Reply to commentary on Bjorkland, Crow, Davis, Dowker
by Jo Dixon
Written by Jennie Dock
[ >> = the BBS commentary, jd> = Jop Dixon]
>> A. Dowker.
>> I argue that the evidence for the central role of spatial ability in
>> mathematical ability, or in gender differences in it, is tenuous at
>> best. This commentary will not deal with the issue of biological
>> versus social origins of gender differences, or with the evolutionary
>> theories that form a basis for part of the author's argument. Rather,
>> it will focus on one topic raised in the target article: the issue of
>> whether gender differences in spatial ability can adequately explain
>> gender differences in arithmetic. Spatial ability has frequently been
>> suggested (e.g. by Casey, Nuttall and Benbow, 1995) to be an important
>> factor in mathematical performance in general, and in gender
>> differences in mathematics in particular.
jd> One surely cannot simply consider this without the consideration of
jd> other factors such as environmental factors.
I agree, there are also other influences that should be taken into
consideration, such as social expectations (for example, if boys are
more competitive than girls in a classroom situation, and the existence
of salient differences in attribution of success or failure: boys'
failure is more likely to be blamed on external factors and girls'
failure on personal limitations).
>> Why indeed should spatial ability be expected to be important to
>> mathematical performance? There seem to be three main ways in which
>> spatial ability might contribute to mathematics: (1) Geometry
>> emphasizes spatial relationships, However, only some forms of geometry
>> emphasize THREE-DIMENSIONAL spatial relationships, which is the aspect
>> of spatial ability that is most gender-differentiated. Trigonometry,
>> for example, predominantly emphasizes two-dimensional geometry. While
>> three-dimensional geometry is one important component of advanced
>> mathematics, it is a relatively small and specialized one. (2) Some
>> degree of spatial ability is necessary for the correct placement and
>> alignment of digits, and as such must play a part in multi-digit
>> arithmetic. Inversions, misplacements and misalignments of digits
>> occur in what is sometimes referred to as "spatial dyscalculia" (c.f.
>> Hartje, 1987). Once again, however, the relevant spatial abilities are
>> two-dimensional. Moreover, as the author points out (pp.22-23), most
>> studies of multi-digit arithmetic have either shown no gender
>> differences or better performance by girls. (3) It is possible that
>> spatial representations of the mathematical relationships in a word
>> problem can facilitate its solution.
jd> But why should there be gender differences in spatial ability?
Gender differences in spatial ability have been attributed to
differences in the development of the brain: greater symmetry of the
hemispheres favours verbal fluency (as can be seen in girls), and
greater asymmetry favours spatial ability (as can be seen in boys) (see
Crow, these commentries). However, just because hemisphere differences
exist, and ability differences exists, it does not follow that they are
cause and effect.
jd> Also, one has to consider the other factors that are involved in
jd> mathematics, surely spatial ability is not the only important factor.
jd> If there are other important factors, then what if they also have
jd> gender differences?
I think that spatial ability is one of the least important factors in
mathematics; as Dowker points out, only Geometry, alignment of digits
and possibly spatial representations of mathematical relationships in a
word problem are areas of mathematics that involve spatial ability, and
these are a very small part of mathematics. To develop this line of
reasoning, perhaps it would be possible to investigate gender
differences in performance in, say, mechanics and statistics. Maybe the
assertion should be that boys are better at spatially orientated
mathematical tasks, but there is either no difference or girls are
better at pure mathematics.
>> T. J. Crow
>> The general finding to be explained is that (with great
>> inter-individual variation) mean performance on indices of verbal
>> fluency is greater in females, and is greater in males on tests of
>> spatial (including mathematical) ability.
jd> These are simply the mean measurements, one must also remember that
jd> there will be variation in the case of male and female ability.
jd> Therefore surely there are high performers who are female and those
jd> who are male.
Yes, this follows on from my previous arguments- a more informative
measure may be to look at the spread of abilities. If it shows that
there is are gender differences in spread, then it can be reasoned that
there are other factors to consider. For example, if boys have a
smaller standard deviation than girls, then one possible explanation is
that boys are encouraged to reach a certain standard regardless of
ability, whereas girls are only given more help if they show a
particular talent for mathematics. This would lead to differences in SD
because boys would all improve irrespective of ability, but only the
more talented girls would get better, leaving the others behind.
