Download - The Gender Confidence Gap in Fractions Knowledge: Gender Differences in Student Belief–Achievement Relationships

The Gender Confidence Gap in Fractions Knowledge: GenderDifferences in Student Belief–Achievement Relationships

John A. RossUniversity of Toronto

Garth ScottKawartha Pine Ridge DSB

Catherine D. BruceTrent University

Recent research demonstrates that in many countries gender differences in mathematics achievement have virtuallydisappeared. Expectancy-value theory and social cognition theory both predict that if gender differences in achievementhave declined there should be a similar decline in gender differences in self-beliefs. Extant literature is equivocal: thereare studies indicating that the male over female advantage in self-efficacy and beliefs about math learning is as strongas ever and there are studies reporting an absence of gender differences in belief. Using data from 996 grades 7–10Canadian students, we found that gender differences in beliefs continued, even though gender differences in achieve-ment were near zero. Gender differences, favoring males, were larger for self-beliefs (math self-efficacy and fear offailure) and weaker for functional and dysfunctional beliefs about math learning. There were also gender differences inthe structure of a model linking beliefs about math, beliefs about self and achievement.

The achievement gap refers to a persistent advantage ofmales over females in student achievement in mathematicsthat begins in the early years and continues into high schooland postsecondary education. Research syntheses con-ducted 20 years ago (Friedman, 1989; Hyde, Fennema, &Lamon, 1990) found that the achievement gap has beengetting smaller over time, and some recent research,reviewed below, suggests that the gap has vanished entirely.The confidence gap refers to a similar male over femaleadvantage in student self-confidence and willingness toengage in mathematical tasks. In contrast with reporteddeclines in achievement, the confidence gap appears tocontinue, despite predictions that the confidence andachievement gaps would decline together (Falco & Crethar,2008; Linver & Davis-Kean, 2005). In this article, we drawon social cognition theory and recent research on genderdifferences in mathematics learning to frame an investiga-tion into the relationship between gender differences inmathematics achievement and beliefs about self and math-ematics learning.

Gender differences in self-beliefs and beliefs in math-ematics learning have short- and long-term implications. Inthe short term, beliefs affect achievement (as shown in theliterature reviewed below): A gender gap in affect mightexplain why some studies find gender differences inachievement. In the long term, beliefs influence studentwillingness to take advanced courses in mathematics(Ansell & Doerr, 1996) and to access careers that requiremathematical training.

Social cognition theory (Bandura, 1997), particularly theconstruct of self-efficacy, provides a mechanism that wouldaccount for changes in student confidence taking place atthe same time as changes in achievement. Self-efficacy isan expectancy about future performance, a set of domain-specific self-beliefs about one’s ability to organize andexecute the actions required to perform particular tasks.Self-efficacy contributes to higher achievement (demon-strated in the literature review below), and higher achieve-ment is interpreted by students as evidence that they havethe ability to perform similar tasks in the future. The reviewof Usher and Pajares (2008) found that mastery experiencesare the most powerful source of self-efficacy information.The reciprocal reinforcement of achievement and self-efficacy produces an upward or downward spiral in whichchanges in one stimulate and sustain changes in the other.From this perspective, if females are becoming as success-ful as males on mathematics tasks, the self-beliefs offemales and other facets of their belief systems should be assupportive of mathematics learning as the self-beliefs andbelief systems of males. If gender differences in achieve-ment are declining, then gender differences in confidenceshould also be declining. The aim of our research was toinvestigate this proposition.

Literature ReviewGender and Mathematics Achievement

Some studies, especially those conducted in the Nether-lands (Seegers & Boekaerts, 1996; Veenstra & Kuyper,

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2004), continue to find that males outperform females onmeasures of mathematics performance, especially on moredifficult items (Duffy et al., 1997). More frequently,gender differences in achievement are near zero (Else-Quest, Hyde, & Linn, 2010; Lloyd, Walsh, & Yailagh,2005; Pajares & Graham, 1999; Silver, Strutchens, &Zawojewski, 1996; Stringer & Heath, 2008). Gender pat-terns vary across countries. For example, analysis of theNational Assessment of Educational Progress data, 1990–2003, found that gender differences in achievement weresmall but consistent over time in the United States,growing larger in the upper grades and with White, eco-nomically advantaged and high-ability students (McGraw,Lubienski, & Strutchens, 2006). Canadian data reported inProgramme for International Student Assessment (PISA)found a male achievement advantage in 1999 (Edgerton,Peter, & Roberts, 2008) and in 2003 (Bassani, 2008; Else-Quest et al., 2010). But studies using samples drawn fromsingle Canadian provinces show a trivial male advantage,typically around .05 standard deviations (SDs), or no dif-ference at all (Lloyd et al., 2005; Ma & Klinger, 2000;Randhawa & Hunter, 2001; Rogers et al., 2006; Ross &Kostuch, 2011; Stringer & Heath, 2008).

