Math performance and its relationship to math anxiety and metacognition. (Report).
North American Journal of Psychology 11.3 (Dec 2009): p.471. (5397 words)
Angela M. Legg and Lawrence Locker Jr.
The current study assessed whether metacognitive skill moderates the effects of math anxiety on performance, reaction time, and confidence on a math task. Metacognition moderated math anxiety and predicted that performance would decrease as anxiety increased, except at high metacognition levels. Further, metacognition predicted confidence in accuracy such that individuals higher in metacognitive processing were more confident in their ability to correctly answer the problems. Psychological and educational implications are discussed.
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Math anxiety is defined as a general fear or tension associated with anxiety-provoking situations that involve interaction with math. In a world in which Eastern cultures reliably outperform Western cultures on math performance tasks, the outcomes associated with math anxiety have far reaching implications (Ginsburg, Choi, Lopez, Netley, & Chi, 1997; Siegler & Mu, 2008; Stevenson & Stigler, 1992). Math anxiety can lead to negative outcomes such as avoidance of college math courses and majors or avoidance of careers that involve frequent math use (Ashcraft, 2002; Chipman, Krantz, & Silver, 1992, Hembree, 1990). For these reasons, additional research on the implications of math anxiety and the cognitive mechanisms associated with math anxiety is essential.
The mechanisms contributing to successful mathematical thinking are complex and diverse, ranging from components operating within the memory system to those contributing to problem solving and use of cognitive strategies. One of the key cognitive mechanisms in math problem solving, and a significant area of research within the math cognition domain, is the utilization of the working memory system (Ashcraft & Kirk, 2001; LeFevre, DeStafano, Coleman, & Shanahan, 2005). A critical issue, therefore, is how and why, math anxiety modulates efficient cognitive processing, as it has been shown that anxiety can tax working memory to such an extent that even individuals with high math aptitude will perform poorly (Beilock & Carr, 2005). Although anxiety can impact processing in a number of ways, one proposal relevant to the current research is the view that anxiety may create a "dual-task" situation that strains working memory resources (Ashcraft, 2002; Ashcraft & Krause, 2007; Eysenck & Calvo, 1992). Individuals might, for example, devote attention to rumination on anxious thoughts that divert resources from the task of problem solving (e.g., devoting resources to worrying about performance rather than application of problem solving strategies).
Given that math anxiety can hinder performance even for individuals with high aptitude, it is important to investigate under what circumstances certain mechanisms are involved in the intrusion of anxiety in performance. A number of potential issues have been explored, such as individual differences in working memory capacity as well as the nature of the math problems themselves (Ashcraft & Krause, 2007; Beilock & Carr, 2005). However, a relatively unexamined area is how one's own thoughts about math ability or cognitive processing may or may not lead to deleterious effects of anxiety. Ashcraft (2002), for example, called for research examining consequences of math anxiety in relation to how individuals perceive their own math competence and performance when solving math problems. One means of exploring this aspect of math cognition is through an investigation of individual differences in metacognition.
Loosely defined, metacognition is often referred to as "thinking about thinking." The abstract nature of metacognition leads to difficulty in developing one all-encompassing, yet meaningful definition that lends itself to empirical investigation (e.g., Schoenfeld, 1992). Explanations of the construct also vary across disciplines, with educational research often viewing metacognition differently than psychological and cognitive domains. However, Schraw and Moshman (1995) proposed a fairly exhaustive description of the components involved in metacognition that is widely accepted in both the educational and psychological fields. These researchers defined metacognition as consisting of two domains; metacognitive knowledge and regulation of cognition. Each domain is comprised of three subdomains.
The first domain, metacognitive knowledge, encompasses all of the knowledge and insight possessed regarding what is already known about cognitions, according to Schraw and Moshman (1995). This domain basically refers to how aware someone is about their own cognitions or thoughts. There are three subdomains of metacognitive knowledge or awareness: declarative, procedural, and conditional awareness. Declarative knowledge is the knowledge about what factors influence learning and affect performance. Knowing that a good night's rest and healthy breakfast can impact test performance is an example of declarative knowledge. Procedural knowledge involves knowing how to perform tasks, using skills automatically, and using strategies efficiently. Driving, for example, benefits from enhanced procedural learning as the skills and strategies used to drive effectively often become automatic and more efficient as one gains more experience with them. Finally, conditional awareness includes knowledge of exactly when and why to use specific strategies and when and why to choose alternates. Conditional awareness might come into play when individuals complete a math task under conditions where time is short. In this situation, individuals will often have to compromise typical solving strategies and replace them with shortcuts to solve problems in order to save time.
