本文是数学专业的留学生作业范例，题目是“Concepts of Gender and Mathematics(性别与数学的概念)”，1896年，查尔斯·达尔文写道“男女智力上的主要区别表现在男人无论从事什么工作，都能比女人获得更高的地位.......如果男人能够在许多学科上明显优于女人，那么男人的平均智力就必须高于女人。”(达尔文见Walkerdine, 1989.第1页)
Introduction介绍
In 1896 Charles Darwin wrote“The chief distinction in the intellectual powers of the two sexes is shewn by the man’s attaining to a higher eminence, in whatever he takes up, than can women…….if men are capable of a decided pre-eminence over women in many subjects the average mental power of a man must be above that of women.” (Darwin see Walkerdine, 1989. p. 1)
After many years of social change we would expect these views on women to have changed. Indeed, it is not common to hear that, mentally, men are better than women, as women have proved themselves to be just as capable. Take Carol Vorderman for example, in my opinion the most famous female mathematician that I know. She is so good at mental arithmetic that she co-hosted Countdown for 26 years (Vorderman, WWW). However, maths is still perceived as a male dominated subject and it is no wonder that women believe they are inferior to men at mathematics. Even though this myth is meant to be a statistical statement, many women interpret it to mean they cannot do mathematics, having a distressing effect on individuals (Gray, 1996. p. 27).
经过多年的社会变革，我们原以为这些对妇女的看法已经改变了。事实上，男人在心理上比女人强的说法并不常见，因为女人已经证明了自己和女人一样有能力。以Carol Vorderman为例，她是我认识的最著名的女数学家。她非常擅长心算，曾与人合作主持《倒计时》节目26年。然而，数学仍然被认为是男性主导的学科，难怪女性认为她们在数学方面不如男性。尽管这一荒诞的说法是一种统计上的说法，但许多女性将其解释为她们不会数学，这对个体造成了痛苦的影响(Gray, 1996)。27页)。
Walkerdine states that“Women, after all, are clearly irrational, illogical and too close to their emotions to be good at mathematics. Or so the story goes.” (Walkerdine, 1989. p. 1)
If certain people actually take on this opinion; that girls are ‘lacking’ in mathematical ability (Walkerdine, 1989), how are girls supposed to have the confidence to believe they can do mathematics, when “girls report less confidence in their mathematical ability even when the girls achieve at the same level of boys”? (Fennema see Orlich et al, 2007. p. 52). It is suggested in Burton (1990) that boys are getting more encouragement and praise in the classroom than girls, which builds a lack of confidence causing a negative effect on the girls ability to learn (Burton, 1990). In this essay I will try and address the biological reasoning behind why girl’s can’t do maths and relate this to the ways in which girl’s learn.
The Biology生物学
There are two hemispheres to the brain, the left and right. These are specialised, to some extent, to perform different tasks. People usually have a preference to one or the other, although certain people are “whole brained” in their thinking, and therefore work just as well in either (Funderstanding, WWW).
大脑有两个半球，左半球和右半球。在某种程度上，它们是专门用来执行不同任务的。人们通常偏爱其中一种，然而某些人的思维是“全脑”的，因此在这两种方式中都能工作得很好(Funderstanding, WWW)。
Abigail Norfleet James (2009) has researched into brain differences in boys and girls. She found that language functions and the memory of certain nouns are lateralized to the left hemisphere, whereas mathematical performance and memory of pictures and topography are lateralized to the right hemisphere. Not only did she find differences in the left and right brain, she also researched into certain parts of the brain which help us to learn mathematics using memory and emotions (Norfleet James, 2009). The hippocampus, located inside the medial temporal lobe, plays an important part in long term memory and spatial navigation (Hippocampus, WWW). The Amygdala, which is also found deep in the medial temporal lobe, performs a primary role in the processing of memory and emotional reactions (Amygdala, WWW). The research showed that as the hippocampus and the left side of the brain develops faster in girls, they excel in language, verbalising and working through situations logically. The right side of the brain and the Amygdala develops faster in boys, so their mathematical calculations and performance are prominent in their learning (Norfleet James, 2009).
