. .E
TR
The importance of breaking set
Socialized cognitive strategies and the gender
discrepancy in mathematics
a na v i l l a l o b o s
University of California, Berkeley, USA
A B S T R AC T
Theories that explain the gender discrepancy in mathematics almost universally
explain why boys are ‘better at math’ than girls while failing to adequately account
for girls’ higher grades in math classes or better performances on tests of computational ability.This article develops a new, more comprehensive theoretical model that
explains girls’ advantages in some areas of math, while also showing how these advantages are a liability in the mathematical realms dominated by boys. Specifically, it
argues that ‘strategy socialization’ in risk-taking and rule-following disproportionately
supports girls in the development of an ‘algorithmic strategy’ and boys in a ‘problemsolving strategy’. As the algorithmic strategy leads to success in elementary school
mathematics, girls’ strategy socialization is rewarded and uncontested. However, the
over-rewarding of this single strategy also leads to difficulties in switching strategies
as demanded by higher mathematics. Boys’ strategy socialization, by contrast, is at
odds with early mathematics, contributing to boys’ underperformance at this stage.
However, boys’ ‘strategic dissonance’ gives them practice in switching strategies,
which aids them in solving unfamiliar problems that require new approaches later in
the curriculum.The implications for educational reform are discussed.
k e y w o r d s gender differences, mathematics, problem-solving, risk-taking, socialization
[It is] considered important to the learning of mathematics to ‘break set,’ to ‘free’ oneself
from the confines of simply following rules or learning by rote in order to discover for
oneself. (Leone Burton, 1986)
Why are girls so good at mathematics? Girls outperform boys in computational mathematics and consistently get better grades than boys in math
Theory and Research in Education
Copyright © 2009, sage publications, www.sagepublications.com
vol 7(1) 27–45 ISSN 1477-8785 DOI: 10.1177/1477878508099748
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classes (Leahey and Guo, 2001; Royer et al., 1999; Willingham and Cole,
1997). Despite math being presented as a ‘male subject’ (e.g. famous mathematicians studied are almost always male, most math teachers are men, and
children are more than twice as likely to receive help on their math homework from their fathers than from their mothers), and despite teachers, parents, textbooks and even toys displaying rampant bias against girls in
mathematics, girls have somehow managed to bridge the gender gap in number of math courses taken and to do better in those courses than boys
(Burton, 1986; Chipman, 1994).
On the other hand – and more in harmony with the scholarly choir – why
are girls struggling in mathematics? Girls are still highly underrepresented in
the science and engineering majors that require high levels of math, and math
aptitude tests in college show gender differences favoring boys even controlling for major (Langenfeld, 1997; Royer et al., 1999). Furthermore, the decadesold, tenaciously stable difference between girls’ and boys’ scores on the math
section of the SAT persists, with an average 34-point difference favoring boys
on the 2005 and 2006 tests (NCES, 2007). How do we explain these facts, particularly in light of girls’ academic achievements in mathematics highlighted
above?
Most theories claiming to explain the gender discrepancy in mathematics
fall into two basic categories: biological theories of inherent difference, and
social theories of gender bias in mathematics. Neither of these theories, which
almost universally overlook girls’ advantages and seek only to explain why
boys are ‘better in math’ than girls, can adequately explain the complexities in
gender and mathematics. The task of this article is to propose an alternative
theoretical model which does explain these complexities.This model does so,
in part, by highlighting girls’ advantages in mathematics, and showing how those
advantages, paradoxically, are crucial to understanding girls’ disadvantages in
math.
First, it is important to understand how prior research and theories fall short.
Biological studies examine such factors as sex differences in brain lateralization
or the effects of testosterone on spatial reasoning (Baron-Cohen, 2003; Geary,
1998; Halpern, 2000; Kimura, 1999).This category of explanation was for many
years the reigning account of boys’ assumed superiority in mathematics.‘Spatial
abilities’ are the most often cited biological advantage for boys in mathematics,
yet the spatial cognition argument has four problematic elements. First, while
spatial ability as a whole has been correlated with generalized intelligence,
among the myriad measures of spatial ability, the only measure in which boys
consistently and significantly surpass girls is in the ability to mentally rotate
three-dimensional objects in one’s head, an arguably limited advantage (Linn
and Petersen, 1985). Second, boys’ putatively biological advantage in mentally
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Villalobos:The importance of breaking set
rotating shapes may not be due to inborn anatomical or hormonal differences,
but may be attributed to their greater exposure to sports than girls, or other
social factors (Beal, 1994). Third, evidence that the ability to mentally rotate
shapes is responsible for sex differences in math achievement is weak and mixed
(Chipman, 1994). Fourth and most importantly, by many indicators of math
achievement, girls surpass boys (Hyde et al., 1990; Klein et al., 1994; Leahey and
Guo, 2001; Lueptow, 1984). Biological theories fail to explain why this would
be so.