The second domain Schraw and Moshman (1995) proposed is regulation of cognition. This domain is implicated in the control of thought processes. There are three subdomains associated with the regulation of cognition: planning, monitoring, and evaluation. Planning involves selecting the appropriate strategies to solve problems and to allocate resources in a manner that allows for efficient and effective problem solving. For example, planning procedures might involve making predictions, such as expecting certain test questions on a test and thus focusing study efforts on those specific topics, while spending less time on unanticipated test material. Whereas planning behaviors tend to occur early before a behavior begins, monitoring, the second subdomain of cognition regulation, is the present awareness of understanding and performance. This subdomain regulates checking behaviors during tasks and self-testing. Monitoring, for example, is employed when a student attempts to paraphrase a paragraph he or she just read, without looking at the page, in order to check for comprehension and retention. Finally, evaluation, the third subdomain, often occurs after a task is completed and involves appraising performance or the regulatory components of cognition. This subdomain can also involve re-evaluating goals or conclusions. For example, rewriting and editing a draft of a manuscript heavily involves the evaluation subdomain as the person must be able to take on the reader's perspective in order to reevaluate the efficacy of the writing.
From the above description, it is evident that the active application of metacognitive processes such as monitoring and planning could be quite beneficial in the context of math problem solving. However, if processes such as self-monitoring or evaluation of strategies were negative or directed to anxiety-focused thoughts or doubts, then metacognitive processing could conceivably lead to negative outcomes. For example, overt awareness of the correct strategies to apply problem-by-problem on a timed test such as the GRE would lead to better performance. However, ruminating on whether a prior problem was solved correctly or excessively questioning whether a problem solving strategy is correct could conceivably lead to hindered performance.
Although the literature reviewing metacognition as it relates specifically to math anxiety is fairly sparse, research does provide some indication of metacognitive processes or perceptions that do relate to math performance. Lucangeli, Coi, and Bosco (1997) conducted a study examining metacognition of math difficulty in elementary school children in relation to the nature of problems presented. Consistent with Ashcraft and Krause's (2007) explanations of characteristics associated with creating difficult math problems (number size, number of steps required to solve, etc.), the fifth graders viewed problems containing large numbers as more difficult than problems with smaller numbers. In this study, individuals were classified in terms of the extent to which they were higher or lower in metacognitive skill as well as the extent to which they were better or poorer problem solvers. Not surprisingly, those individuals classified as poor problem solvers demonstrated lower metacognitive awareness and made more errors when problem solving, suggesting a relationship between metacognition and effective problem solving.
There is also evidence that suggests a link between anxiety and metacognition, albeit within the verbal domain. Everson, Smodlaka, and Tobias (1994), for example, assessed metacognitive word knowledge and reading comprehension in college-age students. However, they found that individuals with low anxiety were better able to use metacognition in a positive way than their highly anxious counterparts. When anxiety was high, metacognition was associated with poorer performance. Although this experiment was conducted within the verbal domain, it may offer some indication of how metacognition might relate to anxiety and performance in other domains.
As the studies discussed indicate, there appears to be a link between math problem solving and metacognition as well as some relationship between anxiety and metacognition, at least within the domain of verbal processing (Everson et al, 1994; Lucangeli et al., 1997). However, no study to our knowledge has directly examined both metacognition and anxiety within the domain of math performance. The current study attempted to bridge this gap in the literature though an investigation of how individual differences specific to math anxiety, as well as the use of metacognitive skill, relate to math performance. That is, individuals were assessed in terms of both their pre-existing levels of math anxiety as well as their use of metacognition during processing. Math performance was defined in terms of both the accuracy and speed of problem solving. Participants' judgments of the accuracy of their performance were also assessed to determine whether differences in anxiety and metacognition relate not only to actual performance, but also to confidence in performance. If, as some research has indicated (Everson et al., 1994), high metacognition is related to more deleterious effects of anxiety, it would be expected that individuals who are high in metacognitive use and also experience high math anxiety levels would tend to display poorer accuracy. Alternatively, individuals with low math anxiety and high metacognition would be expected to display better accuracy.