Gender differences in achievement are moderated byassessment methods. Females receive higher report cardgrades in mathematics than males, even when there are nogender differences on standardized mathematics tests(Kenney-Benson, Pomerantz, & Ryan 2006; Lekholm &Cliffordson, 2009).A plausible explanation for the discrep-ancy in assessments is that teachers reward females withhigher grades than their test scores warrant because femalesexert more effort on assigned tasks (Lekholm & Clifford-son, 2009) and are more compliant with classroom rules.Females tend to score better on open-ended than onmultiple-choice items; the reverse is the case for males,although findings are mixed (Willingham & Cole, 1997).Gender, Beliefs About Self, and Achievement

In social cognition theory, self-efficacy mediatesbetween goals and actions. Self-efficacy contributes tohigher achievement in mathematics (e.g., Bandura, 1997;Borman & Overman, 2004; Bussiere, Cartwright, &Knighton, 2004; Kenney-Benson et al., 2006; Lee, 2006;Marsh, Dowson, Pietsch, & Walker, 2004; Pajares &Graham, 1999; Stringer & Heath, 2008). The key mecha-nisms linking self-efficacy and achievement are that stu-dents with high self-efficacy, in contrast with students withlow self-efficacy, adopt mastery goal orientations (Schunk,1996), persist through obstacles (Multon, Brown, & Lent,1991), and maintain better control of their emotions(Bandura, 1997). Achievement is further strengthened

through the links of self-efficacy to causal attributions (highself-efficacy students attribute success to ability) and otherpsychological constructs related to achievement such asmathematics anxiety and mathematics self-concept(Pajares, 1996). Because high self-efficacy students believethat they will be successful, they are less likely to bediscouraged by fear of failure, a construct inversely relatedto achievement (Caraway, Tucker, Reinke, & Hall, 2003;Eaton & Dembo, 1997; Heinze, Reiss, & Rudolph, 2005;Rao, Moely, & Sachs, 2000).

Pajares’ (1996) review of self-efficacy research foundthat males are consistently more confident than femalesabout their mathematical ability. More recent research ismixed: Some studies continue to find mathematics self-efficacy to be higher in males than females (Bussiere et al.,2004; Fredricks & Eccles, 2002; Friedel, Cortina, Turner,& Midgley, 2007; Preckel & Freund, 2005), while othersreport no gender differences (Kenney-Benson et al., 2006;Rao et al., 2000; Tapia & Moldavan, 2007). Because therelationship between self-efficacy and achievement isreciprocal, if the achievement gap disappears, it is possiblethat the confidence gap would decline and perhaps evendisappear. In an analysis of national data from the Trendsin Mathematics and Science Study and the PISA 2003,Else-Quest et al. (2010) found that countries with genderdifferences in achievement also had gender differences inmathematics confidence and related self-beliefs; correla-tions were in the r = .30s and .40s. The achievement–beliefs relationship is positively correlated on anindividual student basis; however, when we move to largerscale analyses, they have been found to be negatively cor-related (Shen & Tam, 2008). No studies have reportedgender differences for fear of failure, but given the corre-lation of fear of failure with self-efficacy, it is likely thatgender differences will be in the same direction.Gender, Beliefs About Mathematics, and Achievement

General attitudes toward mathematics, such as liking thesubject, are associated with higher achievement (Andersonet al., 2006; Silver et al., 1996). Researchers who focus onspecific beliefs about mathematics and its learning reportthat students tend to hold beliefs that impede success onmathematical tasks. These dysfunctional beliefs includethe assumption that mathematical ability is fixed (Schoe-nfeld, 1989), that problems are solved quickly or not at all(Lampert, 1990; Schommer-Aitkins, Duell, & Hutter,2005), that mathematical knowledge is separate fromknowledge in other domains (Buehl & Alexander, 2005),that one has to use all the numbers provided by a problemto find its solution (Hart, 1993), that some problemsrequire only calculation while others require thinking

