Nancy C. Jordan

Early Identification and Interventions for Students With Mathematics Difficulties

This article highlights key findings from the small body of research on mathematics difficulties (MD) relevant to early identification and early intervention. The research demonstrates that (a) for many children, mathematics difficulties are not stable over time; (b) the presence of reading difficulties seems related to slower progress in many aspects of mathematics; (c) almost all students with MD demonstrate problems with accurate and automatic retrieval of basic arithmetic combinations, such as 6 + 3. The following measures appear to be valid and reliable indicators of potential MD in kindergartners: (a) magnitude comparison (i.e., knowing which digit in a pair is larger), (b) sophistication of counting strategies, (c) fluent identification of numbers, and (d) working memory (as evidenced by reverse digit span). These are discussed in terms of the components of number sense. Implications for early intervention strategies are explored.

Early Math Matters: Kindergarten Number Competence and Later Mathematics Outcomes

Childrens number competencies over 6 time points, from the beginning of kindergarten to the middle of 1st grade, were examined in relation to their mathematics achievement over 5 later time points, from the end of 1st grade to the end of 3rd grade. The relation between early number competence and mathematics achievement was strong and significant throughout the study period. A sequential process growth curve model showed that kindergarten number competence predicted rate of growth in mathematics achievement between 1st and 3rd grades as well as achievement level through 3rd grade. Further, rate of growth in early number competence predicted mathematics performance level in 3rd grade. Although low-income children performed more poorly than their middle-income counterparts in mathematics achievement and progressed at a slower rate, their performance and growth were mediated through relatively weak kindergarten number competence. Similarly, the better performance and faster growth of children who entered kindergarten at an older age were explained by kindergarten number competence. The findings show the importance of early number competence for setting childrens learning trajectories in elementary school mathematics.

A Longitudinal Study of Mathematical Competencies in Children with Specific Mathematics Difficulties versus Children with Comorbid Mathematics and Reading Difficulties.

Mathematical competencies of 180 children were examined at 4 points between 2nd and 3rd grades (age range between 7 and 9 years). Children were initially classified into one of 4 groups: math difficulties but normal reading (MD only), math and reading difficulties (MD-RD), reading difficulties but normal math (RD only), and normal achievement in math and reading (NA). The groups did not differ significantly in rate of development. However, at the end of 3rd grade the MD only group performed better than the MD-RD group in problem solving but not in calculation. The NA and RD only groups performed better than the MD-RD group in most areas. Deficiencies in fact mastery and calculation fluency, in particular, are defining features of MD, with or without RD.

Performance across Different Areas of Mathematical Cognition in Children with Learning Difficulties.

The performance of 210 2nd graders in different areas of mathematical cognition was examined. Children were divided into 4 achievement groups: children with difficulties in mathematics but not in reading (MD-only), children with difficulties in both mathematics and reading (MD/RD), children with difficulties in reading but not in mathematics, and children with normal achievement. Although both MD groups performed worse than normally achieving groups in most areas of mathematical cognition, the MD-only group showed an advantage over the MD/RD group in exact calculation of arithmetic combinations and in problem solving. The 2 groups did not differ in approximate arithmetic and understanding of place value and written computation. Children with MD-only seem to be superior to children with MD/RD in areas that may be mediated by language but not in ones that rely on numerical magnitudes, visuospatial processing, and automaticity.

Cognitive Arithmetic and Problem Solving A Comparison of Children with Specific and General Mathematics Difficulties

This study examined problem-solving and number-fact skills in two subgroups of third-grade children with mathematics difficulties (MD): MD-specific (n = 12) and MD-general (n = 12). The MD-specific group had difficulties in mathematics but not in reading, and the MD-general group had difficulties in reading as well as in mathematics. A comparison group of nonimpaired children (n = 24) also was included. The findings showed that on both story and number-fact problems, the MD-specific group performed worse than the nonimpaired group in timed conditions but not in untimed conditions. The MD-general group, on the other hand, performed worse than the nonimpaired group, regardless of whether tasks were timed or not. An analysis of childrens strategies in untimed conditions showed that both the MD-specific and the MD-general groups relied more on backup strategies than the nonimpaired group. However, children in the MD-specific group executed backup strategies more skillfully than children in the MD-general group, allowing them to achieve parity with children in the nonimpaired group when tasks were not timed. The findings suggest that children with specific MD have circumscribed deficits associated with fact retrieval, whereas children with general MD have more basic delays associated with problem conceptualization and execution of calculation procedures.