We also predicted that the variables would be related to reaction time. Some research suggests that anxiety leads to avoidance behaviors where individuals try to solve problems as quickly as possible in an evasive attempt to end the anxiety-inducing task (Ashcraft, 2002). For this reason, we expected that individuals with high math anxiety and high metacognitive use would display shorter reaction times, whereas those with low math anxiety and high metacognitive use would display the longest reaction times. Low anxiety and high metacognition would presumably lead to greater utilization of checking behaviors, strategies, monitoring behaviors, and accurate evaluation behaviors, thus resulting in the longer time reaction times.
Confidence in perceived performance was also assessed. We predicted that participants with high anxiety and high metacognition would display lower perceptions of their ability to accurately perform relative to individuals with low anxiety.
Fifty-six Georgia Southern University undergraduates participated in this study for course credit in an Introduction to Psychology course. The mean participant age was 19.77 (SD = 2.45 yrs.). The participants consisted of 41 (73.20%) women and 15 (26.8%) men. Most of the participants were sophomores (48.20%); 26.80% were first-year students, 19.60% were juniors, and 5.40% were seniors. The majority of participants (92.20%) had completed at least three high school math courses and 80.30% of the participants had completed at least one college level math course.
The Revised Math Anxiety Rating Scale (RMARS; Plake & Parker, 1982) was used to assess participants' levels of math anxiety. This is a 24-item scale that assesses two factors (anxiety for learning math and anxiety due to evaluation of math performance) and has been shown to have good validity and reliability. Participants rate their level of anxiety on a 0-4 Likert-type scale (0 = no anxiety at all and 4 = extreme anxiety). Participants are asked to provide their level of anxiety during such tasks as "reading the word 'statistics,'" "working on an abstract mathematical problem," and "taking a final examination in a math class." The mean RMARS score for the participants in this study was 1.51 (SD = .68).
Metacognition was measured using the State Metacognitive Inventory (SMI; O'Neil & Abedi, 1996). The SMI is a measure of metacognition in the context of a task and was used as a measure of metacognitive application during math performance. The SMI is a 20-item scale assessing planning, checking, monitoring, and evaluating behaviors. Specifically, the SMI asks participants to respond to statements such as, "I checked my work while I was doing it," "I determined how to solve the task problems," and "I almost always knew how much of the task I had left to complete." A 1-4 Likert-Type scale is used with this measure (1 = did not engage in this behavior at all and 4 = engaged in the behavior "very much so."). Participants were instructed to reflect upon their thinking processes during the math task and indicate how much they engaged in various metacognitive processes. The mean SMI score for the participants in this study was 2.68 (SD = .48).
Following completion of the informed consent, participants completed the RMARS. They were then provided instructions concerning the math task. A modular arithmetic task was used because it has been found in previous literature (Beilock & Carr, 2005) to be robust to the effects of mathematical training and because the task is easy enough to solve without a calculator or pen and paper. Modular arithmetic involves judging whether the answer to a problem is a whole number or a fraction. Participants were presented problems such as 60-10 (mod 5). To accurately solve the problem, participants must subtract 10 from 60 and then divide 50 by 5. The resulting number (10) is a whole number so the original statement is true. An example of a computationally difficult problem utilized in the task is 1527-349 (mod 4). Participants were asked to indicate whether the solution to each problem was a whole number or not by indicating "Y" (yes, it is a whole number) or "N" (no, it is a fraction) on the keyboard. Prior to the experimental trials, participants completed seven practice trials. After completing the practice trials participants were asked whether they had any questions or required any clarification regarding the task.