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(Callejo & Vila, 2009), among others. Less attention hasbeen devoted to functional beliefs about mathematics,such as participation in student discussions helps one learn(Jansen, 2006). Some studies have reported that malestend to have more positive attitudes toward mathematicsthan females (Ercikan, McCreith, & Lapointe, 2005), butgender differences on specific beliefs about mathematicsand its learning reported to date have been very small(Schoenfeld, 1989; Tapia & Moldavan, 2007).Gender and Affect–Achievement Relationships

Although there is considerable research investigatinggender differences in mathematics achievement and affect,little attention has been given to whether there are genderdifferences in the relationships among achievement andaffect. Seegers and Boekaerts (1996) found the relation-ship between self-beliefs and achievement to be the samefor males and females, in a sample of 10- to 11-year-oldDutch children, as did Meece, Wigfield, and Eccles (1990)for grade 7–9 American students.Explanations for Discrepancies Between Beliefsand Achievement

Stereotypes that females lack mathematical ability arepersistent, even with growing evidence that there are few ifany differences in recent mathematics achievement.Gender differences in confidence occurring simultaneouslywith gender equivalence in achievement have been reportedin other stereotypical male domains such as science (Har-gittai & Shafer, 2006), engineering (Chachra & Kilgore,2009), and technology (Andre, Whigham, Hendrickson, &Chambers, 1999). According to social cognitive learningtheory, girls who observe women in their culture or societynot becoming engineers or scientists or mathematiciansavoid, and are anxious about, these subjects because theyare perceived as outside the realm of what is possible(Bussey & Bandura, 1999). Females who hold stereotypicalbeliefs about mathematics as a male domain have lowerperformance on examinations than females who do not holdsuch views and are less likely to seek careers requiringmathematics competence (Kiefer & Sekaquaptewa, 2007;Schmader, Johns, & Barquissau, 2004). The discrepancybetween confidence and achievement might be attributableto support for gender stereotyping.

Social cognition theory also provides a second explana-tion for the discrepancy between confidence and achieve-ment, despite their correlation: Self-efficacy beliefs are aconsequence of student reflection on their achievement. It isnot the absolute level of performance that matters but thestudents’ interpretation of it. For example, the runner whocomes last in the race may view her performance positivelyif it establishes a personal best. The misalignment of con-

fidence and achievement might be based on the internalcomparisons that females make between their perceivedperformance in mathematics and in subjects emphasizingliteracy skills.

Research QuestionsOur study was guided by two research questions:1. To what extent do gender patterns in mathematics

achievement on fractions tasks match gender patterns inself-beliefs and beliefs about mathematics learning? Wepredicted that gender differences would be near zero, but ifthere were differences, males would score higher thanfemales. Gender differences might be visible in oursample because we measured performance on number andoperations tasks associated with gender differences inNational Assessment of Educational Progress (McGrawet al., 2006), and the sample had a high proportion ofWhite students that were relatively advantaged economi-cally. However, individual ethnic and socioeconomicstatus information was not available. We anticipated thatgender differences in affect would be aligned with genderdifferences in achievement, that is, that the differenceswould be small or nonexistent and, if present, would favormales over females. Specifically, we predicted that maleswould have higher self-efficacy and lower fear of failurethan females. We also anticipated that males wouldexpress greater support than females for functional beliefsabout mathematics (i.e., beliefs that contribute to studentlearning) and would hold dysfunctional beliefs aboutmathematics less strongly than females. In contrast, weanticipated that females would report exerting greatereffort in mathematics class than males, in line with previ-ous findings about the greater compliance of females thanmales with classroom requirements (Cole, 2010).

2. To what extent is the relationship between studentbeliefs and achievement on fractions tasks moderated bygender? Because prior research was limited on this issue,we made no specific predictions.