Arithmetic fact mastery in young children: a longitudinal investigation.

Children with poor arithmetic fact mastery (n=45) at the end of third grade were compared to grade-level peers with good arithmetic fact mastery (n=60) in competencies related to reading and mathematics. Children were assessed longitudinally across second and third grades. When predictor variables such as IQ were held constant, the poor fact mastery and good fact mastery groups performed at about the same level and progressed at a comparable rate on math story problems and on broad reading achievement. The groups also progressed at a comparable rate on broad math achievement, although children with poor fact mastery performed at a significantly lower level. Children with poor fact mastery showed remarkably little growth on timed number facts during the study period, despite normal growth in other areas of mathematics. Deficits in fact mastery are highly persistent and appear to be independent of reading and language abilities.

Mathematical Thinking in Second-Grade Children with Different Forms of LD

Based on their performance on a standardized achievement test, second-grade children (N = 49) were classified as having mathematics difficulties with normal reading achievement (MD only), both mathematics and reading difficulties (MD/RD), reading difficulties with normal mathematics achievement (RD only) and normal mathematics and reading achievement (NA). Each child was given a series of tasks so that we might assess their thinking across four areas of mathematics: number facts, story problems, place value, and written calculation. Children with MD/RD performed significantly worse than NA children in most areas of mathematical thinking, whereas children with MD only performed worse than NA children only on complex story problems. The MD-only group outperformed the MD/RD group on story problems and written calculation. No significant differences were found between the RD-only and NA groups on any of the tasks. The results suggested that among children with mathematics difficulties, the MD/RD subgroup is distinct from the MD-only subgroup, with the former being characterized by pervasive deficiencies in mathematical thinking and the latter by more specific deficits in problem solving.

Using Kindergarten Number Sense to Predict Calculation Fluency in Second Grade

Childrens number sense in kindergarten was used to predict their calculation fluency in second grade (N = 198). Using block entry regression, usual predictors of age, reading, memory, and verbal and spatial cognition were entered in the first block and number sense measures were added in the second block. Number sense measures contributed a significant amount of variance over and above the more general predictors (26%—42%). Uniquely predictive subareas were active memory for numbers, number knowledge, and number combinations, with number combinations standing out as the strongest single predictor. Number sense screening in kindergarten, using “at-risk” versus “not-at-risk” criteria, successfully ruled out 84% of the children who did not go on to have calculation fluency difficulties and positively identified 52% of the children who later showed fluency difficulties. The relation of early number skills to later calculation fluency has important implications for math screening and intervention.

A mental model for early arithmetic.

The authors examined young childrens ability to solve nonverbal calculation problems in which they must determine how many items are in a hidden array after items have been added into or taken away from it. Earlier work showed that an ability to reliably solve such problems emerges earlier than verbal calculation ability but did not examine when it first appears. The authors propose that the ability to solve such problems involves domain-general symbolic processes similar to those involved in symbolic play and the use of physical models. Hence the ability to calculate should appear at about 2 years and should be related to overall level of intellectual competence. The authors show that the ability to reliably solve nonverbal calculation tasks emerges only after 2 years of age and that performance on nonverbal calculation problems is highly related to overall level of intellectual competence in children between 3 and 4 years of age.

Development of Calculation Abilities in Young Children.

This study investigates the development of skills for solving verbally and nonverbally presented calculation problems in children between 4 and 6 years of age. Identical addition and subtraction calculations were presented in three problem-type formats: nonverbal problems, story problems, and number-fact problems. The nonverbal problems involved presenting sets of physical referents that were then transformed either by adding or removing elements. The child saw the initial set and the number of elements that were added or removed, but not the final set. The task was to construct an array that contained the number of elements in the final set. The story problems and number-fact problems were presented orally, without props. Results indicate that children as young as 4 years of age have some success on the nonverbal problems, showing that they can transform sets by adding or subtracting elements. In contrast, children do not achieve comparable levels of success on the story problems or number-fact problems until 5 1/2 to 6 1/2 years of age. Moreover, throughout the age range tested, children performed better on nonverbal problems than on either story problems or number-fact problems. These results suggest that childrens earliest ability to add and subtract is based on experiences combining and separating sets of objects in the world and that this ability precedes the development of conventional verbal methods of calculating.