Secondary School Mathematics in Perspective: Conceptions of its Nature and Relevance Sandra Frid Science and Mathematics Education Centre Curtin University of Technology This study addresses a vital but neglected factor of the effect of recent curriculum changes in secondary mathematics: the conceptions of the recipients of the curriculum. Specifically, it investigates the nature of students' conceptions of what mathematics is, the intentions of school mathematics, and the outcomes of school mathematics. Interviews with a selection of Years 10, 11 and 12 students were conducted to obtain detailed information on individuals' perceptions of mathematics and related curricula. The sample was taken from a large government secondary school and an independent girls' school, both in the metropolitan region of Perth, Western Australia. Results indicated a number of conflicts in the expectations and needs of students in relation to mathematics learning. More specifically, there was incongruity between conceptions of the intents of mathematics in a learning versus a school or personal context. INTRODUCTION Publication of the Finn Review (1991), Mayer Committee Report (1992) and the Carmichael Report (1992) has moved the national agenda for mathematics education firmly in the direction of competency based training. Among these and other recent efforts (eg. Australian Education Council, 1991) to relate aims and achievements in mathematics education to societal contexts, employment and national needs, are research reports that indicate school mathematics and students' learning of it are not perceived to meet vocational needs, nor the goal of mathematical study as an intellectually broad, rich, and personally relevant endeavour (for example, Foyster, 1988; Praegar, 1993; Bradley & Kemp, 1993). However, high school mathematics is claimed to be an important foundation for later learning at university and TAFE institutions, and for employment. Research at the school level has indicated that learners' experiences of mathematics influence their attitudes to and outcomes in learning mathematics (Clarke, 1985; Crawford, Gordon, Nicholas & Prosser, 1993; Resnick, 1987). In particular, there are indications that students' conceptions of mathematics affect the quality of related cognitive activities and learning outcomes. The ways in which students interpret the context of their mathematical learning and hence the ways in which they relate to in and out of school mathematics endeavours has also been shown to influence their mathematics performance (Crawford, 1990; Cobb, Yackel & Wood, 1992; Lave, 1988; Steffe & Cobb, 1988). In fact, recent studies in Western Australia have helped make apparent the wide variety of conceptions, even amongst high ability students, about the nature and purpose of mathematics and its study in and out of school contexts (Miller, Malone & Kandl, 1992; Frid & Malone, 1993; White & Taylor, 1994). These studies have highlighted the need to better understand the individuality of students who are presently studying mathematics in high school. For curriculum developers and teachers, the simplicity of the notion that upper secondary mathematics students can all be treated similarly can no longer be entertained. The cohort of students entering upper secondary school has expanded and changed in the last few years. Although new mathematics curricula have been developed and implemented, there has been little attempt to examine the views of this group of students in terms of their conceptions of the mathematics they are studying, their motivations for studying mathematics or their approaches to learning mathematics. The magnitude of the issues involved in the motivation for, and design and implementation of new curricula can be seen in the recent professional debates and media publicity about endorsing a national mathematics curriculum. This study will address a neglected factor in these discussions. It will focus on what has been indicated in the previous paragraphs as a vital component of the effect of any curricula - the conceptions of the recipients of the curricula. Thus, the main aim of this study is: To investigate the nature of high school students' conceptions of mathematics. Specifically, the study focused on the following three research questions: 1. What do students think mathematics is? 2. What do students think are the intentions of mathematics study and why mathematics is included in school programs? 3. What are the predominant affective components of students' mathematics experiences? The importance of addressing these questions lies in the fact that until mathematics educators address and come to understand what mathematics is in students' eyes we are not in a position to understand the effect mathematics courses are having on students. For example, we do not know what is happening to students' personal views of the purpose of mathematics. What curriculum developers, teachers, educational administrators, parents and the general public understand as mathematics and what students see as its nature and purpose are likely two different things. In light of the competency/numeracy push of present school aims, it must be asked if educators and/or society are either asking or expecting something from schools that schools are not in a position to deliver. Thus, the outcomes of this study will put educators in a better position to meet students' expectations and thereby influence them. In addition, it is well documented that once students graduate from high school they do not "know" the