MethodSample

Complete data sets were obtained from 996 grade 7–10students (aged 13–16 years) (N = 500 males and 496females) participating in a technology-based unit on frac-tions in 2007–2008. Eighty-two percent of the sample wasin grades 7–8, with the remainder in grades 9–10. Thespecific features of the fractions instruction are notdescribed because they are not relevant to the researchquestions addressed here (see Bruce & Ross, 2009 forinformation on the fractions instruction).The students were


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drawn from two school districts, one public and one Catho-lic, sharing the same geographic area in central Ontario,Canada. The two districts served virtually identical studentpopulations: 98% were Canadian born, less than 1% spokea language other than English at home, 26% were identifiedas special needs, and average family income in the districtwas near the mean for the province of Ontario.Instruments

The instruments used in the study are in Appendix A(surveys) and Appendix B (achievement). Studentachievement consisted of five fractions items on the pretestand seven on the posttest. Pretest and posttest were onemonth apart. The items included procedural tasks, forexample, “Write two fractions that are equivalent to16/18” and conceptual tasks, for example, “2/10 is lessthan 2/5. How do you know?” All items were scored 0–2by trained markers; chance adjusted inter-rater reliabilityon a sample of items was high (kappa = .86 and .92 for thetwo tests). Student achievement was operationalized ateach test occasion as the mean item score.

Student self-beliefs consisted of three measures. Fouritems from Ross, Hogaboam-Gray, & Rolheiser (2002)measured mathematics self-efficacy (e.g., “as you workthrough a mathematics problem how sure are you that youcan . . . solve the problem”; there were six responseoptions, anchored by “not sure” and “really sure”). Fouritems from Turner, Meyer, Midgley and Patrick (2003)measured the fear of failure (e.g., “I worry a lot aboutmaking errors on my math work”; there were six responseoptions, anchored by “not at all true” and “very true”). Fiveitems from Ross et al. (2002) measured self-reported effort(e.g., “how hard are you working to learn about math?”).There were six response options, anchored by “not hard atall” and “as hard as I can.”

Functional beliefs about mathematics learning consistedof three statements about participating in mathematicaldiscussions from Jansen (2006) and Schoenfeld (1989).The items used Likert scales, and all were negativelyworded (i.e., agreement with the item indicated a low scoreon the construct; the coding on these items was invertedprior to analysis); for example, “I just panic and can’t thinkstraight if the teacher asks me share my ideas.” There weresix response options, anchored by “strongly agree” and“strongly disagree.” Dysfunctional beliefs consisted of sixitems from Schommer-Aitkins et al. (2005), measuringbelief in quick/fixed learning (i.e., that learning occursquickly or not at all and that intelligence is fixed rather thanincremental); for example, “if I cannot understand some-thing quickly, it usually means I will never understand it.”There were six response options, anchored by “strongly

agree” and “strongly disagree.” All belief items were mea-sured at the pretest.Analysis

We compiled means, SDs, and correlations for eachvariable. We addressed research question 1 by conductinga multivariate analysis of variance of all affect variables(General Linear Model [GLM] in SPSS/PASW 17.0 [IBM,Armonk, North Castle, New York, USA]), followed byunivariate analysis with sequential Bonferroni adjustmentof alpha levels. We conducted a mixed analysis of varianceof achievement data in which pre- and post-fractionsachievement were repeated measures and gender was abetween-subjects factor. We represented the differencebetween males and females as effect sizes (Cohen’s d).The smallest detectable difference for our sample with twoequal groups, 80% power and 5% type I error was d = .18or r = .09 (Dennis, 1994).

To address the second research question, we used struc-tural equation modeling (AMOS 7.0). We divided oursample into two randomly selected sets. The first was anexploration sample in which we developed our model. Thesecond was a validation sample in which we confirmed themodel developed in the exploration sample. Using thisstrategy, we reduced the possibility that the trimming andbuilding of our structural models would be distorted byunique characteristics of the sample. After developing andvalidating the model linking affect to achievement, weexamined its fit for males and females. Goodness of fit wasdetermined by root mean square error of approximation(RMSEA), <.08 (Arbuckle & Wothke, 1999), and chi-square/degrees of freedom (df), <3.0 (Kline, 2005). Bothgoodness-of-fit indices consider the number of parametersin the model, rewarding parsimony.

ResultsResearch Question 1: To what extent do gender patterns

in achievement on fractions tasks match gender patterns inself-beliefs and beliefs about mathematics learning?

Table 1 provides the means and SDs for all study vari-ables by gender. We conducted a mixed analysis of vari-ance in which pre- and post-achievement were repeatedmeasures and gender was a between-subjects factor. Therewas a statistically significant main effect (F[1,994] =238.56, p < .001 and an achievement–gender interaction(F[1,994] = 4.86, p = .028], which slightly favored malesover females. The effect size was tiny, partial eta2 < .01,and the 95% confidence intervals for Cohen’s d crossedzero. This result matched our prediction.

