Mark H. Ashcraft

The relationships among working memory, math anxiety, and performance.

Individuals with high math anxiety demonstrated smaller working memory spans, especially when assessed with a computation-based span task. This reduced working memory capacity led to a pronounced increase in reaction time and errors when mental addition was performed concurrently with a memory load task. The effects of the reduction also generalized to a working memory-intensive transformation task. Overall, the results demonstrated that an individual difference variable, math anxiety, affects on-line performance in math-related tasks and that this effect is a transitory disruption of working memory. The authors consider a possible mechanism underlying this effect--disruption of central executive processes--and suggest that individual difference variables like math anxiety deserve greater empirical attention, especially on assessments of working memory capacity and functioning.

Cognitive arithmetic: a review of data and theory

The area of cognitive arithmetic is concerned with the mental representation of number and arithmetic, and the processes and procedures that access and use this knowledge. In this article, I provide a tutorial review of the area, first discussing the four basic empirical effects that characterize the evidence on cognitive arithmetic: the effects of problem size or difficulty, errors, relatedness, and strategies of processing. I then review three current models of simple arithmetic processing and the empirical reports that support or challenge their explanations. The third section of the review discusses the relationship between basic fact retrieval and a rule-based component or system, and considers current evidence and proposals on the overall architecture of the cognitive arithmetic system. The review concludes with a final set of speculations about the all-pervasive problem difficulty effect, still a central puzzle in the field.

Math Anxiety: Personal, Educational, and Cognitive Consequences

Highly math-anxious individuals are characterized by a strong tendency to avoid math, which ultimately undercuts their math competence and forecloses important career paths. But timed, on-line tests reveal math-anxiety effects on whole-number arithmetic problems (e.g., 46 + 27), whereas achievement tests show no competence differences. Math anxiety disrupts cognitive processing by compromising ongoing activity in working memory. Although the causes of math anxiety are undetermined, some teaching styles are implicated as risk factors. We need research on the origins of math anxiety and on its “signature” in brain activity, to examine both its emotional and its cognitive components.

The development of mental arithmetic: A chronometric approach☆

Abstract The development of mental arithmetic is approached from a mathematical perspective, focusing on several process models of arithmetic performance which have grown out of the chronometric methods of cognitive psychology. These models, based on hypotheses about the nature of underlying mental operations and structures in arithmetic, generate quantitative predictions about reaction time performance. A review of the research suggests a developmental trend in the mastery of arithmetic knowledge—there is an initial reliance on procedural knowledge and methods such as counting which is followed by a gradual shift to retrieval from a network representation of arithmetic facts. A descriptive model of these mental structures and processes is presented, and quantitative predictions about childrens arithmetic performance at various stages of mastery are considered.

Mathematics anxiety and mental arithmetic performance: An exploratory investigation

Abstract Two exploratory studies were conducted to determine if mathematics anxiety, as assessed by the Mathematics Anxiety Rating Scale (MARS), is related to the underlying mental processes of arithmetic performance. MARS scores were higher when the test was administered by computer, vs. the standard paper-and-pencil format, and were higher for female than male college students. Small but significant processing differences in simple addition and multiplication were found when subjects were divided by quartiles into anxiety groups. Much larger differences in processing speed and accuracy were found with complex addition problems and a set of difficult problems (e.g. 9 × 16 = 134, true or false) that tested all four arithmetic operations. Overall, the low anxiety group was consistently the most rapid and accurate, the medium high was consistently the slowest, and the high anxiety group the most prone to errors. The results suggest that genuine performance differences exist among the several levels of mathe...

Menatal addition: A test of three verification models

Three explanations of adults’ mental addition performance, a counting-based model, a direct-access model with a backup counting procedure, and a network retrieval model, were tested. Whereas important predictions of the two counting models were not upheld, reaction times (RTs) to simple addition problems were consistent with the network retrieval model. RT both increased with problem size and was progressively attenuated to false stimuli as the split (numerical difference between the false and correct sums increased. For large problems, the extreme level of split (13) yielded an RT advantage for false over true problems, suggestive of a global evaluation process operating in parallel with retrieval. RTs to the more complex addition problems in Experiment 2 exhibited a similar pattern of significance and, in regression analyses, demonstrated that complex addition (e.g., 14+12=26) involves retrieval of the simple addition components (4+2=6). The network retrieval/decision model is discussed in terms of its fit to the present data, and predictions concerning priming facilitation and inhibition are specified. The similarities between mental arithmetic results and the areas of semantic memory and mental comparisons indicate both the usefulness of the network approach to mental arithmetic and the usefulness of mental arithmetic to cognitive psychology.

Mathematics anxiety and working memory : Support for the existence of a deficient inhibition mechanism

A current theory of anxiety effects in cognition claims that anxiety disrupts normal processing within the working memory system. We examined this theory in the context of a reading task, for participants who were high or low in assessed mathematics anxiety. The task was designed to measure the ability to inhibit attention to distracting information and the effects of this ability on explicit memory performance. The results suggested that math-anxious individuals have a deficient inhibition mechanism whereby working memory resources are consumed by task-irrelevant distracters. A consequence of this deficiency was that explicit memory performance was poorer for high-anxious individuals. Based on these results, the recommendation is made that Eysenck and Calvos (1992) processing efficiency theory be integrated with Connelly, Hasher, and Zacks (1991) inhibition theory to portray more comprehensively the relation between anxiety and performance.

Property norms for typical and atypical items from 17 categories: A description and discussion

A description is presented of normative data for property responses to 121 words—17 category labels, three typical and three atypical members of each category, and the words “plant” and “animal.” The production frequency of properties is considered a measure of property dominance or semantic relatedness, and has been validated for the present data as a significant predictor of reaction time to property statements. Additional data include measures that support definitions of typicality in terms of property overlap between member and category, criteriality or dominance of the superordinate term, and the average number of properties generated to the category member. In reverse order, these three variables provide the best prediction of rated typicality. Average number of properties and superordinate dominance were the more important variables in this prediction, were virtually independent statistically, and were approximately equal in their contribution. Implications for semantic memory models are discussed.

Simple and Complex Mental Addition across Development.

Abstract Students in Grades 1, 4, 7, and 10 were timed as they solved simple and complex addition problems, then were presented similar problems in an untimed interview. A manipulation of confusion between addition and multiplication, in which multiplication answers were given to addition problems (3 + 4 = 12), revealed evidence for the hypothesized interrelatedness of these operations in memory only in 10th graders. The overall pattern of results suggests a strong reliance on memory retrieval, even in the first-grade group, with discernible time differences when “procedural” knowledge of carrying is required for problem solution. The results were judged consistent with a fact retrieval model which invokes explicit procedural information when problem difficulty is high or when processes like carrying and estimating magnitudes are required. In agreement with several other reports, the overall slowing of performance to larger problems is best explained in terms of normatively defined problem difficulty or associative strength in memory.

Working memory, automaticity, and problem difficulty.

Summary Two complimentary topics are of special interest in the study of cognitive skills, first the involvement of working memory resources in successful performance, and second the role of automaticity in the component processes of such performance. While these questions figure prominently in contemporary cognitive research, they have only recently begun to receive direct investigation in the area of mental arithmetic. In this chapter, we review the existing research that bears on these issues, then present two experiments. Experiment 1 focused on the deployment of working memory resources during arithmetic processing, using a standard dual-task method. Experiment 2 investigated automatic and conscious processing as revealed by a priming task. The results of both experiments are viewed in terms of the basic problem difficulty variable, and the relationship between this variable and manipulations that tap automatic and effortful aspects of performance. The chapter concludes with some remarks on the central construct of problem difficulty.