## PHILOSOPHY OF MATHEMATICS EDUCATION JOURNAL 11 (1999)
Lim Chap Sam and Paul Ernest University of Exeter
Mathematics is a mysterious subject, and a number of myths are
associated with it. These myths include commonly expressed views including:
"mathematics is just computation", "mathematics is only for clever people
(and males)"; "your father is a maths teacher so you must be good at it
too". Such myths and images are widespread and seem to be present in many countries,
and among all classes of people. Moreover, most of these myths are negative (Buxton, 1981;
Ernest, 1996; Peterson, 1996). It is a matter of concern to us that these negative images
of mathematics might be among the factors that have led to the decrease in student
enrolment in mathematics and science in institutions of higher education, in the past
decade or two. However, there are relatively few systematic studies conducted on the
subject of
A review of past literature shows that there is not yet a consensus on
the definition of 'image of mathematics'. This term has been used loosely and
interchangeably with many other terms such as conceptions, views, attitudes and beliefs
about mathematics. However, in this study, I choose to adopt both Thompson's (1996) and
Rogers' (1992) suggestions and define the term
Interpretative approaches for data collection and analysis employing both quantitative methods and qualitative methods were applied to data collected from a questionnaire in stage one, and semi-structured interviews by telephone, in stage two. The questionnaire consists of two open-ended questions and nine structured questions. The open-ended question asked for respondents' images of mathematics and learning mathematics while the nine structured questions elicited responses on the attitudes, beliefs and images of mathematician of respondents. The semi-structured interview consisted of four sections. It aimed to probe (i) reasons for liking or disliking mathematics; (ii) memories of salient mathematics learning experiences in school; (iii) change of view after leaving school and (iv) other self-reported possible factors of influence on the participants' images of mathematics. These telephone interviews lasted between 4 to 30 minutes each, with a mean length of 12 minutes per interview. All interviews were transcribed and analysed using the qualitative computer software, NUD*ist version 4.
548 'adults' (aged 17 years old and above) in the UK responded to the questionnaires and 62 of them were interviewed in the follow-up telephone interviews. These respondents came from four occupational subgroups: professionals; managerial and technical; skilled; and unskilled workers; as well as teachers and students (both mathematics and non-mathematics specialists). (Brief details are given below of a parallel Malaysian sample).
Some of the major findings of the study are highlighted and discussed below.
Slightly more than half of the UK total sample reported a liking of mathematics but one third of them reported a disliking of mathematics. The percentage disliking mathematics is highest among the youths of age group 17-20 years (44.0%) and among the students of non-mathematics options (49.6%). These alarming negative attitudes towards mathematics among the youths and the non-mathematics students raise concern, because these groups represent the future workforce of the nation.
Most respondents did not seem to differentiate their image of mathematics from their image of learning mathematics. This finding suggests the close relationship between these two types of images. Perhaps most people's image of mathematics is derived from their experiences of learning mathematics in school. In addition, based on our data we found that these images are often unique and personal, and multifaceted and diffuse. Nevertheless, from the analysis, five views that were commonly shared by the UK sample are:
For example, this view is characterised by
'... in maths, you got the right answer or the wrong answer. There isn't anything in between' (R115, text-unit 15, student adviser, age group 41-50, female, dislikes maths)
'maths is fun and [has] definite answer to work to' (R313, insurance sale assistant, age group 31-50, female, likes maths) According to this view, mathematics appears to have a power of certainty, such that,
'I just like the fact that you could get a solution and that nobody could say that you have done wrong. (R113, text-unit 7, science teacher, age group 51-60, female, likes maths) Due to this certainty, many people find learning mathematics rewarding and a source of achievement.
'Obviously if I can get the right answer, I feel achieving and rewarding' (R409, text-unit 11, retired soldier, age group over 60, male, likes maths) However, this view of definite right or wrong answer has hindered some people from liking mathematics. This is because they found mathematics lacks of creativity and it is not discussible.
For example, Mathematics is viewed as:
' a human tool to calculate and predict' (R047) 'an essential tool for everyday life' (R128) While this view of mathematics as a practical and useful tool might hold for those who reported liking, for others who dislike the subject, mathematics is perceived to be irrelevant and comprises
'a lot of things which I never used' (R492) Mathematics is regarded as important and essential to learn because mathematical knowledge is necessary and useful in both daily life and at work. Some reported images of mathematics strongly identified with particular uses such as 'banking account' (R494) or 'VAT receipts' (R459).