We searched for gender differences for the five beliefmeasures, using multivariate analysis of variance. There



was a statistically significant multivariate effect (F[5,990]= 10.818, p < .001], and each of the univariate effectsshown in Table 1 was statistically significant. Becausemultiple tests inflate type I error and pure Bonferroniprocedures (in which alpha is divided by the number oftests) exaggerates type II error, we used a sequential Bon-ferroni method (Holm, 1979) to adjust the alpha levels. Werank ordered the belief variables from smallest to largestp-values; set the alpha level as .05/5 for the largest, .05/4for the next largest, and so on. Only one finding changed:Effort fell slightly short of the adjusted alpha level (p =.026; the criterion was .025).

Males had more positive scores than females on theself-belief scales: mathematics self-efficacy was higher,and fear of failure was lower for males than females.Gender differences on the self-belief scales were small,and Cohen’s d was in the .20s. Self-reported effort wasslightly higher for females, as predicted by previousstudies of female compliance and gender differences ingrades, but the difference was not statistically significantafter the Bonferroni adjustment. For the beliefs aboutmathematics learning measures, the differences were very

small and mixed. Males were slightly more likely thanfemales to subscribe to functional beliefs about mathemat-ics learning, as predicted, but contrary to prediction, maleswere also slightly more likely to support dysfunctionalbeliefs, that is, that mathematics learning occurs quicklyor not at all and mathematics ability is fixed.

Research Question 2: To what extent is the relationshipbetween student beliefs and achievement on fractionstasks moderated by gender?

Table 2 displays the correlations among the variables inthe study. Bivariate correlations of measured variables areabove the diagonal, and latent construct correlations arebelow the diagonal. The diagonal shows the reliability ofthe measured variables. The scales met the alpha = .70criterion, except for the functional beliefs scale (alpha =.54), which comprised only three items. Table 3 shows themeasurement model for the exploratory sample.The chi2/dftest was not met, but the RMSEA criterion of <.08 wasadequate (but not good) for six variables and poor for one.

We tested six models: (a) a two-variable achievementmodel in which pretest achievement was the sole predictorof post-achievement; (b) a three-variable model in which

Table 1Means and Standard Deviations of Study Variables by Gender

Variable Males (N = 500) Female (N = 496) ES Univariate Results

Mean SD Mean SD

Effort 4.32 .97 4.45 .88 –.14 F(1,994) = 4.95, p = .026 (.025)Fear of failure 3.02 1.18 3.3 1.03 –.25 F(1,994) = 15.94, p<.001 (.01)Quick/fixed learning 2.35 1.04 2.16 .94 .19 F(1,994) = 9.38, p = .002 (.017)Functional beliefs 4.15 .77 4.06 .72 .12* F(1,994) = 3.93, p = .048 (ns)Self-efficacy 4.28 .92 4.08 .93 .22 F(1,994) = 4.95, p = .001 (.013)Pre-achievement 1.16 .67 1.21 .66 –.08* See textPost-achievement 1.45 .52 1.43 .57 .04* See text

Notes. ES = effect size (Cohen’s d) * the 95% confidence levels for the effect sizes include 0. Sequential Bonferroni adjustmentsto the alpha level are shown in brackets in the univariate results column.ES = effect size; ns = not significant; SD = standard deviation.

Table 2Correlations of Bivariate and Latent Constructs for the Exploratory Sample (N = 498)

Pre-Achievement Self-Efficacy FunctionalBeliefs

Quick/FixedLearning

Fear ofFailure

Effort Post-Achievement

Pre-achievement .81 .45** .30** –.18** .07 .17** .61**Self-efficacy .51** .86 .51** –.22** .03 .44** .36**Functional beliefs .37** .84** .54 –.24** –.18** .29** .24**Quick/fixed learning –.32** –.33** –.30** .80 .12* –.24** –.14**Fear of failure .02 .03 .20 .10 .76 .23** .01Effort .21** .49** .63** –.44** .24** .85 .09*Post-achievement .73** .39** .22** –.22** –.02 .14** .83

Notes. Bivariate correlations are above the main diagonal; latent construct correlations below the main diagonal; the diagonal showsCronbach’s alpha of the scales.* p < .05; ** p < .001.


