingers), to add and subtract (Fuson, 1992). Formal learning of addition and subtraction begins in most countries in kindergarten.
Curricula typically introduce multiplication and division later than addition and subtraction (usually in second or third grade, e.g., National Governors Association Center for Best Practices and Council of Chief State School Officers (2010) in USA, and Israel Minster of Education (2006)), and explain them in terms of repeated addition, repeated subtraction and partitions of sets. At first, children solve multiplication problems by executing procedural strategies such as repeated addition or derived facts (e.g., 7 × 8 = 7 × 7 + 7). With age and schooling, they form an associative network of problems and solutions to problems in long-term memory, and then retrieve the solutions for most problems from memory (Baroody, 1999; Cho, Ryali, Geary, & Menon, 2011; De Brauwer & Fias, 2009; Lemaire & Siegler, 1995; Sherin & Fuson, 2005). Consequently, children's performance gradually becomes fast and accurate. Children perform better in terms of accuracy and speed on problems involving 1, 0 and 10 (termed rule-based problems; e.g., 3 × 0), problems involving numbers smaller than 5 (termed small problems; e.g., 2 × 3), problems involving duplicate numbers (termed tie problems; e.g., 4 × 4), and problems involving 5 (termed five problems; e.g., 5 × 3) than on problems involving numbers larger than five (termed large problems; e.g., 8 × 9) and numbers involving one number larger than five and one number smaller than five (hereafter termed medium problems; e.g., 4 × 8) (De Brauwer & Fias, 2009; De Brauwer, Verguts, & Fias, 2006). According to a current model of mental multiplication (Verguts & Fias, 2005), small, tie and five problems are easier to solve than medium and large problems because their products have more consistent and less inconsistent digits in the same position (i.e., decade, unit) with their neighboring products. Neighboring products are products of problems in which one of the operands is changed by ±1 or ±2 (e.g., the problem 6 × 4 and its neighbors 7 × 4, 5 × 4, 6 × 3, 6 × 5). Consistent neighbors (e.g., 28 and 24 for the problem 6 × 4) cooperate with the true product and facilitate its retrieval, thus decreasing the difficulty of the problem. Inconsistent neighbors (e.g., 18 and 24 for the problem 6 × 4) compete with the true product and interfere with its retrieval, thus increasing the difficulty of the problem (see Verguts & Fias, 2005, Table 2). With age and schooling, children master other problems as well and the differences between easy and difficult problems decrease (De Brauwer et al., 2006). Yet, these differences never completely disappear and they are still robust in adults (Ashcraft, 1992; Campbell, 1995; Domahs & Delazer, 2005), so procedural strategies, which are sometimes used to solve single-digit problems, are used mainly on difficult problems (LeFevre et al., 1996). It is conceivable that similarities in performance patterns (e.g., similar speed / accuracy patterns between problem types) might suggest a similar underlying memory network (Ashcraft, 1992; Campbell, 1995; De Brauwer et al., 2006; Domahs & Delazer, 2005; Verguts & Fias, 2005). By sixth grade, children already have a multiplication memory network similar to that of adults (De Brauwer et al., 2006).
Children with MLD have a robust arithmetic facts deficiency that persists into secondary school years (Mabbott & Bisanz, 2008; Mazzocco, Devlin, & McKenney, 2008; Ostad, 1997, 1999). They keep on solving most single-digit multiplication problems by using procedural strategies rather than by direct retrieval from memory, and as such, their performance continues to be slower and less accurate than that of typically achieving (TA) age-matched peers (Mabbott & Bisanz, 2008; Mazzocco et al., 2008). Moreover, they use mainly basic procedures such as repeated addition, and they do not spontaneously invent or execute a derived facts strategy (Swanson & Cooney, 1985). Previous studies usually compared the performance level of children with MLD to the performance level seen in their age-matched peers (e.g., Mazzocco et al., 2008), or one year younger TA children (e.g., Mabbott & Bisanz, 2008; Rousselle & Noël, 2008), or to the curriculum expectation level (Calhoon, Emerson, Flores, & Houchins, 2007). These comparisons did not enable one to clearly determine which performance level, relative to the actual performance of TA children, could be achieved by older children with MLD for the various problem types.