Some examples are mathematics is viewed as comprising or represented by:
'numbers and equations' (R005) 'figures and sums' (R340) 'multiply, minus, add, divide' (R453) For many of these participants, mathematics is seen as sets of rules and procedures to be followed and memorised. For some people, this is a pleasure because mathematics is
'formulae, involved and exciting' (R038) and some of them just
'like playing around with numbers, equations, finding solution to problems' (R119, text-unit 3, likes maths). But to others, mathematics is
'rules, formulae learnt before understanding' (R109) This is well described by a middle-aged housewife who reported disliking mathematics.
'Sometimes if you can't remember the formula then you don't know how to get the answer,...' (text-unit 5) and '...you got to stare on the wall. You are mentally blackout - it is all gone!' (text-unit 13) As a result, she said, ' I haven't have any interest at all. I find it [mathematics] boring, doing numbers and things like that' (text-unit 37) (R267, housewife, 31-50, dislikes maths). Many people perceived mathematics as a collection of numbers and symbols, rules and procedures that are needed to be followed and memorised, that is, the symbolic view of mathematics. Many of them indicated that they do not understand mathematics and do not see the purpose of this repetitive process. This perceived irrelevance has resulted in boredom and sometimes confusion for some of them.
The main characteristic of this view is that mathematics is taken as a set of problems to be solved. The enjoyment of learning mathematics then lies in the exploration of these problems and the derivation of solutions. Many reported a sense of satisfaction and achievement when they found the solution to a mathematical problem.
'...there is always enormous pleasure in manipulating numbers a kind of problem solving activity in its own right. So, if I have a mathematical problem that I basically, provided that I understand the rules, then I have quite a lot of pleasure in manipulating that out of whatever it might be. Although I am largely a verbal person, I am also have great fun in solving mathematical problems ' (R061, text-unit 5, university lecturer (psychology), age group 51-60, male, likes maths) In the process of solving mathematical problems, many believe that mathematics stimulates logical thinking and functions as an analytical tool. This is because mathematics is
'logical stimulation' (R100) as well as 'logical - organises things in order' (R122). Subsequently, mathematics is viewed as a means to model the world. A few respondents expressed this as: maths is
'a way to model the physical world' (R301) and
'problem solving - explaining physical processes' (R113). Associated with these images is the view that mathematics learning is 'learning to think correctly and logically' (R384) and it is possibly hierarchical in the sense that, one needs to 'take small steps to understand difficult problems' (R122). Moreover, mathematics learning is all about 'making order out of chaos' (R118). This problem solving view is more often held by those who reported liking than those who reported dislike of mathematics. For the former, this was also given as one of the main reasons for liking mathematics. Perhaps they enjoyed the challenge in searching for solutions to mathematical problems, and felt a sense of satisfaction when they found a solution. In contrast, for those reporting dislike mathematics, learning mathematics is like 'solving a complicated puzzle: there is an answer but it takes long time to find it' (R361) Many of them find mathematics 'difficult' (R434) and learning mathematics is more like 'passing hurdles' (R470) than doing something enjoyable for them.
For these respondents, mathematics is seen as mysterious, foreign and incomprehensible but yet, it is also
'like a sunset - unique and beautiful' (R168). There are on one hand, those who like mathematics because, as they report:
'...I like the elegance of mathematics. The proofs and theories are very elegant. There is ...like recognising the patterns of mathematics, I found it very interesting' (R193, text-unit 3, IT trainer, male, 21-30, like maths). Due to the elegance and aesthetic appeal of mathematics, and for others, the mysterious nature of mathematics, learning mathematics becomes 'an exploration into another world' (R113) or ' a voyage of discoveries' (R116) that is 'fun and challenging' (R140) for those who reported a liking of mathematics. On the other hand, the complexity and abstract nature of mathematics also drives away some people's interest in mathematics because they found mathematics incomprehensible and confusing. They found themselves like 'groping through fog' (R412) or 'wandering in a desert - with the odd oasis of understanding' (R363). The view of mathematics as an enigma was expressed by a small minority of the sample, particularly those who reported liking of mathematics and those who has direct involvement in mathematics such as mathematics students and mathematics teachers. Perhaps this might be one of the main reasons that have attracted these people to undertake mathematics-related studies and careers. However, in general, most people were inclined to hold a composite image made up of elements from several of these five common shared views rather than subscribing to a single view.