“To oversimplify a complex issue, the left hemisphere in most humans is primarily concerned with language based behaviour and with the cognitive skills we might crudely characterize as analytical or logical. It has become apparent recently that the right hemisphere is far superior to the left in most visual and spatial abilities” (Davies and Hersh, 1995. p. 346)
“把这个复杂的问题简单化，大多数人的左半球主要与基于语言的行为和我们可能粗略地定义为分析或逻辑的认知技能有关。”近来，右半球在大多数视觉和空间能力方面明显优于左半球”(Davies and Hersh, 1995)。p . 346)
If we look at certain traits from the opposite sides of the brain, we can see that the left brain is notorious for being analytical and sequential, rational and thinking objectively where as the right side is identified with spatial intelligence, thinking randomly and using an intuitive approach to situations (Funderstanding, WWW). Research suggests that the male brain holds an advantage with making quick decisions from lists where as the female brain works inductively and needs much more information to make that same decision (Gurian et al, 2001). It can be said therefore, that women are more inclined to think in a left brained way and men, a right brained way. As a result of this boys tend to have significantly better spatial skills and find visualizing abstract objects easier. When it comes to learning maths we need to use the whole brain; the left and right hemispheres and the frontal lobe (Gurian et al, 2001. p. 51).
Primary and Secondary Schools tend to have more female than male teachers (Statistics, WWW). In my opinion, female teachers are more likely to teach using the traits found in the left side of the brain as this is what they naturally excel at. This could possibly be seen as the preferred way of teaching as the traits linked to the right side of the brain are usually connected with impulsive actions and general disruption in class (Gurian et al, 2001). In schools, students are taught to think in a successive way, where they build on previous knowledge. They are taught logical steps, which gives a method they then apply to a question to gain an answer. If, however, the student comes across a problem they have not faced before, they may lack the intuitive skills that would allow them to solve this particular, difficult problem. This method of teaching is sometimes seen as being biased towards the female way of learning, as it develops the skills which girls are already, naturally gifted with. However, to study maths to a higher level, rules and methods can only get you so far. The right hand side of the brain allows you to look at problems as a “whole” rather than in individual, single steps. As boys naturally develop this part of their brains, they are already able to think subjectively (to relate their problems to personal experiences or previous challenges they have faced), a skill girls have to learn. Thus, with the schooling system developing the boys left hemispheres, they are capable of answering much more difficult, unseen questions, which may require more than just the taught rules.
Girls are generally left brained and are therefore disadvantaged, as they are educated to think in one way, their right hemisphere being overlooked. The right brained boys, however, are taught these left brained skills, and coupled with their natural way of thinking allows them to be “ whole brained” and much more efficient at the more difficult maths problems. (Fennema & Leder, 1990; Gurian et al, 2001; Norfleet James, 2009)
If schools are focusing on a certain method which disadvantages either girls or boys in ways of their brain functions, are they also biased in the way they are teaching, with regards to how boys and girls learn?
Learning Styles学习风格
Along with having a stronger side of the brain, girls are also inclined to use their minds in a specific way. This is usually related to the way they think. Research has shown that there are two types of reasoning; abstract and concrete.
除了有较强的大脑侧，女孩也倾向于以特定的方式使用他们的大脑。这通常与他们的思维方式有关。研究表明，推理有两种类型;抽象的和具体的。
Abstract is“not seeing or touching the thing and yet still being able to calculate it. For example, when mathematics is taught on a blackboard, boys often do better at it than girls.” (Gurian et al, 2001. p. 45).
If information was to be taken from the blackboard and put onto, for instance, number lines, which are inevitably more concrete, girls tend to thrive. For girls to understand the more abstract parts of maths, for example geometry, they must bring these aspects to life. In Primary School learning, the idea of Logo (a computerised turtle which moves round the screen following directions) makes the abstract ideas of direction and angles become more concrete. In fact many IT programs used in the classroom have been designed to give a concrete illustration of an abstract idea (Skrimshaw, 1993).
