Working memory, worry, and algebraic ability
Introduction
Interest in the impact of emotional states on learning and cognition has a long history in developmental psychology (Fletcher, 1934, Mandler and Sarason, 1952). A widely held view is that emotional states (e.g., anxiety) are relatively stable within a learning domain (e.g., math) and affect problem solving similarly across the domain (i.e., emotion is a trait) (see Pnevmatikos & Trikkaliotis, 2013). In the context of math problem solving specifically, it has also been argued that math anxiety (MA) affects working memory (WM), which together affect math problem solving (Ashcraft & Krause, 2007). Nonetheless, others suggest that emotional states (anxiety), cognition, and learning relationships may differ within and across learning domains (i.e., emotion is a state) (Pnevmatikos and Trikkaliotis, 2013, Punaro and Reeve, 2012;). One difficulty in deciding between state/trait interpretations, as well as MA/WM relationships, in math is that previous relevant research has focused on arithmetic problem solving (Ashcraft and Faust, 1994, Friso-van den Bos et al., 2013, Miller and Bichsel, 2004, Raghubar et al., 2010). It is possible that MA/WM problem solving relationships may differ as a function of math domain (e.g., algebra). In the current study, we investigated the relationships among 14-year-olds’ algebraic anxiety/worry, algebraic WM, and algebraic problem solving abilities.
Although algebra is often considered as the generalization of arithmetic, it is also different from arithmetic cognitively. Algebra involves working with unknown values and requires a structural, rather than a procedural, understanding of mathematical expressions (Alibali et al., 2007, Christou and Vosniadou, 2012, Humberstone and Reeve, 2008, Kieran, 1992, Knuth et al., 2006). The absence of research examining the relationship between WM and MA in algebra is problematic for at least three reasons. First, claims for a link between MA and problem solving may have been overstated (i.e., claims may be restricted to arithmetic). Second, we have little understanding of how anxiety and/or cognitive factors interact to affect problem solving in complex/abstract mathematical domains. Third, from an applied perspective, algebra is typically introduced early in high school, and failure to grasp its intricacies may be a stumbling block to the acquisition of higher level math (Alibali et al., 2007, Stacey and MacGregor, 1999).
Some studies that have examined the MA–arithmetic association found that individuals with high MA are less accurate and slower at solving problems compared with students with low MA (Ashcraft and Faust, 1994, Ashcraft and Kirk, 2001, Faust et al., 1996), whereas other studies found no difference in problem solving accuracy between high and low MA groups (Cates & Rhymer, 2003). However, findings are difficult to reconcile because these studies used different MA and arithmetic problem solving measures. For example, Ashcraft and Faust (1994) used a true/false verification of addition, multiplication, and mixed arithmetic problems, Ashcraft and Kirk (2001) used solving addition problems within a dual task paradigm, and Cates and Rhymer (2003) required participants to solve complex arithmetic and simple algebraic equations. Indeed, the findings of Cates and Rhymer suggest that MA might not impair advanced mathematics problem solving. Moreover, Wu, Barth, Amin, Malcarne, and Menon (2012) examined math performance on a standardized general abilities test and found that MA has a pronounced effect on math reasoning but not on arithmetic computation subtasks. In sum, research suggests that MA may impair simple arithmetic performance; however, the relationship between MA and more advanced mathematics is unclear.
Nearly all indexes of MA are based on responses to questions about anxiety experienced solving math problems (e.g., “How stressed do you feel solving math problems?”; see Capraro, Capraro, & Henson, 2001) rather than the anxiety that occurs while solving math problems. The interpretation of questionnaire responses may be problematic for at least five reasons. First, assessing general perceptions of competence is open to influences (e.g., self-concept, gender stereotypes; see Bull et al., 2008, Monti et al., 2012). Second, commonly used MA measures (e.g., Mathematics Anxiety Rating Scale; Richardson & Suinn, 1972) assess math test anxiety and not MA per se. Third, items on MA scales focus on arithmetic referents and do not refer to more complex mathematical problems (e.g., algebraic problems) (Capraro et al., 2001). Fourth, few studies assess cognitive factors (e.g., memory, attention) that might affect self-reports. Fifth, MA questionnaires tend to tacitly assume that MA is an enduring anxiety (trait) rather than an anxiety state experienced solving particular problems. These limitations suggest that retrospective and global self-report measures should be interpreted with caution and that more proximal indexes of anxiety would be desirable (i.e., assessing anxiety while solving problems).