In order to increase performance incentive, participants were informed prior to the task that the 10 participants with the highest scores on the math task would receive restaurant gift cards. The participants were told that both accuracy and reaction time would be factored into how well they performed on this task. That is, participants with the highest accuracy within the shortest amount of time would be classified as the top performers. Participants were then asked to complete the experimental math trials consisting of 20 modular arithmetic problems. Math problems were presented one at a time in randomized order on a computer screen using E-Prime (Schneider, Eschman, & Zuccolotto, 2002). After completing each problem, participants were prompted to judge their accuracy on the preceding problem (the preceding problem was removed from the screen when participants judged their accuracy). They did so by selecting a number on a seven-point Likert-type scale indicating their confidence that they provided the correct answer for the previous problem (1 = Not confident at all, 7 = Extremely confident). Participants were not provided feedback in terms of their actual performance. All participants completed the task individually.
After completing this math task, participants completed the State Metacognitive Inventory followed by a demographics survey that included questions about age, gender, ethnicity, and number of high school and college math courses taken. Participants were then debriefed and told that they had an equal chance of winning one of the 10 restaurant gift cards and that their performance on the math task would have no bearing on their chances of winning.
To assess whether demographic characteristics related to the dependent measures as covariates, six MANOVAs were conducted with accuracy, reaction time, and confidence entered as dependent variables. The independent variables analyzed were age (dichotomized into 19 and below, and 20 and above), ethnicity, gender, year in school, number of math courses taken in high school, and number of math courses taken in college. Six separate analyses were conducted in order to optimize the chances that any significant covariates would be identified. None of the independent variables significantly related to participants' performance on the modular arithmetic task in terms of accuracy, reaction time, or confidence ratings. As none of the MANOVAs produced significant results, no covariates were entered into the primary analyses.
The average proportion correct for the modular arithmetic task was .80 or 16 out of 20 problems correct (SD = .12). Participants had a mean reaction time for each problem on the math task of 13.11 seconds (SD = 4.83). The mean confidence rating was 5.32 (SD = .87).
Three multiple-regression equations were used to analyze the data. The first analysis assessed the relationship between the predictors and accuracy, the second assessed the relationship between the predictors and reaction time, and the third assessed the relationship between the predictors and confidence ratings. Centered SMI and RMARS scores and the interaction term were entered as predictors sequentially such that SMI and RMARS were entered in the first block and the interaction term entered in the second, according to the methodology set forth by Aiken and West (1991). Centered predictors (RMARS and SMI scores) and the interaction term were calculated according to the procedure created by Baron and Kenny (1986). ModGraph was used to graph the prediction equations (Jose, 2008).
The analysis for accuracy revealed that the overall regression model was significant, [R.sup.2] = .21, F (3, 55) = 4.66, p <.01. Math anxiety significantly predicted performance, B = -.06, [beta] = -.35, t (55) = -2.75, p <.01. Individuals with higher anxiety performed worse than those with low anxiety. State metacognition also predicted performance, B = .08, [beta] = .31, t (55) = 2.41, p <.05 in that higher metacognition was related to higher accuracy. A moderating relationship between metacognition and anxiety was also significant, B = .12, [beta] = .33, t (55) = 2.53 p < .05, in that at high anxiety levels, individuals performed increasingly worse as their SMI scores decreased. Accuracy, however, did not differ at low anxiety levels regardless of state metacognitions. Figure 1 illustrates the pattern of results.
[FIGURE 1 OMITTED]
The regression model for reaction time revealed no significant relationships. Participants at varying levels of anxiety and metacognitive skill showed no differential time performance based on these individual differences.
The overall regression equation for confidence was significant, [R.sup.2] = .18, F (3, 55) = 3.91, p < .05. The results revealed a relationship between metacognition and confidence, B = .78, [beta] = .43, t (55) = 3.28, p < .01 such that individuals with high levels of state metacognition reported greater confidence in their ability to correctly solve the math problems. No other significant relationships emerged. Figure 2 illustrates the pattern of results.
[FIGURE 2 OMITTED]
The results of this study demonstrate that metacognition has a moderating relationship with math anxiety that relates to accuracy in math performance. In addition, increased metacognition is associated with greater confidence in performance. In regard to the relationship between metacognition and anxiety, the results would suggest that individuals with higher anxiety benefit from higher levels of metacognition, as their math performance was similar to those individuals with low math anxiety. As noted, the literature investigating the relationship between metacognition and anxiety is rather sparse. However, some research has suggested that metacognition may have a negative impact on those individuals with higher anxiety (Everson et al., 1994). The opposite pattern was observed here. One possibility that should be considered is that the relationship of metacognition and anxiety may be largely state-dependent relative to such factors as the consequences of the outcome, the nature of the material presented as well as the general context.