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self-efficacy was included as a predictor of pre- and post-achievement; (c) a six-variable model in which beliefsabout self and beliefs about mathematics learning pre-dicted achievement in the exploratory sample; (d) thesame six-variable model in the validation sample; and thesame six-variable model for the entire sample for (e) malesand (f) females. Table 4 shows the fit statistics and thestandardized path coefficients. Figure 1 displays the six-variable model.

All of the models were an adequate fit of the dataaccording to the RMSEA test (<.08). Including self-efficacy in (b) the three-variable model was an improve-ment over (a) the two-variable achievement model (chi2

difference = 148.2, df = 48, p < .001). Using the modifi-cation indices and testing alternate directions for the pre-dictors of self-efficacy and achievement, we generated asix-variable model showing the antecedents of self-efficacy and its achievement consequences, as shown inFigure 1. Fear of failure was dropped from the modelbecause it did not predict either pre- or post-achievement,as shown in the correlation matrix in Table 2. The six-variable model (c) was an improvement over (b) the three-variable model (chi2 difference = 1,078.8, df = 326, p <.001). Model (c) developed in the exploratory sample wasconfirmed in the validation sample, shown as model (d) inTable 4. The fit statistics for the six-variable model in thevalidation sample were slightly better than in the explora-

tion sample. The standardized path coefficients were verysimilar: The range of differences for five of the paths was.04–.09. The sixth (functional beliefs to self-efficacy) wasstronger in the validation sample (.82) than in the explor-atory sample (.63).

FixedLearning

EffortPath A

FunctionalBelief

Path B

SelfEfficacy

PreFractions

PostFractions

PathC

Path D PathF

Path E

Figure 1. Six-variable model linking beliefs about self and math learning toachievement.

Table 3Measurement Model for Exploratory Sample (N = 448)

Latent Construct # Items Standardized PathCoefficients

Chi2 df Chi2/df RMSEA

Effort 5 .65, .78, .75, .80, .68 43.9 5 8.78 .078Fear of failure 4 .72, .62, .77, .53 1.7 2 .85 .044Quick/fixed learning 6 .56, .54, .67, .64, .70, .70 43.7 9 4.86 .068Functional beliefs 7 .61, .41, .58, .64, .21, .26, .33 82.4 14 5.88 .054Self-efficacy 4 .62, .86, .88, .79 21.9 2 10.95 .102Pre-Achievement 5 .44, .56, .91, .87, .48 19.5 4 4.88 .054Post-Achievement 7 .71, .60, .64, .69, .77, .51, .56 61.9 12 5.16 .071

Table 4Model Fit and Standardized Path Coefficients for Belief–Achievement Models

Model Path A Path B Path C Path D Path E Path F Chi2/df RMSEA

(a) Two-variable achievement .72 4.75 .068(b) Three-variable achievement and self-efficacy .55 .71 .06 3.78 .054(c) Six-variable exploratory –.31 .55 .63 .58 .67 .08 3.42 .069(d) Six-variable validation –.44 .62 .82 .51 .71 .04 3.04 .065(e) Male –.39 .62 .86 .64 .63 .12 3.28 .068(f) Female –.35 .59 .59 .46 .76 .03 2.87 .061

df = degrees of freedom; RMSEA = root mean square error of approximation.



All the path coefficients in Table 4 are statistically sig-nificant at p < .001, except for those in path F (self-efficacyto post-achievement). Most (83%) of the effect of self-efficacy on post-achievement was indirect, through theinfluence of self-efficacy on prior achievement. Anincrease of 1.0 SDs of self-efficacy would increase post-achievement by .47 SDs.

Allowing direct paths from each of the other affect vari-ables did not improve the model nor did allowing pathsfrom each affect variable to self-efficacy. Model (c) indi-cated that beliefs about mathematics learning affectedstudent achievement through their impact on students’self-efficacy beliefs. Students who did not subscribe todysfunctional beliefs, that is, the belief that mathematicsability is a fixed endowment and that success on math-ematical tasks occurs quickly or not at all, were morelikely than students who held such views to report exertingeffort in mathematics class. Students who saw themselvesas effortful were more likely to hold functional beliefsabout learning mathematics, such as participation in classdiscussions helps you learn. Although not measured in themodel, it is probable that these learning-enhancing beliefsdeveloped from experience; students who worked harderdiscovered the benefits of class participation. Functionalbeliefs in turn contributed to stronger self-efficacy thatresulted in achievement. The indirect effects on achieve-ment of an increase of 1.0 SD of each belief variable were.30 SD for functional beliefs, .16 SD for effort, and –.05SD for quick/fixed learning.