mathematics they are purported to know (for example, see Frid, 1992; Schoenfeld, 1992). Much research in mathematics education in relation to this problem has focussed upon designing new teaching methods or environments and has not examined the broader context of the nature of students' mathematics learning which is addressed in this study. That is, there has been little questioning of the role of secondary schools for what they are preparing students, whether it be university or vocational studies, or employment. Assumptions about the schooling system might be narrowing in that schools might not have an appropriate context to adequately meet tertiary or TAFE institutions' needs, nor those of industry or daily living. Previous research indicates this might be the case (Bishop, 1988; Lave, 1988; Rogoff, 1990). However, until we know how students, as the most direct recipients of school practices, experience and interpret both the content and context of school programs we are not in a valid position to evaluate the effects of recent trends and changes in our schools. This study will address this challenge. METHOD AND DATA SOURCES Since the project proposed here is concerned with the context of learning and related descriptions of and learners' mathematical interpretations, qualitative research methods were employed. This approach is in line with current educational research practices as they shift away from purely quantitative, quasi-scientific experiments so that researchers can more explicitly document and analyse the experiences of teachers and learners in the broad encompassing social and academic complexities of classrooms (Moss, 1994). An inductive reasoning approach for data analysis was adopted (Glaser and Strauss, 1967; Powney and Watts, 1987), and phenomenographic techniques were also incorporated (Marton, 1988). In particular, the study used in-depth semi-structured interviews with students. The interview protocol asked the students to respond to a range of questions about their thoughts on: why mathematics is taught in schools, what is important about learning mathematics, what mathematics is and whether it is created or discovered, reasons why people choose particular mathematics courses, factors that facilitate mathematics learning, and specific experiences with mathematics learning. Interviews were recorded on audio tape and later transcribed. Two high schools in the metropolitan region of Perth, Western Australia, were selected for the study: a large government school and an independent all female school. A total of 40 students, 20 teachers, 2 career councillors and two administrators (one principal and one deputy principal) were interviewed. Only the student interview data will be examined here. Nine students were in Year 12, sixteen were in Year 11 and fifteen were in Year 10. The sample of students was representative of high, middle and low achievement in mathematics, and at the government school was balanced in terms of gender. Approximately half the students were of a multi-cultural background (generally Asian), which is characteristic of the general student population in the region. Some students were interviewed individually, while others were interviewed in groups of two or three. The reason for this difference was to establish if students would speak more openly and comfortably individually or with their peers. Approximately half the interviews were conducted near the end of the 1993 school year, while the others were conducted in the middle of the 1994 school year. The differing times enabled reliability of the data to be established in relation to end of year examinations and their potential influences on attitudes and related conceptions of mathematics experiences. RESULTS AND RELATED DISCUSSION What is mathematics? Students initially gave many 'expected' responses to the question: What do you think mathematics is? These responses were 'expected' in that they reflected views of mathematics reported in other studies (for example, see Crawford et al., 1993; Chant and Galbraith, 1993), including views that mathematics is: (1) numbers, rules and formulae, and (2) a logical process or way of thinking. Many students also spoke of the applied nature of mathematics, and how numbers, rules or formulae, or logic and a way of thinking can be applied to solve problems. An additional conception of mathematics a connected hierarchy in which studies relationships or patterns was spoken of by some students, and these were generally students achieving at average or above average levels. The question that elicited further insight into students' conceptions of mathematics was: Where did mathematics come from and what is it for? Many students responded to this question in a way that pointed to a view of mathematics as a sort of 'technology', that is, a human endeavour for addressing human needs and solving human problems. These needs might be due to a desire to understand one's environment, or they might arise from wanting to accomplish a specific task. Some examples of the sorts of responses students gave in this regard are: Brenda:It's based on our curiosity understand and explain things. Lesley:. . . It just came from a need to understand things. Tin:I think they created it as a way of explaining relationships, like the creation story is just a way of explaining how we got here. To say how it happened. I think they created a way in which we study relationships. . . . The relationships were there and they had to create a method of studying them. Anna:I think, sort of