The claim that MA negatively affects problem solving is partially consistent with attentional control theory (ACT) proposed by Eysenck and colleagues (Eysenck and Calvo, 1992, Eysenck and Derakshan, 2011, Eysenck et al., 2007). ACT focuses on the effects of state anxiety and worry. Worry has traditionally been identified as the cognitive component of anxiety and is thought to be the aspect of anxiety that affects WM and performance (Deffenbacher, 1980, Eysenck and Calvo, 1992, Hayes et al., 2008, Liebert and Morris, 1967). In ACT, it is proposed that worry reduces WM capacity, which in turn negatively affects problem solving ability (Eysenck and Calvo, 1992, Eysenck and Derakshan, 2011, Eysenck et al., 2007). Worry is also claimed to have a facilitative effect (sometimes referred to as a motivation or arousal effect), resulting in more effortful processing that limits the impairment effect of worry and may increase performance effectiveness (Eysenck and Calvo, 1992, Eysenck and Derakshan, 2011, Eysenck et al., 2007). The inhibitory and facilitative effects of worry suggest a U-shaped function in performance. ACT also suggests that when WM capacity is reduced by task demands, the negative effects of anxiety on performance increase. In other words, worry, WM, and task demands interact to affect problem solving ability.
However, findings in support of ACT are mixed (Ashcraft and Kirk, 2001, Hoffman, 2010, Hopko et al., 1998, Miller and Bichsel, 2004). Mattarella-Micke, Mateo, Kozak, Foster, and Beilock (2011), for example, found that only high WM individuals, compared with those with low WM, are affected by MA. Miller and Bichsel (2004), in contrast, found that problem solving accuracy (assessed by math subtasks of the Woodcock–Johnson III Tests of Achievement) had a negative relationship with MA. In a second analysis, they found that problem solving accuracy was associated with high WM (compared with low WM), but this was examined only in high MA individuals. In contrast, Owens, Stevenson, Hadwin, and Norgate (2014) found that the effects of trait anxiety on performance of a combined math and general reasoning measure differed depending on WM capacity. High trait anxiety was associated with (a) impaired reasoning performance for low WM individuals, (b) no effect for moderate WM individuals, and (c) improved performance for high WM individuals (Owens et al., 2014). Differences in findings may be due to variation in the WM tests used in the studies; Miller and Bichsel (2004) assessed verbal and visual WM using language and paper folding tasks, Hoffman (2010) measured WM with an operation span task that involved a mathematical equation component (Turner and Engle, 1989, Unsworth and Engle, 2005), and Owens and colleagues (2014) used a battery of WM tests.
Although WM is hypothesized to affect all aspects of math problem solving (LeFevre, DeStefano, Coleman, & Shanahan, 2005), there is disagreement about how to define and/or assess WM (Libertus, Brannon, & Pelphrey, 2009). It has been argued that the concept of WM in math research needs to be better defined (Hicks et al., 2001, Raghubar et al., 2010). WM in math research is hypothesized to comprise visual–spatial and verbal components and a domain-general central executive component (Andersson and Lyxell, 2007, Gathercole et al., 2004, Holmes and Adams, 2006, Imbo and Vandierendonck, 2007, Swanson, 2006, Wilson and Swanson, 2001, Zheng et al., 2011). However, evidence suggests that WM may include more domain-specific abilities (Knops et al., 2006, Libertus et al., 2009). Indeed, Libertus and colleagues (2009), Oberauer, Süß, Schulze, Wilhelm, and Wittmann (2000), and Zheng and colleagues (2011) have all proposed that WM may have number/math-specific components. These domain-specific components may be less evident in children but appear in adolescents and adults (Libertus et al., 2009), suggesting possible age-related changes in domain-specific WM.
Raghubar and colleagues (2010) suggested that WM resources in math reasoning depend on several factors. First, the math domain itself may affect WM; standardized math tasks (e.g., the Applied Problems subtest from the Woodcock-Johnson III Tests of Achievement) often cover a broad range of math skills to assess mathematical cognition but are likely to be too general to assess math–WM relationships (Raghubar et al., 2010). Second, the strategies used to solve math problems may also affect WM because they place different demands on WM. For example, counting and performing operations in complex addition are more WM dependent than tasks such as multiplication, which involves recall from long-term memory (De Rammelaere et al., 1999, Imbo and Vandierendonck, 2007, Raghubar et al., 2010). Third, individuals’ skill level in a math domain may affect WM. For example, arithmetic is associated with visual–spatial WM in young children but tends to be associated with verbal WM in adolescents and adults. Raghubar and colleagues (2010) suggested that the acquisition of new math skills may depend on visual–spatial WM initially but become dependent on verbal WM with the acquisition of expertise.
Compared with arithmetic, algebraic competence depends more on abstract reasoning and is thought to reflect formal reasoning ability (Piaget and Inhelder, 1969, Tolar et al., 2009). Arithmetic problem solving largely involves numerical operations; algebra, by contrast, is an abstract structural representation of numerical relations that partly builds on arithmetic principles (Knops et al., 2006, Tolar et al., 2009).