Lakoff and Johnson (1980) point out that "metaphor is pervasive in our everyday life, not just in language but in thought and action" (p.3). Perhaps it is then not unusual for people to express their images in the form of metaphors. In this study, 27% of the respondents expressed their image of mathematics in the forms of metaphors, while 66% of them gave their images of learning mathematics in metaphoric terms. It is interesting to find the variety and diversity of these metaphors, besides the commonalities that they shared. Some common metaphors used by most respondents are described and discussed as follows:
This was the most common metaphor given by the sample. Some examples are: maths is a ' 'challenging journey - rewarded by arrived at your destination' (R255) Learning mathematics is like
' an easy stroll on a windy day' (R034) or 'running uphill - difficult but you get there' (R376) Implicitly, the journey metaphor highlighted the close relationship between images of mathematics and images of learning mathematics. For many, mathematics as a challenging journey elicits the experience or process of learning mathematics. For some people, the experience of learning mathematics might be like a struggle in a journey such as,
'walking through mud' (R155) or an uphill struggle' (R417). These metaphors indirectly indicate the difficulty and frustration that were experienced by these people, especially those reported a disliking in learning mathematics in school. Some of them felt that learning mathematics is like
'being stuck in a bus queue' (R268) or 'climbing a topless mountain' (R104). They felt helpless and anxious about their inability to understand mathematics. A salesman expressed his image of learning mathematics as 'driving a Boeing 707 ( I don't fly)' (R462). Later in the interview, he explained that,
It is an expression that I used when someone say it is easy, all you got to do is this. It is easy if you can fly and you know how to drive a Boeing 707. But if you sit in front of a 707, where do you start? You know that you want to take off. You know that you want to start the engine. You want to go forward but you don't know how. So, you have the interest. Sometimes the interest is there but the lack of comprehension of how the whole thing work, it means you don't progress any further. ' (text-unit 19, male, over 50, dislikes maths) Thus it is sad to notice that some people might be full of interest before starting the journey of learning mathematics, but their interest was killed off by their lack of understanding. Whose fault is this? In contrast, particularly those who reported a liking of mathematics viewed these journeys as explorations or discoveries. For them, learning mathematics is like
'exploring - there is always something new to know' (R331) or 'being an explorer-finding new paths and worlds' (R364) For these people, mathematics is a journey to discover new things, new knowledge and new insights. A middle-aged mathematics teacher described his images of learning mathematics as 'the best sort of travelling in a new land' and he explained that,
'Well, when you are studying a new area and you are having to grasp it sometimes, you know, because you haven't done that sort of mathematics before and you begin to realise why some statements, some theorems in mathematics are true or you begin to see the use of that theorem can have, you know, the statement is making connection without the thing, and that I find interesting' (R293, text-unit 9, male, mathematics teacher, 41-50, likes maths) These results suggest that it was the joy of discovering new understanding in mathematics that attracted them to get interested in mathematics. Even though many of them also found learning mathematics a difficult journey like,
'a journey through a dark tunnel with a light at the end' (R139) or 'walking through sand - hard work but put in effort, you'll get there' (R136) Therefore, there is this sense of achievement and satisfaction that
encourage these people to work hard and to strike for the solution. Implicitly these
metaphors indicate that there is a definite solution for each mathematics problem.
Learning mathematics is There was also a young mathematics student who uses journey metaphor to illustrate her change of view from absolutist to fallibilitist (Ernest, 1991):
'I mean we always brought up with that of the right and wrong answer
and suddenly we were told that was not the most important part of maths, the most
important part is (R133, text-unit 13) According to her view, she was brought up with an absolutist view, but now she was exposed to an alternative view that there are many strategies and possible answers to a mathematical problem. To her, learning mathematics should be focused on 'process' rather than 'product'. This fallibilist view, however, was only shared by very few of the respondents. It is interesting to read that some undergraduate students and tutors in Allen and Shiu's (1997) study also gave the metaphor of mathematics as a journey. They categorised these responses under one of their four categories: 'struggle leading to success'. Two very similar responses from the tutors are: learning mathematics is like
'climbing a hill: - hard work where you follow the path you're on - and then the joy and satisfaction of being at the top ' (T3) 'climbing a hill. The higher you get the clearer the view of surrounding countryside - as you can see more the links and layout and connections become more obvious. (T18). (p.10) In short, mathematics as a journey metaphor indicates that mathematics learning is a difficult process that needs a lot of effort and time. However, there are two possible extreme outcomes: either you reach the destination (obtain the solution) and feel happy and satisfied, or the opposite of having failed to solve the problem and feel disappointed and frustrated.