people trying to logically understand everything that's going on around us. And from that comes all these different sorts of things. Rules. I:Would you see mathematics as something that is discovered or something that's created? Anna:Bit of both. . . . It was probably created as necessity, just to - it is - it's sort of - it's created, but . . . Elaine:It can't be created from nowhere. Anna:We discovered things. Elaine:It's there to be created, sort of? Anna:Yes, we discover things which help us more to . . . I don't think you can exactly discover. Elaine:But sort of if you just - you don't really discover a formula or something. AnnaYou create a formula, but then again . . . somebody must have created it, before it was discovered. This last excerpt shows how students found themselves in a state of indecision when they were probed as to whether they thought mathematics was something people created or something people discovered. There were some students who were more definitive than Anna and Elaine, deciding quite quickly that mathematics is something people create. However, their reasons for this decision indicated they equated creation to being "made up". That is, they saw mathematics as something mathematicians have "made up", and the reasons they saw it that way included that they did not see much use for mathematics and did not understand it to any degree. Some additional examples of the creation/discovery debate are given below: Helen:Oh in the start created because . . . Ellen:Why would they create it? I don't know. I think it was just there. . . . They stumbled across it. Helen:It's part of life. It's part of everything. It's not just, you know, you have arts, and you have maths and science. They're all combined. Ellen:It's hard to identify where you use it. It's just - sometimes it comes out, but I don't know. But I do think it is discovered. Tania:I don't know. Discovering it then creating it as you go. You kind of do a bit of both as you go. Yes, because things like the hypotenuse. I don't know. How would he have discovered that? Like, because, that squared plus that equals that squared, or whatever. He couldn't have just created that, like: "Oh yes!" Because it applies to everything that involves the hypotenuse and that. So probably just discovery. It must be noted that these excerpts indicate students have a fairly broad view of mathematics. Their conceptions of mathematics, when they are probed to respond beyond an initial immediate response, include much more than the stereotyped view of maths as numbers, formulae, logic or problem solving. Why is mathematics studied? Analysis of interview data revealed four major interwoven and overlapping contributors to conceptions of the intentions of mathematics studies: social status of mathematics, utility of mathematics, career aspirations, and intrinsic challenge and interest in mathematics. These components appeared to form a web of beliefs that are influential in determining the nature of perceptions of school mathematics and what it means to 'understand' mathematics. Each of these four components will now be more clearly defined and each will be elaborated upon and supported in relation to primary data from the interview transcripts. Social Status of Mathematics Students' perceptions reflected a social norm that mathematics is an 'important' and essential subject to study. They saw it as a subject with much prestige in the eyes of the community, especially employers. Many had questioned the validity of this status, but had accepted that it was a social value or convention that they must acknowledge in their choice of subjects for upper secondary or tertiary studies. Some examples of what students said in this regard are: Elaine:Because lots of employers look at what you got for maths. It's seen as very important, I think. . . . In university, if you have a look at the pre-requisites, a lot of them are maths, to get into anything else. So it is a matter of having to do it. . . . Science and maths especially. More emphasis is placed on doing well in them. If you can do well in English it's not, there's not many things you can do in English. You can do journalism and things like that, but there's more areas to study if you do science and maths. Like medicine and all that kind of thing. Cathy:Maths and science are always the two subjects that are made paramount. Lisa:I think it's - people, you know, sort of think maths and science are important because they're hard. Because smart people do them. Lisa's and Cathy's words reflect an aspect of general community views that mathematics success is highly valued, and is something people should strive for. Students were clearly aware of the status of studying abstract mathematics as opposed to "vegie" maths, as can be seen in Tracy's words: Tracy:You get Foundations in Maths, and then Intro Calc, and G and T, and you get - it's the general - like people say: "What are you doing?" "Oh, I'm doing Foundations in Maths." "Oh, I see." And then people, other people say: "I'm doing G and T. . . . and it's like a very class sort of thing. It sort of divides people up. Some students were as accepting as other students' of the necessity of studying mathematics and this diversity is highlighted in the following two interview excerpts: I:Why do you thinks mathematics is taught in high school? Nancy:To bore the kids. . . . I don't know. I guess it's because it was taught through primary school and then high school, so you might as well keep going. Tracy:It's good to know but I don't use it. Grace:The