The concept of equivalence differs in algebra and arithmetic—a possible reason for the negative transfer between arithmetic and algebra (Humberstone and Reeve, 2008, Jones et al., 2012, McNeil and Alibali, 2005, McNeil et al., 2010). Studies of algebra have identified difficulties with the equivalence sign, symbolized as “=”, as a major stumbling block in the acquisition of algebra (Alibali et al., 2007, Kuchemann, 1981). There are two interpretations of the meaning of the equivalence sign; one is operational and the other is relational (Alibali et al., 2007, Kieran, 1981). An operational view of the equivalence sign indicates the answer or the end of the problem (Humberstone and Reeve, 2008, Kieran, 1981, Kuchemann, 1981, McNeil and Alibali, 2005). In relational understanding, the sign suggests that the left side of the problem is equivalent to the right side (Humberstone and Reeve, 2008, Kieran, 1981, Kuchemann, 1981). Students with an operational understanding tend to read equations from left to right, which is less sophisticated and is similar to arithmetic processing (Humberstone and Reeve, 2008, Kuchemann, 1981, McNeil and Alibali, 2005). Conversely, the relational understanding of equivalence is recognized as more advanced algebraic thinking (Kieran, 1981, Kuchemann, 1981). Students’ algebraic ability can be assessed using these properties of the equivalence sign. Equations that have properties that favor an operational or relational view would be recognized as “easy” or “hard” algebraic equations, respectively.
The current study examined the role of WM and worry in algebraic problem solving. A sample of 14-year-old female students completed algebraic judgment/worry, algebra problem solving, and algebraic WM tasks. MA appears to be uniformly higher in females than in males (see Kargar et al., 2010, Ma, 1999, Ma and Xu, 2004). Indeed, Devine, Fawcett, Szűcs, and Dowker (2012) found that after controlling for test anxiety, the association between MA and math performance disappears for males but not for females.
In an algebraic judgment/worry task, students made same/different judgments about pairs of algebraic problems that differed from each other in their equation form, after which they rated how worried they felt while making judgments. In the algebraic problem solving task, students mentally solved algebraic problems that varied in difficulty. To avoid issues associated with domain-relevant math–WM relationships, an algebraic span task assessed algebraic WM. Basic reaction time (RT), nonverbal IQ (Raven’s matrices), and standardized math ability (Woodcock–Johnson III Tests of Achievement) were also assessed. Several studies have found that RT is related to arithmetic abilities (Bull and Johnston, 1997, Floyd et al., 2003, Kail, 2007). In addition, general intelligence has been found to be associated with math performance (Rohde & Thompson, 2007) as well as with working memory span (Conway, Kane, & Engle, 2003).
We used latent profile analysis to identify different worry–WM subgroups to examine claims from ACT. We were interested in how algebraic problem solving performance differed among identified groups to assess the relationships among worry, WM, and algebra performance. We anticipated that we would identify four subgroups (high/low WM × high/low worry) using latent profile analysis. We hypothesized that the WM–worry profiles would predict algebraic problem solving accuracy. According to ACT, the effect of worry on performance varies depending on WM capacity, so we expected that low WM with high worry would be associated with low/poor algebraic performance compared with profiles characterized by low WM with low worry. High WM profiles, with low or high worry, were predicted to be associated with high algebraic performance. Finally, we hypothesized that these worry–WM profiles would predict algebraic performance even after we statistically controlled for processing speed, nonverbal IQ, and general math performance.
Section snippets
Participants
A sample of 80 female students (mean age = 14 years 7 months, SD = 8 months) attending girls’ schools in a large Australian city participated. The sample consisted of students from diverse multicultural and socioeconomic backgrounds, as is typical of Australian urban high schools. According to school personnel, none of the students had identified learning difficulties, and all had normal or corrected-to-normal vision. Participants were excluded from analyses if they performed below chance accuracy on
Initial analyses
Descriptive statistics are reported in Table 2, Table 3, Table 4. Algebraic problem solving performance was correlated with both the Calculations and Applied Problems subtests of the Woodcock–Johnson III, indicating that the task was related to, but distinct from, general math ability.
To determine whether judging the equivalence equation pairs affected worry ratings, a 2 (Difficulty: easy or hard) × 2 (Equivalence: equivalent or nonequivalent) analysis of variance (ANOVA) was conducted using
Discussion
Our study sought to identify algebraic worry–WM profiles in a sample of adolescent girls and to determine the relationship between these profiles and algebraic problem solving performance. Four subgroups were identified from LPA that differed in worry and WM. However, whereas the HM/LW and LM/HW groups were identified, the other two groups had moderate WM and differed only in terms of worry, namely, MM/LW and MM/HW. The profiles suggest that WM and worry levels covary. Consistent with previous
Conclusion
In this study, we examined algebraic worry and WM in female high school students. We identified four groups that differed in worry level–WM capacity relations that were differentially related to algebraic problem solving ability. More specifically, we found a nonlinear relationship between algebraic performances and differences in worry level and WM capacity, consistent with predictions derived from ACT. Nevertheless, we acknowledge that our research is based on a static model of the
Acknowledgments
We thank the editor and three anonymous reviewers for their helpful and insightful comments; the article is much improved as a result of their input. We also thank our colleagues in the Cognitive and Neuropsychological Development Laboratory in Psychological Science at the University of Melbourne for comments on earlier drafts of the article.
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