Furthermore, people have a predisposition to relate to a particular learning strategy. It has been suggested that girls tend to conform to the serialist, or analytic, model of thinking (Clark and Millard, 1998). These are “one step at a time learners” (Scott-Hodgetts, 1986. p. 68) who work through problems methodically, leading to instrumental understanding. In an article on the different types of understanding, Skemp (1976) described instrumental as ‘rules without reasons’. What he did not realise was, “that for many the possession of such a rule and the ability to use it, is what they mean by ‘understanding’” (Skemp, 1976. p. 2).
Aside from serialistic understanding we have the holistic, or intuitive, model of thinking. Scott-Hodgetts claimed that holists like to take “an exploratory way, working first towards an understanding of an overall framework” (Scott-Hodgetts, 1986. p. 68). The approach that holists take of looking at the whole framework and then filling in the gaps is a way of relational understanding, not only knowing which method works, but why. So although it may take longer for a pupil to become a relational learner, as there is more content, it is never the less, easier, for the holist thinker to then adapt this method and apply it to unknown problems (Skemp, 1976). This is seen as the preferable method to learning as it allows students to link together different concepts of mathematics. Research shows that boys coincide with this manner of thinking (Clark and Millard, 1998).
A lucky few tend to be able to switch between both the holist and the serialist approach. These students are called versatile learners. In higher level mathematics it becomes very important to be able to switch your view point, from looking at a problem analytically to globally, in order to see the problem as a whole.
少数幸运的玩家能够在整体模式和连续模式之间切换。这些学生被称为多才多艺的学习者。在高等数学中，为了把问题看成一个整体，能够转换你的观点，从分析的角度看问题，到全局的角度看问题，是非常重要的。
“… pupils are expected to do more than simply reproduce items of knowledge, as they have been taught. They must, for example, also be able to restructure bodies of knowledge in ways appropriate to different problems – a difficult task for the serialists because of their inclination to learn sequentially, without necessarily forming an overall picture of the relationships involved. … whilst holists are busy speculating about relationships, and discovering the connections between initially disjoint areas of mathematics, it may not even occur to serialists to begin to look for such links.” (Scott-Hodgetts, 1986. p. 73)
If you are capable therefore of using both of these techniques then surely you gain great advantage over your fellow workers?
There are a few issues regarding these learning strategies. In Primary schools, students should have
“the freedom to develop their ideas using their preferred learning strategies, however, teachers do sometimes impose their own strategies upon pupils” (Scott-Hodgetts, 1986. p. 70).
I believe at such a young age, it is easier for the teacher to teach rules and for the pupil to learn these, even if they have no understanding of them. Take long multiplication for example. I was taught a step by step procedure which gave me an answer. At this age I had very little idea about why we added a zero at the end of the second line of computation, or why we ‘carried a one’; I was just told that is how it is done. This method of learning, remembering and applying, confirms to serialists that this approach is best and leads to success. Even in Secondary school it is known that “teacher exposition tends to be serialistic in style” (Scott-Hodgetts, 1986. p. 70),
Scott-Hodgetts (1986) claims that children who are predisposed to a serialist approach are less likely to become versatile learners than those who think more holistically, purely because of the way that they have been taught (Scott-Hodgetts, 1986). However, it has been discovered that if serialists are exposed to a holistic style of teaching they are just as capable of gaining the same understanding, at the time, as the holists. Although, Pask and Scott, claim that in the long term, such teaching has a “genuine effect on reducing efficiency”(Pask and Scott see Scott-Hodgetts, 1986. p.72). This inconsistency of teaching styles could explain why certain pupils are capable of working well in class, but then not performing well in mathematics examinations. (Scott-Hodgetts, 1986)
At the same time that serialists are convincing themselves that learning and remembering their method is the way to gain the top grades, holistic learners will be shown the effectiveness of a different strategy. They have then begun to be a versatile learner before they reach secondary education.