Possibly because of these limitations, many scholars have dropped any biological component from their explanations of the gender disparity in mathematics, and those who have not frequently espouse dual-stranded theories
that incorporate both biological and social influences (Benbow, 1988; Halpern,
2000;Wilder, 1997).
The second category of explanation of the gender disparity in mathematics points to social bias in mathematics socialization. These studies reveal the
many ways in which mathematics is guarded intellectual terrain, coded ‘male’
through such mechanisms as textbook bias, differential rewards or rebuke for
boys’ and girls’ attempts to pursue or opt out of the mathematics curriculum,
and stereotype threat (Love, 1993; Ma, 2001; Quinn and Spencer, 2001; RiegleCrumb, 2005).The conclusions of these bias studies are far less contested than
biological conclusions; however, they share some of the same limitations. Most
notably, unless girls’ inherent abilities in math are vastly greater than boys’ (a
possibility few discuss), it is difficult to understand how the current generation of girls would manage to take the same number of math classes as boys
and do better than boys in those classes despite a biased social milieu that disproportionately rewards boys’ success in math and continues to give students
the message that girls are good at ‘verbal’ tasks and boys at ‘mathematical’ ones.
Specifically, how can mathematics’ ‘male’ reputation, differential rewards, and
other forms of bias against girls in mathematics be so well documented, yet
girls tenaciously maintain certain mathematical advantages over boys? To make
sense of these highly complicated facts, one must search beyond both biological and gender bias in math explanations, neither of which adequately
explains the described patterns.
This article draws on psychosocial, gender, mathematics and education literatures to develop an alternative theoretical model that is consistent with the
many nuances of the gender discrepancy in mathematics. It argues that boys and
girls receive different reinforcement for rule-following and risk-taking behaviors within a wide array of institutional and personal settings, and the subsequent
behavioral adaptations support different thinking strategies. Specifically, those children conditioned to follow rules and avoid risk – disproportionately girls – have
ideal preconditions for developing an algorithmic strategy, and those children for
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Theory and Research in Education 7(1)
whom rule-breaking and risk-taking is more tolerated or even rewarded –
disproportionately boys – have ideal preconditions for developing a problemsolving strategy. This ‘strategy socialization’, experienced primarily outside the
mathematics classroom, predisposes a gendered split in the types of mathematical
proficiencies at which boys and girls excel, which is reinforced in the classroom
by math-specific bias, creating a sense of inevitability and naturalness to the
bifurcation.
Furthermore, as the algorithmic strategy and its psychosocial preconditions
match both teachers’ desires and the approach required of early mathematical
content, many girls experience strategic alignment in the elementary mathematics classroom, creating an uncontested sense that theirs is the winning
strategy.This wins them grades, but comes at a cost in deeper learning, as will
be explained. By contrast, the preconditions of the problem-solving strategy,
more highly reinforced by boys’ socialization, neither align with elementary
teachers’ desires nor match the approach required to solve most elementary
school mathematical problems, and therefore boys experience strategic dissonance in early mathematics education.This mismatch of boys’ broader socialization and the expectations in the elementary mathematics classroom creates
difficulties for boys early in their mathematics education. However, being
forced to navigate between conflicting strategies gives boys more practice than
girls in recognizing strategic differences and selecting the appropriate one,
which serves as an advantage in higher mathematics.
This article will explicate this new theoretical framework by: 1) clarifying
the multifaceted gender disparity in mathematics, 2) developing the concept
of ‘strategy socialization’ in which children are socialized with the psychosocial precursors of two distinct forms of mathematical strategizing, 3) showing how strategy socialization – which is generally different for boys and girls –
affects mathematical outcomes, 4) elaborating a theory of strategic alignment
for girls and strategic dissonance for boys, which further explains gender differences in mathematics, and finally 5) discussing the problematic aspects of
the described strategic differences between boys and girls, and proposing solutions.
t h e g e n d e r d i s pa r i t y i n m at h e mat i c s
Before theorizing about strategy socialization, we must better understand the
male/female disparity in mathematics. Despite many studies making the general claim that ‘males perform better than females in mathematics’ (Ramos and
Lambating, 1996), it is not so one-sided as these studies suggest.Though boys
generally score higher on ability tests in secondary school, girls generally
receive higher grades, even in such stereotypically ‘male subjects’ as math
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Villalobos:The importance of breaking set
(Kimball, 1989; Klein et al., 1994; Lueptow, 1984).And throughout elementary
school, girls outperform boys even on math ability tests, which are primarily
computational at that phase in the curriculum (Leahey and Guo, 2001).
Indeed, girls’ superior computational abilities continue throughout their
schooling (Hyde et al., 1990). Thus, contrary to popular misconception, girls
have an advantage in certain mathematical arenas and during the early phase of
their schooling.