For example, in this study, the modular arithmetic task was used. This task is used primarily to avoid practice effects relative to math backgrounds as well as be computationally within the abilities of the participants without a calculator. Further, by using a math task within the capabilities of the participants, we reduced the potential confound of assessing math competence rather than metacognitive skill and math anxiety levels. Thus, the task was not designed to necessarily exceed the capabilities of the participants. Furthermore, given that it was an experimental situation, the consequences of failure to perform were relatively minimal (e.g., as opposed to the GRE or SAT). It is therefore likely that participants perceived the math task as not being beyond their capabilities and the cost of not performing at peak was not as threatening.
Still, it should be noted that performance did decrease for some individuals with higher math anxiety. Thus, the task was sensitive to effects of math anxiety. However, the nature of the task was such that individuals who were high in use of metacognition, even if high in math anxiety, were able to effectively utilize the beneficial aspects of metacognition. Such aspects may have included checking behaviors, strategic use of problem solving, or effective deployment of strategies at appropriate times per the conditional awareness subdomain of Schraw and Moshman's (1995) metacognitive conception. If this is the case, these strategies potentially mitigated anxiety-related influences, possibly by allocating mental attention to metacognitive processes, rather than anxiety-related thoughts. However, it is possible that had the context been more analogous to a high-stress testing situation (e.g. the SAT or a final exam), the highly anxious individuals might have utilized metacognition in a negative fashion by ruminating on the situation and potential outcomes, or worrying about poor performance rather than planning or problem solving.
The notion that highly metacognitive individuals in the present study were utilizing these processes in a positive fashion is supported by the finding concerning the relationship between metacognition and judgments of performance. Overall, higher metacognition was associated with perceptions of better performance. Notably, a secondary analysis indicated that confidence in accuracy was positively correlated with actual performance on the task (R = .43, p = .001). It would appear that these participants were not only devoting mental resources to the task at hand in an efficient manner, but were generally more confident in the performance. Thus, it is possible that both the application of beneficial problem solving strategies, and associated confidence as a result of doing so, may have diminished or diverted attention away from negative anxiety-related cognitions.
An educational implication of this finding would be to advocate metacognitive training. Kruger and Dunning (1999) found evidence that addressing metacognitive processes such as strategy use and checking behaviors increased college students' ability to perform well on varying tasks. Furthermore, much of the educational literature suggests that metacognitive training is beneficial to individuals in elementary, middle and high school (Cardell-Elawar, 1995; Kramarski & Mevarech, 2003; Teong, 2003).
Notably, metacognitive training has been shown to be a very effective method in which to overcome mathematics problem-solving difficulties. Metacognitive training is usually based on the principals set forth by Polya (1945) and involves directing student and participant attention to metacognitive thinking such as strategy use, problem solving, and time and accuracy monitoring. Metacognitive training also involves encouraging individuals to monitor their confidence in their abilities or lack of confidence. Kramarski and Mevarech (2003), for example, examined students' performance interpreting a linear graph unit. Some students received metacognitive training whereas others received traditional teaching, either in groups or individually. Individuals who received the metacognitive training performed significantly better than those who received the traditional teaching method, regardless of whether they received the metacognitive training in groups or individually.
Furthermore, Kruger and Dunning (1999) showed that even if students are examined in terms of differences in high and low achievement, metacognitive training does have positive benefits, although greater benefits seem to occur for low-achieving groups. These researchers found that high achievers benefit most from apprehending the superiority of their own answers by viewing other individuals' responses to problems. However, individuals at low-achievement levels benefit from instruction regarding the skills necessary to correctly evaluate themselves as well as how to positively use metacognitive strategies.