与此同时，系列学习者正在说服自己，学习和记住他们的方法是获得最高分数的方法，全面学习者将显示出不同策略的有效性。在他们进入中学教育之前，他们已经开始成为一个多才多艺的学习者。
Conclusion结论
After looking at the brain and the mind in relation to learning mathematics, I feel that it is not that girls cannot do maths, on the contrary girls have many skills that would make them adept at the subject, it is more that girls are not taught the right techniques, which would broaden their minds when it comes to tackling harder mathematical problems. I believe that if girls were taught in a holistic way they would, like the boys, learn how to look at problems as a whole and become more proficient at their mathematics. However, as Pask and Scott (see Scott-Hodgetts, 1986) pointed out, mixing learning techniques can become a disadvantage in the long run to the serialistic learners. But if Holists are able to learn from a serialistic point of view, why can’t serialist learn from a holistic point of view? Surely this would increase the number of versatile learners?
Drawing on my own experiences as a woman, if I look at the skills I have developed to reach my current level in mathematics, I know that I have needed to be analytical in most of the problems I have encountered, as well as being able to observe the problem in full. Surprisingly, being able to think randomly as well as logically can sometimes help to solve the most difficult problems. Evidence shows that even though boys are more naturally intuitive and seem to have many of the skills needed in progressing in mathematics, girls are analytical and sequential which are evidently needed for mathematics. The natural abilities that most girls seem to inherit are the key components when first learning maths. It is only the boy’s ability to understand this new way of learning, which automatically makes them versatile learners, that gives them the edge over girls. Being able to look at problems analytically and form an algorithm to obtain an answer is vital to any problem solving subject, and therefore it is easy to see why girls can enjoy, and can be good at mathematics. We can see from the following statistics that boys excel when the maths becomes more complicated. In Teresa Smart’s article on Gender and Maths in England and Wales she explains that even though there are fewer boys than girls taking GCSE mathematics, more boys continue maths on to A-Level. Only 35% of pupils taking A-Level in 1992 were girls, which shows that the percentage of girls studying mathematics decreases as the level of mathematics increases (Smart, T. 1996). This reiterates that girls can do maths; they are just not taught the necessary skills which are needed to continue the subject to A-Level or even university.
However, in researching this topic I found that it is not only our brain or the way we learn which causes us to think we, as girls, are not capable at maths, but also society and the perception they have on girl’s and mathematics. Until recently girls have very much been considered for different jobs than boys. Careers advice to girls in the fifth form in the 1980’s was based on “retail and clerical types of employment” (Burton, 1986). If girls were adamant that they wanted to study what was considered to be a more male subject, they were expected to achieve higher grades or in some cases, not considered for the jobs at all. In fact, one fifth form girl, in which the section of Burtons book is written, was told by her interviewer that he discouraged women from taking opportunities which would better their career (taking day-release) as they “tend to leave and have babies” (Burton, 1986). In today’s job market women are considered equal to men, however, I feel, it will take a few years before we see equal numbers of women and men in male dominated occupations.
It is also important to point out that the content of this essay relates to the general assumption of boys and girls. I know myself, that after extensive research I would consider myself to be a holistic thinker, contrary to what the research suggests I should be as a girl. Does this mean that those female students who study university mathematics and indeed go further, think in a more male way, or are they just fortunate enough to have become versatile learners despite what they have been taught at school?
The statement; Girl’s can’t do Maths: Myth or Fact, could be argued to be both myth and fact. We have seen that girls are less likely to be better at maths biologically but if our teaching system was different would we gain different results?
该声明;女孩不会数学:荒诞的说法还是事实，可以认为既是荒诞的说法又是事实。我们已经看到，女孩在生物学上的数学成绩可能不太好，但如果我们的教学体系不同，我们会得到不同的结果吗?
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