>> Annett (1985) has proposed that a single gene (the right-shift-factor)
>> can explain the transmission of handedness within families and
>> postulates that this gene (with a random element) determines which
>> hemisphere shall be dominant for language.
jd> Again, there are other factors that affect handedness. Consider the
jd> times when left handedness was considered to be wrong and children
jd> were forced to be right handed. Also, does this suggest that there
jd> are differeneces according to handedness for language ability? Has
jd> this been proven?
Handedness is certainly not random (ninety percent of people are
right handed, admittedly some social influences are brought to bear
on this, but social influences alone cannot account for the large
difference), so what is there to say that the gene that determines
the dominant hemisphere for language is random?
>> D. F. Bjorkland
>> As cognitive developmental psychologists, we have never been very
>> interested in sex differences. In fact, we admit to often treating sex
>> differences only as error variance in our analyses, or treating sex as
>> a "nuisance" variable -- something that must be controlled, like
>> stimulus presentation order, but nothing that is of interest for its
>> own sake. There have been several reasons for our disinterest. First,
>> in agreement with Geary, most differences in cognitive abilities
>> between the sexes are small in magnitude and are getting smaller with
>> time. Second, and more important, we have ignored sex differences in
>> our empirical research to date because we felt that there was no
>> driving theoretical reason to assess them. We do not feel that
>> male-female differences by themselves, with no theoretical basis,
>> promote an understanding behind the differences.
jd> Exactly. One has to consider other factors too, such as the
jd> environmental factors. However, is sex simply a 'nuisance' variable?
jd> Surely, it must play a part in some way.
First, I do not agree with the implication that Bjorkland seems to
be making that sex differences are getting smaller with time due to
evolution. Either he has based this assertion on empirical data from
psychologists, which will only have been available over the last few
hundred years, or postulation from woolly evidence dating from human
history. In the former case, the time frame is not large enough to
allow for any measurable difference in ability (evolution happens
over hundreds, if not thousands, of years), and in the latter case,
the evidence is not concrete enough, or sufficiently control to make
any valid comparisons. Any gender difference in cognitive ability
that can be reliably measured is surely due to social influences
ranging from the upbringing of young girls to equal rights for women
in the workplace.
Second, I disagree that we have ignored sex differences because
there is no driving theoretical reason to assess them- if this were
the attitude of all scientists, then nothing would ever be discovered
or invented, for example penicillin, which was discovered by
accident. I think the main reason that sex differences have not been
investigated is that it is only in the last few decades that women
have been accepted as people in their own right, and up until now,
sex differences may have been assumed to be a product of factors
other than physical arrangements of the brain. Now, it is more viable
to compare women to men because there are fewer confounding
variables, such as others'Ò expectations on performance, that would
have an effect on observed ability.
>> H. Davis
>> In short, counting should mean counting, whether performed by a rat, a
>> 6 year old child, or a mathematics professor.
jd> Is there any proof that a rat can count? Also, consider the
jd> differences in how a six year old and a professor count.
I agree there are differences in how a professor count and how a rat
counts, but at one time in both their lives they started with no
ability, or possibly an implicit ability, to perform this feat, so I
think you are missing Davis'Ò point. He is arguing that there are
differences in the way a rat and a professor count because of factors
such as practice. he argues that if a rat were exposed to the same
influences as a professor was, then a common vocabulary and
analytical system for numerical competence across species should be
>> If primary mathematical ability
>> requires exposure to teachers or other social agents to make itself
>> manifest, then perhaps so does the NC of animals.
jd> Surely one can have mathematical skills present without exposure to
jd> others. Surely there are many skills people have that haven't been
jd> exposed to others.
I agree with Davis in that there may be an implicit numerical
ability, but this is far removed from mathematical competence. I
think mathematical skill is a skill contrived by humans for like the
skill of writing is. I do not think that numerical competence can be
compared between humans and animals for this very reason. This would
be like trying to compare writing ability: in this case it is obvious
to see why an animal such as a rat cannot write- it cannot hold a
pencil- who is to say that there is a similar, if less salient reason
why animals are not as numerically competent?
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