use is that you'll get your TEE score which will get me into uni. Tracy:Like a safeguard. Utility of Mathematics Students saw mathematics as important because they saw it as something needed in their daily lives. However, what they often actually described as relevant mathematical knowledge in this regard was mathematics taught primarily in elementary school. This point is exemplified in the following interview extract. Kath:Because you have to know it I guess. Well, you use it in like lots of things, like even when you go shopping and stuff like that. Some things like algebra I don't think you really need, or - at the moment we're doing the derivatives or something, and unless you're going to be an engineer or something you probably won't need it. But you need it in later life. However, in spite of the fact students frequently commented that it is important to study mathematics because you need in your daily life, they also frequently questioned the extent to which one needs to study mathematics. They saw the basics as essential to all people, but felt that much of what was taught at high school was for university or career purposes only. Further, most saw high school mathematics as relevant to very few careers. They noted that they saw it as unnecessary requirement for most careers, and wondered why it was so frequently a pre-requisite course of study. For example: Grace: But like even basic maths at school, you still have to do algebra and that, and unless you're really going to professionally use it, you're never going to use it again when you leave school. I:Why do you think it's so important for jobs? Karen:I don't know. I've always wondered about that because I don't really like it very much. But they always say you have to have it. I don't know why. Tracy:It's mainly to prepare us for uni, I think. Everything's preparing for something else. Like in primary school we prepare for high school maths, but in high school we prepare for uni maths. . . . I don't know, but I can see the relevance of it in some careers, like engineering and stuff like that, but for other careers I don't see why some of the maths we're taught is taught. . . . Well as, as a nurse, where would you use a quadratic triangle or something? I don't see its relevance if you're a nurse, knowing how to change a quadratic formula to complete a square form or something like that. I mean it's not really going to help you to be a better nurse. Career Aspirations Students were motivated to study mathematics to enhance their prospects for a particular profession or job, to keep their career options open and maximised, and to assist their chances of gaining acceptance at a post-secondary institution. In relation to these points they said such things as: Kath:Because I need it I guess. You need it for - I chose it because of what I wanted to do in university. Nancy:Hardly any of the courses at uni which prepare you for jobs that don't want an amount of mathematics. Elaine:I have to because if I don't do that, then I won't get my full TEE marks. Then I won't be able to get into uni to study what I want to study. Because there's nothing else I want to do and they always say: "Oh, you need maths for something." . . . Some people look at it as another subject in school that they have to go to, like in the classroom. And some people look at it as something they're going to have to do in later life. And there's just all ways of looking at it. I look at it as something that I have to do; not that I particularly want to do. But I know I have to do it anyway. and if I'm going to get somewhere in life I'm going to have to basically know what I am doing there. . . . Because like, they're telling us that: "You're not going to get hired in a job if you don't know like how to do your basic maths," right. Yet, along with recognising the necessity of mathematics studies to gain university entry and thereby attain desired career aspirations, students were frustrated. They felt as though somehow else, or the system had imposed unnecessary or unreasonable demands upon their choices, as can be seen in Tania's words: Tania:Deciding to do maths was one of the hardest things. I mean, you really needed it. There was really no doubt not to do it. But I think to be an actress and to be on the stage, you DON'T need to sit there and go: "Well, if I am going to walk in this direction, then I am going to have an angle of so much in order to - " Things like that. . . . For university entry and to get in. I mean to do a Bachelor of Arts of Performing Arts at university, you still need to have got a suitable score, and mathematics is one of the things you need in high school to get into it. A lot of courses have a pre-requisite that you have to be competent in maths. Al:The only reason I am doing maths is to keep my options open. . . . Yet I don't really know what I want to do. But because, you know, you need it for university. Tania:I never really thought of not doing maths. Mathematical Challenge and Interest Some students expressed an interest in mathematics related to a sense of the intellectual fascination and challenge it provided, and these students were not necessarily the most able students. In general though, few students enjoyed mathematics for its own sake. Some examples of their comments about their interest or lack of interest in mathematics are: Ellen:Like for research assignments or something, for English or something, you like research something and then you find out all these points and then later on after a while you go: "Oh, I