However, we see a more complex picture emerging toward the end of
grade school and during the junior high school years. When almost 300,000
sixth graders took the Survey of Basic Skills, which distinguished math by
problem type, girls were more successful than boys at computations whereas
boys were more successful at problem-solving or ‘insight’ problems that
require thinking beyond what is given in a question (Marshall, 1984). This
gendered bifurcation by skill type continues into junior high and high school
(Willingham and Cole, 1997). It is therefore not surprising that in junior high
school, when the curriculum generally shifts from a computational base to
one of problem-solving, girls’ math ability test scores begin to wane (Leahey
and Guo, 2001). By high school, 78 per cent more adolescent boys than girls
meet proficiency levels in multistep problem-solving (NCES, 1999: 140).
Many scholars believe that the drop in female ability test scores in junior
high is not a coincidence, but in fact that it occurs precisely because of the
new curricular emphasis on problem-solving. But why should girls excel in
one branch of mathematics while boys dominate another? This is the most
critical question, yet it is least discussed. Past studies are based mostly on the
assumption that boys are better at math than girls, undifferentiated by problem type and, that being the case, the research goal is to determine what factors inhibit girls from success. The question here is: What is it about female
socialization that favors strong computational abilities, and what is it about
male socialization that favors success in problem-solving?
s t rat e g y s o c i a l i z at i o n
The proposed theoretical model suggests that gendered socialization affects
the differing psychosocial precursors to algorithmic versus problem-solving
strategizing, which are then either reinforced or challenged in the classroom,
creating different gendered outcomes.
An ‘algorithmic strategy’ describes the approach in which rules and clearly
defined, pre-established methods are utilized to deliver the expected results.
This strategy is what a person draws on to perform computations, such as multiplying 23.4 times 6.7, or to fill in ‘drill and practice’ worksheets where the
method is consistent and must be applied to all of the questions. It is logical
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Theory and Research in Education 7(1)
that children who are most highly rewarded for following rules and being careful would be most successful in developing an algorithmic strategy, which
draws on careful observance of rules and known processes to reach a solution.
By contrast, a socialization in which ‘breaking set’ is more tolerated and in
which there are fewer negative consequences for rule-breaking, risk-taking, or
deviating from or challenging procedural guidelines would leave one less
rehearsed at following step-by-step instructions, which would translate to less
success at algorithmic problems such as computations. At the same time, rulefollowing may also entail asking fewer questions, difficulty in thinking beyond
what is ‘given’ in a problem, and over-dependence on exact models by which
to come up with results. Likewise, risk aversion may create a reluctance to take
wild stabs at inventing one’s own strategies or pathways and may paralyze a
student if there is no self-evident procedure by which to arrive at a solution,
all of which would inhibit the development of a problem-solving strategy.
A ‘problem-solving strategy’ describes the approach in which insight, application to unfamiliar contexts, synthesis, or sometimes trial and error, are utilized to ascertain the very method one should use to attempt a solution. For
example, consider the problem, ‘Beth drove 40 miles/hour to visit her sister,
then drove 60 miles/hour the same distance back home. What was her average speed while driving on the whole trip?’The ‘insight’ required in this problem is to realize that Beth drove 40 miles/hour for more time (since she was
going slower) than she drove 60 miles/hour. Upon realizing this, the question
remains: How can one calculate the average? One could simply guess that the
answer would be closer to 40 than 60 (by virtue of going 40 miles/hour for
more time), and just write an answer such as 48 and hope for the best. Or,
since one is not given how many miles the trip was, one could make up and
insert a distance and calculate using that (and possibly insert another distance
to see if it yields the same results). Or one could introduce a variable for either
the time, distance, or both, and use algebra to solve it.The method is not transparent simply by reading the question. One must puzzle. If one is afraid of or
uncomfortable with puzzling, guessing, just making up one’s own distance and
solving the problem, or creating a more abstract algebraic equation, and if one
does not have the initial insight about the relative durations of driving at each
speed, one might impose a purely algorithmic strategy and try (incorrectly) to
apply the formula for calculating a mean by adding the two speeds and dividing by two, yielding 50 miles/hour. An over-developed algorithmic strategy
and the associated reliance on exact models to follow may actually be debilitating in problems such as the one above, and those children who are relatively
unbound by the algorithmic strategy may have greater potential for success in
such situations where questioning a problem must occur before answering it.
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Villalobos:The importance of breaking set
The proposed association of rule-following and carefulness with the
algorithmic strategy, and of rule-questioning and risk-taking with the problem-solving strategy is itself gender-neutral and should apply equally to boys
and girls. However, girls’ and boys’ socialization experiences are frequently
different.