Similarly, Cardell-Elawar (1995) examined elementary and middle school age children who were considered low-achievers in mathematics. In this study, individuals were randomly assigned to either receive traditional teaching or metacognitive training. The metacognitive training directed students to answer certain questions throughout the problem-solving process that related to metacognitive functioning such as, "Do I understand the words in this problem?" and "With what operations needed to solve this problem do I typically have difficulty completing?" Students receiving the metacognitive training significantly improved their performance compared to students in the control condition. Interestingly, the students in the metacognitive training group also exhibited improved attitudes toward mathematics. This finding supports the notion that one benefit of metacognition may be related to promoting feelings of self-efficacy.
An aim of future research, therefore, should be an examination of math anxiety and metacognition across a broader range of contexts, particularly in high-stress situations such as the SAT or GRE. Importantly, the efficacy of metacognitive training should be examined to determine whether such training could offset effects of anxiety even within such high-stress situations. We have argued that such benefits may occur by shifting processing resources from anxiety-related thoughts to the present problem, as well as perhaps leading to greater feelings of self-efficacy or confidence.
In terms of limitations of the current study, it should be noted that the design was quasi-experimental as metacognition and math anxiety were not manipulated factors. Future research should attempt to explore means by which to experimentally manipulate both metacognition and anxiety in order to make more robust claims regarding possible causal relationships among these variables. One method by which to manipulate math anxiety may be to use the methodology set forth by Beilock and Carr (2005) in which participants are videotaped, told that they have a partner who is relying on them to improve their performance, and that professors will be evaluating the videotapes in the future. Metacognition could experimentally be manipulated by providing some participants with metacognitive training prior to the task. Another way in which to obtain stronger control over metacognition would be to present some participants with metacognitive analysis questions throughout a problem solving task. For instance, if a participant is solving a math problem, at random times throughout the process, different questions might appear on the screen asking the participants to respond to various items such as, "How much time is remaining to complete this task?" or "Are there any other ways to solve this problem that might be more efficient? Future replication or extension of the current research should also be aimed at the identification of the specific metacognitions that may lead to better performance or buffer math anxiety effects. Experimental investigation of the various metacognitive skills influencing performance and math anxiety (checking, planning, monitoring, and evaluating) will provide a more complete picture of the beneficial nature of metacognition, as well as potentially provide more insight into processing mechanisms affected by math anxiety.
Also, the moderating pattern found in the current study (metacognitive skill moderating anxiety in relation to accuracy) may need to be interpreted with some caution due to some participants performing at ceiling. For example, as the task was intended to be within the range of capabilities for all participants, additional benefits of metacognition for low-anxiety individuals may not have been detected due to close to ceiling performance. But, as discussed above, benefits of metacognition may be largely a function of the nature of the problems or the context. Again, the findings presented here may only apply to math tasks in which difficulty levels do not exceed the capabilities of the individuals. Differing patterns may emerge in different contexts (e.g., high-stakes situations such as the SAT or GRE), although the proper use of metacognition may be of benefit in those situations as well. Finally, research should examine the extent to which the patterns obtained in the current study are also observed in other populations. For example, research might examine at what age metacognition begins moderating anxiety. Additionally, adult populations from both math-intensive fields such as computer science as well as non-math-intensive fields such as English education might be examined to determine if there are differences among various professional groups in how metacognition relates to math performance.
Research on math anxiety and metacognition is still in its infancy as the relevant variables, mechanisms, and outcomes are identified. The implications of math anxiety are well known, far-reaching and possibly contribute to the learning gap between Western and Eastern cultures. As basic research explores the components related to this issue, applicable interventions can be identified that mitigate the negative impact of math- related stress or anxiety. Metacognition may provide a viable means by which to approach the problems of math anxiety and math avoidance in our society.
Notes: We thank two anonymous reviewers for their valuable feedback and comments regarding this manuscript. We presented portions of this research at the Fourth Annual Meeting of the Georgia Psychological Society in Macon, GA on April 11, 2009.
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Author info: Correspondence should be sent to: Angela Legg, Georgia Southern University, Department of Psychology, PO Box 8041, Statesboro, GA 30460 email@example.com
Angela M. Legg and Lawrence Locker, Jr.
Georgia Southern University
Legg, Angela M., and Lawrence Locker Jr. "Math performance and its relationship to math anxiety and metacognition." North American Journal of Psychology 11.3 (2009): 471. InfoTrac Humanities & Education Collection. Web. 31 Jan. 2010