The fit of the six-variable model was very similar formales and females as shown Table 4 in row (e) for malesand row (f) for females. The fit reached the chi2/df testcriterion of <3.0 only in the female sample, although theRMSEA criterion was met for both genders. All of thepaths were statistically significant (except for path Fbetween self-efficacy and post-achievement) for bothgenders. The standardized path coefficients were verysimilar for three of the paths (A, B, and F) and were largerfor three others: Path C from functional beliefs to self-efficacy was stronger for males (.86) than for females(.59); path D from self-efficacy to prior achievement wasalso stronger for males (.64) than for females (.46); incontrast, path E from pre- to post-achievement was stron-ger for females (.76) than for males (.63).

DiscussionGender Differences in Achievement

The first contribution of the study is the finding thatgender differences in fractions tasks are near zero.Although there was a statistically significant achievement

difference-favoring males over females in our study,gender accounted for less than 1% of the variance inknowledge of fractions. Our findings, added to extantresearch, suggest that gender differences in achievementare minimal in Canada, more like findings from the UnitedStates than from the Netherlands where fairly large differ-ences have been reported (Seegers & Boekaerts, 1996;Veenstra & Kuyper, 2004).Gender Differences in Beliefs and Achievement

The second contribution of our study is the finding thatgender differences in self-beliefs continued, even whengender differences in fractions achievement were negli-gible. In the literature review framing our study, we foundthat recent studies were mixed, with some researchersreporting that the male advantage in mathematics confi-dence had disappeared while other studies reported that itcontinued. Our investigation found that gender differencesin self-beliefs (self-efficacy and fear of failure) were sta-tistically significant and of small but meaningful size.Females were less confident than males about their abilityto perform well on mathematics tasks, despite there beingminimal performance differences between the genders. Wealso found that the path models linking beliefs to fractionsachievement differed for the two genders.

We found smaller gender differences for beliefs aboutmathematics learning. There was a male advantage on ourmeasure of functional (achievement enabling) beliefs, butmales also reported greater support than females for dys-functional beliefs, including the belief that mathematicsability is fixed. The misalignment of beliefs and achieve-ment was smaller but in the same direction as the mis-alignment of confidence and achievement and can beexplained by similar processes. Our models explained asmall portion of the variance in achievement. The relation-ship between belief and mathematics achievement contin-ues to be complex and requires further investigation.Implications for Schools

Pajares (1996) found in his review that self-efficacy isa better predictor of willingness to take advanced math-ematics courses than prior mathematics achievement; thatis, that the confidence gap contributes to lower participa-tion of women in secure, well-paying occupations thatrequire mathematics skills for entry. This finding is con-firmed in data from the Trends in Mathematics andScience Study for students from Canada, Norway, and theUnited States (Ercikan et al., 2005), which indicated thatthe most powerful predictor of participation in advancedmathematics courses was student attitudes toward math-ematics. In this sense, the confidence gap is an equityissue. Based on the literature, we recommend several


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strategies for reducing the confidence gap: (1) continueefforts to make mathematics attractive to girls and womenby overtly confronting stereotypical beliefs that math-ematics is a male domain by presenting role models offemale mathematicians including watching video epi-sodes of females doing mathematics (Morge, 2007);calling on females as frequently as males to give answers;providing equal time for males and females to explaintheir solutions; creating balanced gender cooperativegroups in the classroom because female participation islower in unbalanced groups (Lee, 1993); and helpingteachers recognize, through collaborative inquiry, thatsome of their instructional strategies may contribute tostereotyping (Wells, 1994); (2) increase the likelihoodthat female students will interpret their performance asbeing successful by providing rubrics that enable studentsto reduce discrepancies between their self-evaluations andschool standards (Andrade, Wang, Du, & Akawi, 2008);(3) provide attributional training in which students learnto attribute success to ability and failure to lack of effort(Heller & Ziegler, 1996); and (4) confer status on femalestudents by recognizing their achievements becauserespect from teachers is a stronger predictor of self-confidence for females than males (Huang & Brainard,2001; Pomerantz, Altermatt, & Saxon, 2002).