Despite the modern social milieu in which assertive, soccer ball-kicking
girls are a socially accepted expression of adolescent femininity, there are myriad forces at play in a girl’s life which still disproportionately support rulefollowing and carefulness. These social factors span institutional settings. For
example, within families, fathers monitor and protect their daughters from
physical risk-taking more than their sons (Hagan and Kuebli, 2007). Within
educational institutions, elementary school age boys are ‘praised for their
knowledge and giving the right answer, [whereas] girls [are] praised for obedience and compliance’ (MacDonald and Rogers, 1995: 1). Likewise, teachers
rarely penalize untidy work from boys, but if the same paper is thought to be
from a girl, she is most often penalized (Spender, 1982: 79). Also when boys
ask questions, protest, or challenge the teacher, they are often respected and
rewarded, whereas girls engaging in the same behavior are often rebuked and
punished (60).
These rewards and rebukes foster different generalized behaviors in boys
and girls. For example, boys have been shown to take more risks than girls in
a number of realms, including driving (Chen et al., 2000), drug use (Tyler and
Lichtenstein, 1997), performance of physical feats (Fetchenhauer and Rohde,
2002), financial decisions (Mittal and Dhade, 2007), and even such everyday
activities as cutting the time closer when trying to catch a bus (Pawlowski and
Atwal, 2008). Boys are also more likely than girls to challenge rules. For
example, boys engage in more disruptive behavior in the classroom (Spender,
1982: 54). In fact, boys make such disruptive protests if the teacher does not
direct most of his or her attention to them, which, in the interest of classroom
control, teachers predominantly do (Parker, 1973). In a recent study of 4572
children, researchers found that boys exhibit ‘lower sociability and compliance
[than girls] and greater … oppositional behavior’ (Lahey et al., 2006: 730).
By contrast, girls appear to more readily comply with adult requests (Forman
and Kochanska, 2001). Girls are three times more likely than boys to raise their
hands in class, and they put more effort into neat handwriting, turning in
complete homework, and eventually getting good grades – which they do
consistently more than boys (Lueptow, 1984; Sadker and Sadker, 1985;
Willingham and Cole, 1997). Indeed, in a study of elementary school children,
Serbin (1990) finds that girls’ better academic performance than boys is a function of their greater social responsiveness and compliance with adult direction.
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Theory and Research in Education 7(1)
In sum, boys are disproportionately rewarded for taking risks and challenging rules, and girls are disproportionately rewarded for being careful and abiding by rules; consequently, gender differences in behaviors emerge over time
in these realms.
h o w s t rat e g y s o c i a l i z at i o n a f f e c t s
mat h o u t c o m e s
These behavioral differences influence academics. For example, there are
trans-disciplinary benefits to rule-following and carefulness, such as the better
generalized academic performance demonstrated by Serbin above.Additionally,
despite most studies of gender differences in mathematics conspicuously
avoiding discussions of the precursors of algorithmic success, it is evident that
rule-following and carefulness would aid in the algorithmic strategy, which is
by definition following rules to reach a solution whose correctness depends
on the carefulness of one’s execution.The inverse claim – that rule-following
and carefulness would be detrimental to the development in either sex of a
problem-solving strategy – may be less obvious. However, Burton notes:
Following the procedural rules of mathematics and the behavioural rules of the classroom
was necessary to successful completion of tasks. However, challenging the internal rules
of the mathematical discourse, relating particularly to the teacher’s authority as guardian
of those rules is important in producing what teachers describe as ‘real understanding.’
That many girls would not dare to make a challenge offers a different explanation of girls’
mathematical development. (Burton, 1986: 145)
Burton does not elaborate further on her intriguing hypothesis that girls’ presumed hesitancy to challenge procedural rules may play a role in their success
or failure in higher mathematics. However, there is evidence to confirm her
hypothesis. In early research relating psychosocial attributes and behaviors to
children’s cognitive abilities, Maccoby finds that ‘acceptance of authority’
(an indicator of rule-following) is intellectually detrimental to both sexes
(Maccoby, 1966: 36).While Maccoby discusses intelligence as a whole, differentiating the type of intelligence is clearly required in any nuanced discussion
of mathematical proficiency.
There are many studies of the problem-solving branch of this differentiation.
The best support for a connection between rule-challenging and problemsolving is a Crombie and Gold study in which the authors directly test how
four- and five-year-old children’s levels of compliance relate to their ability to
‘solve’ an owl toy whose eyes light up if the correct sequence of levers is pulled.
They find that ‘for girls and boys without observable behavior problems, high
compliance is negatively related to problem-solving competence’ (Crombie and
Gold, 1989: 286). The mechanism they propose in this relationship is: ‘Lower
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Villalobos:The importance of breaking set
compliance may provide children with more opportunities to become emotionally independent [of their parents], thus affecting the development of their
problem-solving competence’ (289). This is a different mechanism than the
more direct one proposed here wherein rule-reliance itself makes it difficult to
think beyond what is given in a question and to engage in problem-solving.