Closely linked to a utilitarian view of mathematics, some of mathematics images also see it as an important and necessary skill for daily life and work. Mathematics is
"an essential basic skill for society" (R092). Similarly, learning mathematics is like:
"learning to walk, we've all got to" (R009) Once again, the skill metaphors reflect the view that mathematics is a skill that is not always easy to learn, just like
"learning a musical instrument, some are easier and others are extremely hard" (R542). Nevertheless, at least nine respondents were attracted to learn mathematics because to them, learning mathematics is acquiring a skill, like
"riding a bike - once learnt never forgotten" (R123) There were some respondents who viewed mathematics as a set of skills that is hierarchical, like " brick laying - each brick is the foundation for the next block" (R081). Others believed that mathematics learning is a skill that needs memorisation such as
"learning law: rules and cases to remember in total" (R485) or needing a lot of practice:
"learning to ride a bike - takes plenty of practice" (r520). Likewise, the skill metaphor for mathematics suggests that learning it could be a skill that grows easier for some people, like
"playing the stock exchange - once you get the hang of it, it's ok" (R469) or getting more difficult for others, just like
"riding a bike, simple enough until you come to a mountain" (R066). In summary, mathematics as a skill metaphor suggests that mathematics is viewed in terms of its utilitarian value, while learning mathematics viewed as a skill is seem to be hierarchical, needing memorisation and lots of practice; difficult to be mastered by some but easy for others.
Besides the above three most common metaphors in our sample, mathematics is also commonly given in terms of an experience, in particular, a negative daily life experience. Mathematics as a metaphor of daily life experience have been used to portray a wide variety of feelings, from as enjoyable as
"like playing with my children never tiresome" (R526) to as painful as
"a pain in the arm" (R181) or "a big headache" (R523). More often these metaphors indicate learning mathematics as a negative experience than as a positive one. For example, learning mathematics is as boring as "going to sleep" (R003) or "watching paint dry" (R006) as painful as "having a tooth pulled out" (R078) as frustrating as " being stuck in a bus queue" (R268) or as (presumably) unpleasant as "having to walk to work in the rain" (R110). Nevertheless there were a few respondents indicated positive experience of learning mathematics like,
"watching TV never want to switch off" (R526) or having
"a cold shower - refreshing" (R061). A few respondents experienced mathematics learning as something that is necessary to do even though they was feeling of "unwillingness" or "forced to". For example, learning mathematics is like
"being dragged unwillingly along" (R278) or "like going to see the doctor - horrible but sometimes necessary" (R208). A young librarian gave an interesting metaphor for mathematics as "sitting in a class on a hot day wanting to be somewhere else", and she explained in the interview that,
R207: I think it is something personally for somebody in the classroom you find for every subject, especially if it is the subject that you don't like. That is the last place that you want to be in a maths room when the sun is shining outside. [laughter all along] I: Do you mean you felt that you are in a way forced to do it? R207: Ya. Yes, maths is something that you have to do to do anything later on. You have to get the maths qualification. And if you not good at it or you don't like it, then it is very much, you know, struggle to do it.' (R207, text-unit 11-13, female, age 21-30, dislikes maths) In summary, mathematics is viewed as part of life experience and learning mathematics is viewed as more often related to negative experience rather than the positive one for most people, at least for this sample.
Closely related to problem solving, some respondents viewed mathematics in terms of games and puzzles. It is like:
"a brain teaser - a puzzle to be solved" (R388) and learning mathematics becomes
"finding your way through the maze" (R174); or "playing chess - absorbing and challenging" (R220). Viewing mathematics as a game or a puzzle to be solved has made mathematics learning fun and challenging for many people. Mathematics is
"fun when everything works out but remain a challenge" (R470) or learning mathematics is like playing
" a jigsaw puzzle-slow but relaxing- it makes your mind work" (R389) Interestingly, out of the 10 respondents who expressed their images of mathematics as games or puzzles, nine of them are female and all of them reported liking mathematics. These results indicate that some female respondents like mathematics because they viewed mathematics as solving puzzles and playing games. How widespread this is, is not known.