know about that!" You know, it's interesting. But maths isn't like that. It's just like you learn it and then that's all out the door and you forget about it the rest of your life. Helen:. . . We sort of remember for the test and that's about it. Dean:I just shut it out of my head, just like everybody else. I think it's a boring subject. Faith:It's always been my favourite subject since primary school. Barb:It's challenging. . . . And I like how we have - when we have group discussions. Like you hear everyone's point of view. Anna:I sort of do it because it's challenging, and I think it's - it's just an interesting and challenging thing to do. A strong interest in mathematics was also expressed in relation to career aspirations and the social importance of mathematics, and enjoyment was achieved through being successful in relation to these other key components. Other components of mathematics learning Other features of mathematics learning that students felt were influential in their effect upon learning were: „Mathematics learning is more that rote memorisation; it is necessary to understand what you are doing. Tracy:. . . in logarithms you're set rules and she proves that they actually do work, and the logic behind the rules. So it's not just something you learn by rote. It's something you understand. Grace:Like last year when we didn't - like we had a rule and you didn't know where it came from. And that makes it harder to use and harder to learn than if you understand what's behind the rule. „Mathematics classes are "too fast", proceeding at too quick a pace for people to learn best. Ellen:But now I am going right down hill because I don't understand it. She just goes really fast and everyone in the class I think would agree that she just goes too fast. . . . Because she goes onto the next thing and you're just figuring out the next thing, like the first thing you did, and then you look up and there's like all different stuff and it's like you know, and it takes the whole lesson to try and figure out one thing. „Mathematics learning repetitive and boring and students could learn more if mathematics were taught in a way that made it more fun and more relevant. Alice:Well it's really different because in maths you learn something and in other subjects you learn different things. Because like Nancy said, there's more things to learn. It is not always repeating things. Nancy: Just like in health studies. You can only learn so much about sex and everything. It's just like maths. It's boring. David:It's a pretty good subject, even though most people don't enjoy it. I don't really enjoy any of it. . . . I don't really enjoy doing any of it. It's just a pretty useful subject. CONCLUSIONS Students having 'understanding' of mathematics that is dependent upon context. That is, they define mathematics in a school sense as well as a personal sense, and these are to a large extent independent of each other. More specifically, students generally have well-formed conceptions from a mathematics "discipline" perspective of what mathematics is and where it comes from, yet these conceptions are not taken into account when students are asked to give personal meaning to mathematics in relation to their lives. Students' personal perspectives give meaning to mathematics only in relation to a number of social factors, including the social status of mathematics, the utility of mathematics, career aspirations, and interest (of lack of interest) in mathematics. That is, students reflect a wider cultural and community perspective of mathematics than what mathematics educators generally aim for. More specifically, students put mathematics in a whole school, career and life perspective, but mathematics education research does not presently adequately address this broad viewpoint. Thus, it could be said that 'understanding' mathematics is neither a goal nor a necessary component of students' mathematical learning, at least not in the sense mathematics educators might define 'understanding' (for example, as understanding is conceived of in National Council of Teachers of Mathematics, 1989; Australian Education Council, 1991). Students 'understand' mathematics when they are meeting their goals as described in relation to their career aspirations and sense of the social importance of mathematics. What also is of interest is how the four components identified within the context of the intents of mathematics study conflict with the ideals of how mathematics educators identify problems present in mathematics education practices. For example, understanding generally is associated with a range of interconnecting, cognitive frameworks that can be utilised to explain concepts and solve problems (National Council of Teachers of Mathematics, 1989). Further self-analysis is needed by the research community. More specifically, recognition is needed for how students view curriculum because this study indicates students do not separate mathematics from their personal social contexts. They do not perceive of mathematics as one might describe mathematics as a discipline, but rather, describe mathematics in relation to a range of socially derived components. Educators' capacities are restricted whilst they persist with disregarding the whole school context and a view of mathematics that virtually ignores students' views. REFERENCES Australian Education Council Review Committee (Chair: Finn, B.) (1991). Young people's participation in post-compulsory education and training. 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