Nevertheless, the Crombie and Gold study gives excellent support to the proposition that rule-following and problem-solving are negatively correlated, except
possibly at the extreme low end of rule-following.
There is likewise reason to believe that risk-aversion is detrimental to problem-solving. In a meta-analysis of 150 studies of gender differences in risktaking, upon disaggregating the type of risks tested for, the authors find the
gender gap in risk-taking is most pronounced in the realms of physical skills
and of intellectual risks (i.e. girls are less willing to risk being wrong) (Byrnes
et al., 1999). Unwillingness to be wrong leads students to engage in activities
where they know they can succeed, such as computations, and avoid taking
on new challenges, such as problem-solving (Elliott and Dweck, 1988).We see
this play out on the math section of the SAT, where high-scoring girls are
more likely to use conventional strategies (clearly defined methods) in correctly solving problems whereas high-scoring boys are more likely to use
unconventional strategies (e.g. atypical methods, insight, or estimation)(Gallagher
and DeLisi, 1994). Ramos and Lambating (1996) also find that girls take fewer
risks than boys on such tests and omit more answers, again presumably for fear
of being wrong.They therefore equate risk-avoidance with ‘failure-aversion’,
which is characteristic of the brightest girls, but not the brightest boys (Licht
et al., 1984).
There may be a cycle at play in which risk-avoidance and failure-aversion
create a propensity toward algorithmic strategizing, and repeated utilization of
the algorithmic strategy reinforces failure-aversion. This is because the primary algorithmic challenge is to avoid careless mistakes utilizing a given
method, whereas the primary problem-solving challenge is selecting one’s
method. Thus, on a more abstract level, we could say an algorithmic strategy
is product-oriented (right answer), whereas a problem-solving strategy is
process-oriented (right method – as well as right answer). Dweck (1986) finds
that a ‘performance-oriented’ pattern, which is concerned with turning out a
good product, works against the pursuit of challenge because individuals with
low assessments of their ability often choose tasks they perceive as easy, for
which success is assured. Even students with high assessments of their ability
may sacrifice learning opportunities that involve the risk of ‘failure’ for opportunities to look smart (Elliott and Dweck, 1988). By contrast, the ‘masteryoriented’ pattern, which emphasizes the process of learning, such as in
problem-solving, is associated with challenge-seeking and persistence in the
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Theory and Research in Education 7(1)
face of obstacles. These differences are gendered, and girls are more often
driven by product goals, extrinsically motivated by adult approval for the right
answer, whereas boys more often have the ‘mastery-oriented’ intrinsic motivation derived from the process challenge (Boggiano et al., 1991).This is particularly the case among high achievers, where girls with high grades gravitate
even more toward tasks they know they are good at, whereas boys with high
grades more frequently prefer new and risky problems that might stretch their
abilities (Licht et al., 1984).
Gertrude Stein said ‘If you can do it, then why do it?’ and her point may
have educational implications for girls. It is a pedagogical fact that students of
either sex are unlikely to gain confidence if they stick to tasks that are easy for
them (Relich, 1983) and, without a certain degree of confidence, they are
unlikely to challenge themselves in the first place. So it is another cycle: smart
girls often seek out problems they already know how to do for the rewards
associated with the ‘right answer’; their success does not build their confidence
or broaden their abilities, and so it remains unlikely that they will try new,
potentially confidence-building problems with any level of determination.
Boys’ socialization is somewhat different. A meta-analysis of studies shows
boys to be more impulsive than girls, that is, more likely to give quick, incorrect answers (Lueptow, 1984: 153). This socialized male prerogative to be
‘wrong’ may be debilitating in computational math where carefulness is critical, but may actually facilitate insight. If a problem has no clear process by
which to achieve a solution, an impulsive person will likely just take off in any
direction, right or wrong, and at very least get a feel for the territory, for what
works and what does not in the process of solving problems. By contrast, people who must know in advance that the steps they take are correct may be
paralyzed when there is no marker to that effect.
Differentiating strategic thinking into ‘algorithmic’ and ‘problem-solving’,
we can more clearly understand how girls are funneled into the former type
of intellectual success, boys into the latter. Owing to greater pressures to follow rules and avoid risks, girls who adhere to their socialization should both
get better grades than boys and surpass boys in algorithmic abilities necessary
for success in computations.As already shown, studies confirm they do.At the
same time, boys who adhere to their characteristic socialization should surpass
girls in problem-solving, which comes later in the curriculum. This has also
been confirmed by numerous past studies.
This accounting offers a partial explanation for the dissimilarity in men’s
and women’s math skills without relying on any intrinsically male or female
characteristics. Instead, it traces proficiency differences back to differing strategies, which are strengthened by psychosocial preconditions differently reinforced in boys and girls.