The consistent arithmetic deficiency seen in children with MLD raises the question whether they have at all formed a memory network for multiplication problems. Geary (2004) suggested that the pervasive use of back-up procedures, together with a working memory deficiency, interferes with the formation of an arithmetic memory network in children with MLD. In contrast, Mazzocco et al. (2008) found that eighth graders with MLD performed better on small, tie and five problems than on large problems, indicating at least a partial formation of a multiplication network in memory. Yet, it is not clear whether the characteristics of the multiplication network in children with MLD are similar to those seen in TA children. The observation of Mazzocco et al. that small, tie and five problems were easier for both TA children and children with MLD (though they were not as easy for children with MLD, compared with TA children, since TA children rarely erred on easy problems, while eighth graders with MLD did) might indicate similarities of underlying memory representations (De Brauwer et al., 2006; Domahs & Delazer, 2005).
Because arithmetic facts deficiency is a defining feature of MLD (American Psychiatric Association, 2013; Jordan, Hanich, & Kaplan, 2003), it is important to thoroughly examine this feature. The first objective of the current study is to examine age differences for the various problem types in children with MLD, relative to TA children. The second objective of the current study is to examine whether the characteristics of the multiplication network in children with MLD are similar (as indicated by performance similarities) to the characteristics seen in TA children, even at younger ages.
The term “number sense” reflects a set of understandings and skills that enable a person to efficiently use multiple relationships among numbers and operations, to use benchmarks to judge number magnitude, and to recognize unreasonable results. (Andrews & Sayers, 2015; National Council of Teachers of Mathematics, 2000). The observation that when learning arithmetic facts, children gradually not only commit less errors, but their errors become more plausible, suggests that learning arithmetic facts enables children to refine their number sense. From the second grade onward, most errors are: in-table errors, close to the true result (i.e., having a maximum distance of two “steps” on the multiplication row from the true result) and related to the operands row / column (e.g., 4 × 6 = 28; Butterworth, Marchesini, & Girelli, 2003). From the third grade onward, most errors obey the parity rule (namely, even X even = even, even X odd or odd X even = even, odd X odd = odd; Lemaire & Fayol, 1995). In adults, most errors also obey the five-rule (i.e., the product’s unit-digit of problems involving 5 must be either 5 or 0, but it cannot be 5 or 0 when none of the operands is 5; Lemaire & Lecacheur, 2004) and share the decade-digit with the true result (Domahs, Delazer, & Nuerk, 2006). This pattern of plausible errors suggests that children gradually not only learn the multiplication table, but also refine their number sense.
Relative to age-matched peers, children with MLD have a persistent number sense deficiency (Ashkenazi, Mark-Zigdon, & Henik, 2009; Geary, Hoard, Nugent, & Bailey, 2012; Mazzocco, Feigenson, & Halberda, 2011; Moeller, Neuburger, Kaufmann, Landerl, & Nuerk, 2009; Mussolin, Mejias, & Noël, 2010; Schleifer & Landerl, 2011). When solving multiplication facts problems, children with MLD not only erred more than their TA age-matched peers, but their errors were more distant from the true result (Mabbott & Bisanz, 2008). In a series of verification experiments, we found that sixth graders with MLD were not sensitive to products’ numerical features such as parity (Rotem & Henik, 2013) and relatedness to the operands’ row (Rotem & Henik, 2015a). Sixth and eighth graders with MLD were not sensitive to the approximate size of products (Rotem & Henik, 2015a) nor to the product’s specific decade digit (Rotem & Henik, 2015b). However, in accordance with Mabbott and Bisanz (2008), we found that eighth graders with MLD become sensitive to parity (Rotem & Henik, 2013) and to relatedness of the product and operands via a shared multiplication row (Rotem & Henik, 2015a). These findings suggest some development of number sense in children with MLD, even if late and partial. The third objective of the current study is to examine age differences in number sense in sixth and eighth graders with MLD relative to TA children.
The current study compared the performance of two age-groups of children with MLD on a multiplication facts production task to the performance of three age-groups of TA children, as well as adults, in order to examine whether the multiplication development of children with MLD follows the same path seen in TA children. Specifically, we wanted to examine: (1) the age-related differences in multiplication performance in children with MLD relative to TA children, as a function of problem type; (2) whether children with MLD form a multiplication network similar to that seen in TA children; and (3) age differences in number sense related to the multiplication table.
Section snippets
Participants
One hundred and twenty-eight participants took part in the study. Twenty-four were adults (Psychology and Education undergraduate students), 72 were TA participants and 32 were children with MLD. Descriptive information regarding the experimental groups is presented in Table 1. Children with MLD were recruited from 10 regular public schools in Israel and TA children were recruited from four of these schools. We focused on sixth and eighth graders with MLD because according to De Brauwer et al.