Consistent with past research studies (Cesar, 1995, APU surveys, 1988,1991, Kogelman & Warren, 1978, Burton, 1989, Vanayan et al., 1997), there were two myths of mathematics still commonly shared by the majority of the sample. They are (a) mathematics is difficult, and (b) mathematics is only for the clever "ones", but one traditional myth that, (c) mathematics is a male domain has been challenged and was supported by only 20% of the sample.
The results highlighted five main possible factors of influence on an adult's image of mathematics. Arranging them in the order of decreasing importance (that is by frequency of occurrence), they are: mathematics learning experiences in school, personalities and teaching styles of mathematics teachers parental support and motivation (mostly father) an individual's own personal interest in mathematics (whatever its source), and peer influence and support.
The majority of the UK sample (95.4%) believed that some people are better at mathematics than others. Half of them regarded this mathematical ability as something that is inherited from parents. The mathematics teacher was perceived as the second most important attributing factor, followed by effort and perseverance. These beliefs about mathematical ability and its possible contributing factors were similar across gender, age, and occupational groupings. However, the belief that there are gender differences in mathematical abilities varies significantly between genders, age groups and occupational groupings. The male respondents tend more to believe that men are better in mathematics (even though this a minority view) but the female respondents tend to believe that both men and women are equally good at mathematics. In terms of age groupings, the youth and the older age groups had the lowest level of belief that both men and women are equally good at mathematics. Significant variations among the occupational groupings suggest that careers might have impact on people's belief about gender differences in mathematical ability. The teachers were the most likely to believe that both genders have equal ability in mathematics (although all that we observed was a correlation). But the professionals and the non-mathematics students were the two dominant groups that held the belief that mathematics is a male domain.
As a substudy, we also explored the image of mathematics of a sample of 406 Malaysian students and teachers. A comparison between the Malaysian and UK student and teacher samples show that there are cultural differences in the images of mathematics and learning mathematics between them. These differences are most significant in terms of beliefs about success in mathematics. As shown in Figure 1, 68% of the UK students and teachers listed inherited mathematical ability as the most important or second most important factor contributing to overall mathematical ability, whereas 79% of the Malaysian sample listed effort and perseverance as the most important or second most important contributing factor. This finding is interesting because it indicates there is a cultural difference in perceptions of the factors contributing to a person's mathematical ability. However, inherited mathematical ability is still considered as an important factor in Malaysia, because half of the Malaysian sample ranked it as the most important or second most important factor. This suggests that the belief that "mathematics is more for the clever ones" might also be held by some of the Malaysian students and teachers too.
OMITTED
Besides effort and mathematical ability, mathematics teachers seem to be the next most commonly cited contributing factor in both countries. Moreover, 10 % more of the Malaysians than the UK sample attributed mathematics teachers as the most or second most important factors for mathematical ability. The analysis thus indicates indirectly the important role of mathematics teachers in the sample's image of mathematics. In summary, while the Malaysian student and teacher samples attributed success in mathematics more to one's own effort, their UK counterparts attributed it more to inherent mathematical ability. This finding concurs with other cross-cultural studies (for example, Ryckman and Mizokawa, 1988; Huang and Waxman, 1997), namely that Eastern societies tend to most value effort, perseverance and hard work whereas Western families tend to view mathematical ability and creativity as the most important contributing factors for success in mathematics.
There appear to be significant differences in the images of mathematics between those who reported liking and disliking of mathematics. The findings reported here suggest that, at least in this sample, people's images and beliefs about mathematics are correlated with their liking of mathematics, and with their attitudes towards learning of mathematics. Thus, people's overall images of mathematics may causally impact on people's attitudes (although converse influences are also possible). Therefore, the issue of how to present mathematics in the most appealing way to students, to enhance both images and attitudes, whilst teaching it effectively, remain a challenge for both mathematics teachers and mathematics educators. A more detailed analysis of the study reported here will appear as Lim (1999).
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