psychology

psychology

This article provides an overview of the relationships among working memory, math performance, and math anxiety. We provide examples from the mathematical cognition literature to show: the critical role of working memory in performing arithmetic and math; the relation- ship between math performance and math anxiety, espe- cially on standardized math achievement tests; and finally, the way that math anxiety compromises the functioning of working memory when people do arithmetic and math. We conclude with some predictions concerning the risk factors for math anxiety, and with some of the educational implications of this work. See Ashcraft and Ridley (2005) and Ashcraft, Krause, and Hopko (2007) for full-length treatments of these issues. Excellent summaries of the entire field of mathematical cognition can be found in Campbell (2005).

We begin with a statement concerning just one justifi- cation (of many) for this work. Math and science are in the headlines these days, with research-based reports about the relatively poor job American schools do in teaching math and science, and the depressingly substandard job many students are doing in mastering these topics. No one doubts the importance of math and science to the work- force in a technological society, or their importance in general to an educated populace. So there is a general, undeniable need for investigations about the learning and mastery of math. And from a disciplinary perspective, the rich complexity of math in all its facets suggests that it should be an interesting topic for cognitive psychology to address, and a critical one in any discussion of the rel- evance of cognitive psychology to education.

Working Memory and Math Performance Considerable evidence has appeared in the past 10 to

15 years concerning the vital role that working memory plays in mathematical cognition. In LeFevre, DeStefano, Coleman, and Shanahan’s (2005) view, the literature now supports a clear generalization concerning the important positive relationship between the complexity of arithmetic or math problems and the demand on working memory for problem solving. One aspect of this relationship involves the numerical values being manipulated, and one aspect examines the total number of steps required for problem solution. We take these in turn.

It is now clear that working memory is increasingly involved in problem solving as the numbers in an arith- metic or math problem (the “operands”) grow larger. The benchmark effect in this area is the problem-size effect, the empirical result that response latencies and errors in- crease as the size of the operands increases: For example, 6 7 or 9 6 will be answered more slowly and less accurately than 2 3 or 4 5 (see Zbrodoff & Logan’s 2005 review). Part of this effect, we have argued, is due to the structure of the mental representation of arithme- tic facts in long-term memory, and the inverse relation- ship between problem size and problem frequency—for example, in textbooks (e.g., Hamann & Ashcraft, 1986). That is, larger arithmetic problems simply occur less fre- quently, and hence are stored in memory at lower levels of strength (see Siegler & Shrager, 1984, for a comparable approach); this is similar in most respects to the standard word-frequency effect found in language processing re- search. A second part of the effect, documented in the

243 Copyright 2007 Psychonomic Society, Inc.

Working memory, math performance, and math anxiety

MARK H. ASHCRAFT AND JEREMY A. KRAUSE University of Nevada, Las Vegas, Nevada

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 litera- ture 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 relation- ships 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.

Psychonomic Bulletin & Review 2007, 14 (2), 243-248

M. H. Ashcraft, mark.ashcraft@unlv.edu

244 ASHCRAFT AND KRAUSE

past 10 years, is the increasing tendency for larger op- erand problems to be solved via some nonretrieval pro- cess, whether it be counting, reconstruction from known problems, or other strategies (see, e.g., Campbell & Xue, 2001; LeFevre, Sadesky, & Bisanz, 1996). Because non- retrieval processing is invariably found to be slower and more error prone than memory-based retrieval, the occur- rence of strategy-based trials will slow down overall re- sponse latencies, especially for larger problems. Critically for the present discussion, strategy- or procedure-based performance will rely far more heavily on the resources of working memory in comparison with performance based on relatively automatic memory retrieval.

We illustrate this with a series of experiments reported in Seyler, Kirk, and Ashcraft (2003). In this work, we tested college adults on the “basic facts” of subtraction—that is, the inverses of the addition facts 0 0 up to 9 9. As shown in Figure 1, there was a gently increasing problem- size profile on response latency up to 10 n problems, but then a dramatic increase in reaction times (RTs) be- ginning with 11 n problems; error rates jumped from below 5% to the 10%–22% range at the same point. The dramatic change in the performance profiles suggested strongly that the larger subtraction problems were being solved via strategies. To test this possibility, we repeated the study, asking participants to answer the question “How did you solve the problem?” after each trial. The reported incidence of strategy use matched the RT and error pro- files very closely; strategy use was reported an average of 3% of the time on small subtraction problems, but on 33% of the trials with large problems. On this evidence, simple

subtraction is heavily reliant on strategy use, a pattern that should disadvantage participants if they are laboring under limited working memory resources.

To document this final prediction, we tested simple subtraction in a dual task setting: Participants held two, four, or six random letters in working memory while per- forming the subtraction; they then had to report the letters in serial order. The dual task led to a significant decrement in performance, as measured by accuracy of letter recall. Importantly, this decrement was especially pronounced for the large subtraction problems, those that relied heav- ily on strategies rather than on retrieval. And the pattern was exaggerated when participants’ own working memory capacity was considered. There was substantially more in- terference with letter recall for the low-working-memory- span participants (a 56% error rate in the most difficult condition) than for the medium- or high-span groups (re- spectively, 46% and 31% error rates; Seyler et al., 2003, Figure 7). In short, there was an increasing cost of the dual task requirement for participants with lower work- ing memory capacity. Simple subtraction, an arithmetic operation introduced routinely in second grade, has a substantial working memory component to it, especially because even adults continue to rely heavily on strategy- based processing instead of memory retrieval. Such reli- ance disadvantages participants whose working memory is occupied by a secondary task, and also those whose working memory capacity is low.