These suggestions for schools are derived from socialcognition theory. The first set focuses on students’outcome expectancies. Outcome expectancies are beliefthat particular actions will have desired consequences ifproperly implemented; self-efficacy is the belief that onewill be able to implement these actions. The outcomeexpectancies to be generated by our suggestions forschools are the belief that engagement in mathematicslearning will have beneficial outcomes for students andthat these outcomes do not conflict with the gender roles ofeither males or females. An integral part of influencingoutcome expectancies of students is for teachers to changeinstructional practices, and teacher belief systems, whichcommunicate that mathematics is a male domain. Recom-mendations in sets 2–4 address the self-efficacy beliefs ofstudents. The goal is to increase students’ mastery expe-riences, the most salient source of self-efficacy informa-tion (Pajares, 1996). These recommendations are designedto influence student perceptions of their performance, thatis, that they are more successful than they realize, and thatsuccess is due to students’ effort and mathematical ability.By overtly strengthening students’ self-beliefs and beliefsabout mathematics learning, the gender gap in affect, andits negative impact on female participation in engagementin mathematics, can be reduced.

ConclusionSchools have made admirable progress in reducing the

gender gap in achievement. Girls and women are per-forming as well as boys and men in some countries,including Canada, the site of this study. Student achieve-ment is tightly linked to student beliefs about their math-ematics ability and to their beliefs about learningmathematics. But links between the two are complex: Wecannot assume that the confidence gap will decline as anatural consequence of the closing of the gender gap inachievement. The complex web of variables and mecha-nisms that are part of the classroom environment andrelated societal norms make it a challenge to tease outthe specific instances and the broader systemic activitiesthat perpetuate stereotypes in mathematics classrooms.We need to vigorously confront dysfunctional beliefs thatcontribute to the gender gap in course taking and in par-ticipation in occupations that require mathematical skilland understanding. Women’s sense of agency andempowerment in mathematics are of critical impor-tance—the value placed on females having opportunitiesto learn and using powerful mathematics requires closestudy at this time.

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Appendix A: CLIPS Student Survey Scales(See text for response options.)

Section 1: Self-Efficacy

How sure are you that you could solve a math problem?As you work through a math problem how sure are youthat you can?

• Understand the math problem• Make a plan• Solve the problem

Section 2: Functional Beliefs

You’ve got to say what you think so you can hear whatother people have to say about what you are thinking.

If you are there throwing out your ideas, you could find anew way of doing a math problem.

Don’t be afraid to ask questions.If I’m wrong, it doesn’t really bother me.You may be solving a problem OK but then another

student could show you an easier way that’s really fastand quick.

Sometimes when you answer something, it just clicks inyour head and then you know what you are talkingabout.

When the teacher asks me a question I can’t answer rightaway, someone else will be asked.

Section 3: Fixed Learning

If I cannot understand something quickly, it usually meansI will never understand it.

Working hard on a difficult problem only pays off for thereally smart students.Some people are just born smart, others are born dumb.Students who are “average” in school will remain

“average” for the rest of their lives.The really smart students don’t have to work hard to do

well in school.Successful students understand things quickly.

Section 4: Fear of Failure

If I gave the wrong answer to my teacher’s math question,I would feel terrible.

If I were to get a low grade in math, it would make me feelvery sad.



I worry a lot about making errors on my math work.I would get very discouraged if I made errors on a math

assignment.

Section 5: Effort

How hard are you working to learn about math?How hard are you working to solve math problems?As you work through a math problem how hard are you

working to understand the problem?As you work through a math problem how hard are you

working to make a plan?As you work through a math problem how hard are you

working to solve the problem?

Appendix B: Achievement MeasuresCLIPS Pre-Fraction Variables

1. 2

10is less than 2

5. ________________________

How do you know?4. Circle all the fractions that are greater than 1.

3

8

8

3

19

17

38

3

8. Write two fractions that are equivalent to

3

4. ________________________

9. Write two fractions that are equivalent to

16

18. ________________________

10. Fill in the boxes to write a fraction that is equivalent

to .30.

CLIPS Post-Fraction Variables3. Find as many equivalent fractions as you can from

this picture.

6. If 1

4of a figure is shaded what percentage is shaded?

7. Write an equivalent fraction for 2

3.

9. Change the mixed number 43

5to an improper

fraction.

11. Write two fractions that are equivalent to 5

9.

16. Fill in the following chart.

FRACTION

1

2

DECIMAL PERCENT

17. Fill in the following chart.

FRACTION

3

4

DECIMAL PERCENT


288 Volume 112 (5)

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