Results
To trace the performance level of each group for each problem type, mean accuracy and mean RT were subjected to a separate two-way analysis of variance (ANOVA) with a between-participant factor of group (sixth graders with MLD, eighth graders with MLD, TA second graders, TA fourth graders, TA sixth graders, and adults) and a within-participant factor of problem type (small, tie, five, medium, and large). Mean accuracy was analyzed after an arcsine transformation to an approximate normal
Discussion
The current study focused on comparing the multiplication development and related number sense in sixth and eighth graders with MLD and TA participants in various age groups. Specifically, we wanted to find 1) the age-related differences in multiplication performance in children with MLD relative to TA children, as a function of problem type; (2) whether children with MLD form a multiplication network similar to that seen in TA children; and (3) age differences in number sense related to the
Funding
This work was conducted as part of the research in the Center for the Study of the Neurocognitive Basis of Numerical Cognition, supported by the Israel Science Foundation (Grant 1799/12) in the framework of their Centers of Excellence, and by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant agreement no. 295644.
Declaration of Competing Interest
None.
References (55)
Cognitive arithmetic: A review of data and theory
Cognition
(1992)- et al.
Numerical distance effect in developmental dyscalculia
Cognitive Development
(2009) The roles of estimation and the commutative principle in the development of third graders` mental multiplication
Journal of Experimental Child Psychology
(1999)- et al.
The representation of multiplication facts: Developmental changes in the problem size, five, and tie effects
Journal of Experimental Child Psychology
(2006) - et al.
The development of children`s mental multiplication skills
Journal of Experimental Child Psychology
(1991) Developmental dyscalculia and the brain
- et al.
Basic number processing deficits in developmental dyscalculia: Evidence from eye tracking
Cognitive Development
(2009) - et al.
Symbolic and nonsymbolic number comparison in children with and without dyscalculia
Cognition
(2010) - et al.
The development of product parity sensitivity in children with a mathematics learning disability and in typical achievers
Research in Developmental Disabilities
(2013) Diagnostic and statistical manual of mental disorders
(2013)
Identifying opportunities for grade one children to acquire foundational number sense: Developing a framework for cross cultural classroom analyses
Early Childhood Education Journal
Attentional networks in developmental dyscalculia
Behavioral and Brain Function
Basic multiplication combinations: Passive storage or dynamic reorganization?
Computational fluency performance profile of high school students with mathematical disabilities
Remedial and Special Education
Production, verification and priming of multiplication facts
Memory & Cognition
Mechanisms of simple addition and multiplication: A modified network-interference theory and simulation
Mathematical Cognition
Mathematical learning disorder in school-age children with attention-deficit hyperactivity disorder
The Canadian Journal of Psychiatry
How does a child solve 7+8? Decoding brain activity patterns associated with counting and retrieval strategies
Developmental Science
A longitudinal study of children`s performance of simple multiplication and division problems
Developmental Psychology
Some assumptions and facts about arithmetic facts
Psychology Science
What makes multiplication facts difficult: Problem size or neighborhood consistency?
Experimental Psychology
Research on learning and teaching addition and subtraction of whole numbers
Mathematics and learning disabilities
Journal of Learning Disabilities
Mathematical cognition deficits in children with learning disabilities and persistent low achievement: A five-year prospective study
Journal of Educational Psychology
Assisting students struggling with mathematics: Response to intervention (RtI) for elementary and middle schools (NCEE 2009-4060)
Math curriculum for elementary schools
Meitzav – A standard mathematics test for fourth grade, November version
Cited by (7)
Developing mental number line games to improve young children's number knowledge and basic arithmetic skills
2022, Journal of Experimental Child PsychologyCitation Excerpt :As argued by researchers, a well-planned interventional teaching model can improve the effectiveness of mathematics learning among young children (Rotem & Henik, 2020).
The Number of Different Digits Determines Solution and Verification of Multiplication Problems
2023, Journal of Experimental Psychology: Learning Memory and CognitionHow children learn math: The science of math learning in research and practice
2022, How Children Learn Math: The Science of Math Learning in Research and PracticeSpatial Reasoning Profile of the Students with Good Number Sense Ability
2021, Journal of Physics: Conference SeriesResearch on children's cognitive development for learning disabilities using recommendation method
2021, Concurrency and Computation: Practice and ExperienceNumber sense ability of junior high school students
2021, Journal of Physics: Conference Series