The important point here is that strategy-based solu- tions are not just slower, but far more demanding on work- ing memory, whereas memory retrieval is usually found to be a fast and relatively automatic process, with little or no demand on working memory resources. Reports consistent with this generalization are now common—for example, work showing the dramatic decline in latencies and working memory involvement as a function of prac- tice on difficult math (Beilock, Kulp, Holt, & Carr, 2004; Tronsky, 2005).

Similarly, the number of steps in a problem solution is generally strongly correlated with response times, and with the working memory resources necessary for correct solutions; this is roughly analogous to the increase in pro- cessing load with an increase in the number of clauses in a sentence, for instance. As an example, Hecht (2002) found that a concurrent articulatory task disrupted addition tri- als performed via counting far more than it did trials per- formed via retrieval (see comparable results in a test of sequential adding by Logie, Gilhooly, & Wynn, 1994).

In our test of the relationship between number of steps and working memory (Ashcraft & Kirk, 2001), we rea- soned that the carry operation in addition should require additional working memory processing, because carrying adds yet another step to the processing sequence. We pre- sented participants with addition problems ranging from basic addition facts up to two-column additions; half of all problems required a carry (e.g., 27 14; see also Fürst & Hitch, 2000). The results were clear cut (see Ashcraft & Kirk, 2001, Figure 1). Carry problems were considerably slower than their noncarry counterparts, fully 1,200 msec slower for the largest problems. Likewise, carry problems

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Figure 1. Mean reaction time (RT) to simple subtraction facts from 0 0 to 18 9 by minuend; for example, in 14 8, the “minuend” is 14. From “Elementary subtraction,” by D. J. Seyler, E. P. Kirk, & M. H. Ashcraft, 2003, Journal of Experimental Psy- chology: Learning, Memory, & Cognition, 29, p. 1341, Figure 1. Copyright 2003 American Psychological Association. Adapted with permission.

WORKING MEMORY, MATH PERFORMANCE, AND MATH ANXIETY 245

invariably had higher error rates (from 5.2% to 9.4%) than their noncarry counterparts (0.2% to 2.1%).

It appears that working memory processing is integral to arithmetic and mathematics performance whenever a procedure other than direct memory retrieval is operating. That is, when simple one-column addition or multiplica- tion is being performed, the underlying mental process responsible is principally retrieval from memory, in which case working memory plays a minor role, at best. But when performance relies on algorithmic procedures— say, carrying—or when other reconstructive strategies are used, then working memory is crucial. Likewise, for multi- step problems, there is an increasing reliance on working memory as the number of steps increases (see, e.g., Ayres, 2001), and at points in problem solving when the need for retaining intermediate goals and values is highest (see, e.g., Campbell & Charness, 1990).

Math Performance and Math Anxiety Serious research on math anxiety began to appear in

the early 1970s, when a suitable objective instrument for measuring math anxiety became available. Since that time, scores of articles have appeared on the various psychometric properties of the original scale and its descendants, and on the relationships between math anxiety and a host of other characteristics. The best summary of this work remains the Hembree meta-analysis (1990), which, for the most part, is the source of the following correlations between math anxiety and various aspects of math performance.

The story told by the correlations is sad indeed. The higher one’s math anxiety, the lower one’s math learning, mastery, and motivation; for example, a math anxiety correlation of .30 with high school grades, .75 with enjoyment of math, .64 with motivation to take more math or do well in math, and .31 with the extent of high school math taken. The overall correlation between math anxiety and individuals’ math achievement, as measured by standardized tests, is .31. Thus, highly math-anxious individuals get poorer grades in the math classes they take, show low motivation to take more (elective) math, and in fact do take less math. They clearly learn less math than their low-anxious counterparts.

These correlations mean, simply but importantly, that as math anxiety increases, math achievement declines. This seemingly inherent relationship between math anxiety and achievement poses a genuine interpretive quandary: Is lower performance on a math task due to math anxiety or to lower mastery and achievement in math? Fortunately, our work suggests a partial way out of the quandary. That is, we collected scores from some 80 undergraduates on a math-anxiety assessment and also on the Wide Range Achievement Test (WRAT), a standard math achievement test. The correlation between math anxiety and the com- posite WRAT score was .35, very close to the value in Hembree’s (1990) meta-analysis. But we then rescored the WRAT performance, taking advantage of its line-by- line increases in difficulty (e.g., whole-number addition in Line 1, multiplication of fractions in Line 5, solving for two unknowns in Line 8). When the test is scored in this fashion, the impact of math anxiety is much clearer;

see Figure 2. Simple accuracy is at ceiling for all groups on the initial lines of the test, suggesting no evidence of lower achievement per se for math-anxious individuals on the whole-number arithmetic taught in elementary school (i.e., even high-anxious individuals can answer whole- number problems correctly). Likewise, when we gave un- timed paper-and-pencil tests of our whole-number arith- metic stimuli, we found no math-anxiety differences on accuracy, even though these same stimuli generated online anxiety effects in an RT task (Faust, Ashcraft, & Fleck, 1996). But group performance on the WRAT does start to diverge around Line 4 or 5: On the most difficult line of the test, the high-anxious group averages fewer than one in five problems correct. Thus, the lower achievement of math-anxious individuals seems limited to more difficult math, the math taught at or after late elementary school.

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