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Villalobos:The importance of breaking set
Strategy socialization comprises the first layer of the proposed theoretical
model, and already explains a great deal. It is quite different from prior socialization explanations which reveal bias against girls in mathematics through
such mechanisms as wait-time differences in the classroom, differing representation and depictions of girls and boys in mathematics textbooks, or even
talking Barbie dolls that say ‘Math is hard!’The multitudes of bias studies reveal
what we could characterize as an unsupportive ‘mathematics socialization’ for
girls. By contrast, the theory of strategy socialization is unique in accounting
for both boys’ and girls’ respective advantages (and disadvantages) in distinct
mathematical realms. However, as is also becoming evident, girls’ strategy
socialization, while contributing to their algorithmic advantage, ultimately
favors good grades over further learning. Furthermore, while strategy socialization is theoretically distinct from accounts of gender bias in mathematics,
these two effects are synergistic. Despite girls’ better academic performance in
math than boys, the world is still rife with the message that boys are better at
math. When girls experience that message, that is when stereotype threat is
high, girls perform less well on word problems (often problem-solving) but
not on equivalent numerical problems (algorithms)(Quinn and Spencer,
2001). Thus the combination of girls’ biased strategy socialization and gender
bias in mathematics serves to exacerbate and further naturalize the difference.
t h e o r y o f s t rat e g i c a l i g n m e n t o r d i s s o na n c e
Even if we were to eliminate gender bias from mathematics, however, there
would still be a problematic convergence between young girls’ strategy socialization outside the mathematics classroom and the expectations issued within
the classroom. This comprises the second layer of the proposed theoretical
model.
As already argued, elementary-age girls’ non-classroom experiences disproportionately foster rule-following and risk-avoidance, which reinforce an
algorithmic strategy. Entering elementary school with some degree of this
strategic preference, girls (and boys) encounter a curriculum that highlights
algorithmic learning through its emphasis on computational math.Therefore
girls’ psychosocial precursors to the algorithmic strategy are further enforced
by this being the strategy required to solve elementary math problems.
Furthermore, mathematics assessment is frequently staunchly divided between
‘right’ and ‘wrong’. Unlike the assessment for, say, composing an essay where
style matters and where one can have a ‘bad’ (poorly written, unconvincingly
argued) essay but not a ‘wrong’ one, in mathematics, missteps are often not
recognized as valuable unless they lead to the correct answer.This may foster
a product goal orientation, an external locus of motivation, and an algorithmic
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Theory and Research in Education 7(1)
strategy. Finally, the teacher wishes students to complete these computational
problems successfully. That is, the authority figure in the classroom not only
enforces rule-following by virtue of issuing rules, but the content of those
rules is in the direction of algorithmic strategizing.
Thus there is perfect strategic alignment between: 1) elementary-age girls’
more general socialization, 2) the intellectual approach required by the mathematical problem type most frequently found in the elementary classroom, 3)
the evaluation methods often used in mathematics, and 4) the authority figure’s wishes for students.This perfect alignment favoring an algorithmic strategy reinforces for girls that this is the winning strategy. Because of this strategic
alignment, many girls master the algorithmic strategy with aplomb, evidenced
by their computational advantage.
However, it is their very mastery of the algorithmic strategy that, though
helpful in grade school mathematics, leads to difficulties in problem-solving.
This is because an over-reliance on exact models by which to format results
is debilitating if there is no such model. Thus, by too fully mastering what it
takes to succeed in early mathematics, many girls inadvertently pave the way
to their own undoing in the later mathematics.Their quickness to learn ‘what
works’ leads not to success, but to debilitation.
Kessel and Linn (1996) discuss this backfire in girls’ learning by proposing
that girls’ underperformance on standardized math tests may occur precisely
because they learn more in math courses than boys.The authors conjecture that
girls’ utilization of classroom-learned procedures leads to great success in the
classroom, but over-learning these procedures can cause difficulty in unfamiliar settings where those procedures may not be so appropriate.
If we apply Kessel and Linn’s over-learning argument to the sudden shift of
emphasis from computational to problem-solving mathematics in junior high,
it suddenly makes sense that girls who are the ‘best students’, the ones with
the highest grades who have lived in greatest accord with prior (algorithmic)
expectations, are the hardest hit by the shift. In general, of students with A, B,
C and D grade-point averages, girls with A-averages show the greatest debilitation in response to failure, whereas boys with A-averages are the only group
to show any facilitation (Licht et al., 1984). Bright girls, it seems, learn the
ropes the quickest, but the ropes they are given in elementary school mathematics classrooms are built to fray when higher-level thinking enters the curriculum.
