Children's mathematical performance: five cognitive tasks across five grades.

Children in elementary school, along with college adults, were tested on a battery of basic mathematical tasks, including digit naming, number comparison, dot enumeration, and simple addition or subtraction. Beyond cataloguing performance to these standard tasks in Grades 1 to 5, we also examined relationships among the tasks, including previously reported results on a number line estimation task. Accuracy and latency improved across grades for all tasks, and classic interaction patterns were found, for example, a speed-up of subitizing and counting, increasingly shallow slopes in number comparison, and progressive speeding of responses especially to larger addition and subtraction problems. Surprisingly, digit naming was faster than subitizing at all ages, arguing against a pre-attentive processing explanation for subitizing. Estimation accuracy and speed were strong predictors of children's addition and subtraction performance. Children who gave exponential responses on the number line estimation task were slower at counting in the dot enumeration task and had longer latencies on addition and subtraction problems. The results provided further support for the importance of estimation as an indicator of children's current and future mathematical expertise.

Three brief assessments of math achievement.

Because of wide disparities in college students' math knowledge-that is, their math achievement-studies of cognitive processing in math tasks also need to assess their individual level of math achievement. For many research settings, however, using existing math achievement tests is either too costly or too time consuming. To solve this dilemma, we present three brief tests of math achievement here, two drawn from the Wide Range Achievement Test and one composed of noncopyrighted items. All three correlated substantially with the full achievement test and with math anxiety, our original focus, and all show acceptable to excellent reliability. When lengthy testing is not feasible, one of these brief tests can be substituted.

Cognitive processes of numerical estimation in children.

We tested children in Grades 1 to 5, as well as college students, on a number line estimation task and examined latencies and errors to explore the cognitive processes involved in estimation. The developmental trends in estimation were more consistent with the hypothesized shift from logarithmic to linear representation than with an account based on a proportional judgment application of a power function model; increased linear responding across ages, as predicted by the log-to-lin shift position, yielded reasonable developmental patterns, whereas values derived from the cyclical power model were difficult to reconcile with expected developmental patterns. Neither theoretical position predicted the marked "M-shaped" pattern that was observed, beginning in third graders' errors and fourth graders' latencies. This pattern suggests that estimation comes to rely on a midpoint strategy based on children's growing number knowledge (i.e., knowledge that 50 is half of 100). As found elsewhere, strength of linear responding correlated significantly with children's performance on standardized math tests.

Working memory, math performance, and math anxiety.

The cognitive literature now shows how critically math performance depends on working memory, for any form of arithmetic and math that involves processes beyond simple memory retrieval. The psychometric literature is also very clear on the global consequences of mathematics anxiety. People who are highly math anxious avoid math: They avoid elective coursework in math, both in high school and college, they avoid college majors that emphasize math, and they avoid career paths that involve math. We go beyond these psychometric relationships to examine the cognitive consequences of math anxiety. We show how performance on a standardized math achievement test varies as a function of math anxiety, and that math anxiety compromises the functioning of working memory. High math anxiety works much like a dual task setting: Preoccupation with one's math fears and anxieties functions like a resource-demanding secondary task. We comment on developmental and educational factors related to math and working memory, and on factors that may contribute to the development of math anxiety.

The effects of anxious responding on mental arithmetic and lexical decision task performance.

Anxiety-related responding and skill deficits historically are associated with performance-based problems such as mathematics anxiety, yet the relative contribution of these variables to substandard performance remains poorly understood. Utilizing a 7% carbon dioxide (CO2) gas to induce anxiety, the present study examined the impact of anxious responding on two performance tasks, mental arithmetic and lexical decision. Independent variables included math anxiety group, gender, and gas condition. Dependent variables included task performance and physiological and self-report indices of anxiety. A total of 64 university undergraduate students participated. Physiological and verbal-report measures of anxiety supported the utility of 7% carbon dioxide-enriched air as an anxiety-inducing stimulus. Behavioral disruption on performance tasks, however, did not differ as a function of carbon dioxide inhalation. Performance did differ as a function of math anxiety. High math anxious individuals generally exhibited higher error rates on mathematical tasks, particularly on tasks designed to measure advanced math skill and those requiring working memory resources. These findings are discussed with reference to processing efficiency theory, discordance among anxiety response systems, and the intricacies associated with skill measurement.

Elementary subtraction.

Four experiments examined performance on the 100 "basic facts" of subtraction and found a discontinuous "stair step" function for reaction times and errors beginning with 11 - n facts. Participants' immediate retrospective reports of nonretrieval showed the same pattern in Experiment 3. The degree to which elementary subtraction depends on working memory (WM) was examined in a dual-task paradigm in Experiment 4. The reconstructive processing used with larger basic facts was strongly associated with greater WM disruption, as evidenced by errors in the secondary task: this was especially the case for participants with lower WM spans. The results support the R. S. Siegler and E. Jenkins (1989) distribution of associations model, although discriminating among the alternative solution processes appears to be a serious challenge.