Boys are given the same ropes in elementary school, so the question of why
boys are less debilitated by them merits exploration. Elementary-age boys’
non-classroom socialization, as already shown, has a higher tolerance than
girls’ for rule-challenging and risk-taking, which enables a problem-solving
strategy to develop. However, when those boys enter the elementary math
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classroom, like girls, they are taught the algorithmic strategy through computational mathematics.The skills necessary to succeed in this problem type run
contrary to their broader socialization, as do the wishes of the authority figure teacher who wants students to learn algorithmic strategizing (and to follow classroom rules and be careful in their work). Boys, therefore, experience
more strategic dissonance.This disjuncture between strategy socialization and the
requirements of the elementary school math classroom helps explain why
boys do less well in elementary school math proficiency tests. However,
because of their strategic dissonance, boys do not become as wedded to the
algorithmic strategy – it is not as fully supported in all aspects of their lives –
and this leaves them freer to select between strategies.Thus, when the transition to problem-solving occurs in junior high, boys are more able to shift
strategies as required by the curriculum, whereas girls are less able to do so.
It is analogous to the contrast between monoglots and polyglots.Those who
speak more than one language from an early age have a greater facility learning new additional languages than those who are monolingual, and polyglots
also have a greater understanding of the concept of language as a social construct with options rather than as ‘just the way it is’.
Because the various forces colluding in girls’ ‘monolingual’ strategic alignment are subject to social influences, they are theoretically malleable, so if we
find this alignment stymies girls’ (or boys’) higher mathematical success, this
knowledge may help us better devise curricular and pedagogical environments conducive to all students succeeding.
w h at t o d o a b o u t t h e p r o b l e m
First of all, is it a problem that girls have greater algorithmic abilities while
boys dominate problem-solving? Halpern, who studies both biological and
social forces contributing to cognitive differences between boys and girls,
states that ‘researchers have shown that there are areas in which females, on the
average, excel, and areas in which males, on the average, excel. But differences
are not deficiencies … The problem lies not in the fact that people are different. It is in the value that we attach to these differences’ (Halpern, 2005: 1). In
fact, girls excelling in the algorithmic strategy and boys excelling in the problem-solving strategy is socially problematic and is distinctly not a case of ‘different but equal’ abilities, for a number of reasons. First, the intrinsic,
process-oriented goal of the problem-solving strategy fosters challenge-seeking
and further learning, whereas the extrinsic, product-oriented goal of the algorithmic strategy does not. Second, there is inequality in that, while boys excel
at problem-solving, they are at least solidly introduced to the algorithmic strategy
at an early age, whereas the reverse is not so true for girls. Third and finally,
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Theory and Research in Education 7(1)
because problem-solving is the thrust of higher math, a lack of ability in that
arena thwarts the successful continued pursuit of mathematics, and thereby of
the sciences.Yet on the ‘values attached’ level, dropping math and science is
the last thing women should do if they seek equal employment with men.To
illustrate, in the University of California system, calculus is a requirement for
a major in engineering, computer science or the physical sciences – majors
traditionally dominated by men and generally leading to high-paying careers.
Calculus is not required for degrees in the humanities, social sciences and education – majors associated with women and with low salaries.
Given that these differences are indeed problematic, the question arises:
What can be done as a corrective? Because the psychosocial behaviors associated with greater success in problem-solving are those most reinforced in
boys, there are two immediately apparent routes to greater mathematical
equality. The first route is to socialize girls the same way boys are currently
socialized, including greater tolerance for girls who engage in class disruptions, who lack conscientiousness in their schoolwork, and who are impetuous and disobedient. However, this ‘solution’ assumes that stereotypical boy
socialization is superior to stereotypical girl socialization, that a rebel is favorable to a team player, and that being ‘good’ is actually bad. Any second grade
teacher would beg to differ.The second possible route to mathematical equality is to socialize both girls and boys with a similar mix of rules and tolerance
of questioning the rules. While this may be ideal and is precisely the recommendation for parents and others who engage with young children, it requires
a thoroughly unbiased society, and is therefore difficult to implement through
policy.
A less sweeping solution located solely within educational institutions is
therefore proposed.As has already been evidenced, part of what separates boys’
and girls’ success in higher mathematics is a function of the curricular shift
from computations to problem-solving in junior high.This transition is more
jolting and anxiety-producing than it has to be, and it disrupts girls’ learning
disproportionately more than boys’ because girls are less rehearsed at selecting
between competing strategies, having had a singular strategy previously reinforced in both social and mathematical settings.
To avoid the strategic alignment that over-rewards girls in the algorithmic
strategy, given that this is the strategy most highly reinforced by girl socialization at large, computations should be deemphasized in the elementary
mathematics curriculum. The memorization and drill-and-practice model of
instruction required for computational success reinforces girls’ prior training
toward rule-following and risk-avoidance, and presents a regulations-based
understanding of mathematics and learning that girls too easily master given
their strategy socialization elsewhere. Note that the elimination of algorithmic
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Villalobos:The importance of breaking set
learning from elementary school mathematics curricula is not being suggested.This is partly because of the great practical utility of computations. But,
just as importantly, it is essential that reform attempts aimed at supporting
multistrategic girls do not inadvertently create the reverse problem of monostrategic boys. Eliminating formulaic, rules-based math from elementary
schools would exacerbate boys’ current algorithmic disadvantage, and likely
make it far more difficult for boys to learn to follow instructions, exhibit carefulness in their work, put in the effort to maintain good grades and other
desirables associated with the algorithmic strategy. Maintaining boys’ strategic
options through teaching mathematical rules and carefulness should not come
at the expense of girls, however, and that is why it is imperative not to overaccentuate algorithmic strategizing in elementary schools.
In tandem with this deemphasizing of computations, educators should more
highly emphasize problem-solving in early education, and create a curriculum
aimed at teaching young students to take mental risks, to try out their own
ideas, to question processes, and to extract learning from their failed attempts
at solutions rather than view those attempted pathways as ‘wrong’.This would
create an environment of mathematical risk-taking and rule-questioning in the
elementary classroom where girls who have been socialized to please authority
figures and be ‘good’ could enact that goodness not by rote but by truly thinking and puzzling and questioning assumptions.
This recommendation unfortunately meets resistance in actual classrooms.
The National Council of Teachers of Mathematics (NCTM) made its own
recommendations in its ‘Agenda for Action’, drafted in the late 1970s. It
advised: ‘The mathematics curriculum should be organized around problem
solving [and] … [a]ppropriate curricular materials to teach problem solving
should be developed for all grade levels . . .’ (NCTM, 1980: 1–3).Yet despite
this agenda and despite attempts of teacher education programs around the
country to effect changes in how elementary mathematics is taught, on the
ground we see that elementary teachers still ‘have a tendency to avoid thought
about reasons in mathematics’ and to focus instead on ‘computational skill
through the rote application of procedures, routines and algorithms’ (Newton
and Newton, 2007: 69). This is due, at least in part, to the ‘accountability
movement’, which includes the No Child Left Behind act.‘[T]he “high stakes
testing” mandated by the No Child Left Behind act has resulted in many states
creating straightforward, skills-oriented assessments. A universal trait, not just
in the US, is that teachers will “teach to the test.” Hence, there are political
pressures toward a narrowing of the curriculum, with the direction toward an
emphasis on skills rather than concepts and problem solving’ (Schoenfeld,
2007: 538). Thus, if we wish to see a greater focus on problem-solving in
elementary school, we might need to reconsider not just assessment at the
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Theory and Research in Education 7(1)
micro-level (i.e. ‘Is this student’s solution “right” or “wrong”?’) but at the
macro-level (i.e.‘How are we assessing whether schools are imparting students
with appropriate capacities?’).
The recommendation of this article is that all levels of math education deemphasize the rigid, right-or-wrong product-orientation in mathematics to
which girls disproportionately fall prey, especially smart girls. Many girls adopt
the algorithmic learning strategy that authority figures request of them and that
works best to solve elementary school mathematics problems, and because that
strategy often resonates so well with their broader socialization, they may conclude that it is the only strategy and therefore have more difficulty than boys in
switching to an alternative strategy later in the curriculum when a problemsolving approach would better serve them. If both the rules and the questioning of the rules were presented on each leg of the journey, it would be less
difficult for girls, and indeed all students, to adjust to the sudden rigors of reason; they would have learned the process of creative puzzling at such a young
age that it might come as second nature, like language. If all students were
encouraged to be multistrategic, many more might successfully grapple with
unfamiliar problems, as well as be contentious about schoolwork, drawing the
benefits of both the problem-solving and algorithmic strategic orientations.
This article is an examination more of the problem than of the solution to
regimented thinking and its effects on the gendering of intellectual strategies.
The question remains: How can we promote insights in all of our children?
There is no set procedure to follow. It will take insights of our own.
ac k n ow l e d g e m e n t s
The author gratefully acknowledges the many formative comments of anonymous reviewers as well as the following non-anonymous ones: Perrin Elkind, Carole
Hesse, Arlie Hochschild, Michele Murphy, Gretchen Purser, Ofer Sharone, and
Tom Villa-Lovoz.
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b i o g ra p h i ca l n o t e
ana villalobos will receive her PhD in sociology in May 2009 from the
University of California, Berkeley, where she researches the relationships between
motherhood, societal insecurity and the ideology of independence in American
society. Before embarking on the PhD program at Berkeley,Villalobos earned a BA
in mathematics from the Honors College at the University of Oregon, and an MA
in education from Stanford University. The theoretical model she presents in this
article stems not only from her training as a sociologist, but also from her prior professional experiences teaching high school mathematics for eight years, where she
first observed some of the phenomena she discusses. [email: [email protected]]
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