Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction

In this chapter we investigate students’ views about learning mathematics. We ­continue our analysis of the series of interviews that we conducted with 22 undergraduate students studying mathematics as a major at an Australian university. In the previous chapter we explored their ideas about the nature of mathematics itself; here we move the focus to their ideas about how they go about studying mathematics. We present a theoretical model based on our research findings, aiming to build on and expand earlier descriptions of students’ learning approaches, such as the surface and deep approach of Marton and Säljö (1976a) and the 3P (presage-process-product) model of Biggs (1999). As for our results in the previous chapter, the small convenience sample that we have used for our interviews could limit the generalisability of our findings to the larger group of mathematics students as a whole. However, our results seem consistent with previous literature on student learning, and also with our findings about students’ conceptions of mathematics.

When we carried out analyses of our interview transcripts we found that students’ conceptions of learning mathematics could be described using three hierarchical levels. At the narrowest level, students learned by focusing on the disparate techniques of mathematics. A broader level was represented by a focus on the subject of mathematics itself. In the broadest and most comprehensive view, students considered the role of mathematics in their lives and talked about developing a mathematical way of thinking and viewing the world, satisfying their intellectual curiosity and helping them to grow as a person. Again, the hierarchical nature implies that students with the broadest ‘life’ conception of learning mathematics were also aware of the ‘subject’ and ‘techniques’ aspects, but those with the narrowest techniques conception did not seem to aware of the broader ideas. There seems to be an obvious parallel between these three conceptions of learning mathematics and the three conceptions of mathematics itself that we identified in the previous chapter.

During their interviews, students commented on various aspects of their mathematics learning. Supported by previous theoretical discussion, we distinguished three separate aspects of their learning: their intentions for learning, their approach to learning, and the outcomes of their learning. We illustrate these three aspects at each of the three levels of conceptions with quotations taken from the interviews – that is, we present our results in the words of the learners themselves. This is a deliberate decision; we believe that teachers of mathematics can benefit from becoming aware of the views of learning held by their students, who are, after all, the people carrying out the learning. Indeed, some of these students show a very sophisticated appreciation of their own learning and the various problems that they have to face and overcome during the process of learning. Towards the end of the chapter we summarise the pedagogical implications of our results, and we return to this point in more detail in Chap. 8 of the book.

Early phenomenographic studies of students’ conceptions of learning were carried out in the context of asking students to read sections of a textbook and then questioning them about it (e.g., Marton and Säljö 1976a, b). From these studies, five hierarchical conceptions of learning were identified, in two groupings. A narrower view of learning as reproducing comprised views of learning as: accumulating knowledge, memorising and reproducing, and applying to other situations. A broader view of learning as primarily seeking meaning included: learning as understanding a situation, and learning as seeing something in a different way. A later study added a sixth, and broadest, conception – learning as changing as a person (Marton et al. 1993) – resulting in the classical outcome space for conceptions of learning (see Marton and Booth 1997, Chap. 3). The well-known distinction between ‘surface approach’ and ‘deep approach’ to learning, that is, between aiming for reproduction and aiming for meaning, was developed in the context of these results.

An interesting feature of these early studies is that the broadest and most holistic levels of student learning seem to include an explicit recognition of the ontological aspect of learning. The broadest conceptions of learning – as ‘seeing something in a different way’ and ‘changing as a person’ – are exemplified by statements from students about their learning: “Opening your mind a little bit more so you see things (in the world) in different ways” and “I think any type of learning is going to have to change you … you learn to understand about people and the world about you and why things happen and therefore when you understand more of why they happen, it changes you.” (Marton et al. 1993, pp. 291–292). These general results concerning students’ views of learning were known at the stage when we commenced our study with students of mathematics. We aimed to look at students’ ideas about learning mathematics in particular, and to investigate how these ideas compared with the general views of learning developed from studies in the context of reading (and writing) texts.

Previous Investigation of Views of Learning Mathematics

Research on students’ learning in mathematics at the tertiary level is a wide-ranging and active field, and its overall directions have been summarised in various handbooks (such as Holton 2001; Bishop et al. 2003; Skovsmose et al. 2009) and reports of international conferences (for example, the ICME – International Congress on Mathematical Education – conferences held every 4 years). Bishop’s team (2003) divide their comprehensive handbook into four main themes of contemporary relevance: policy dimensions (social, political and economic), responses to technological developments (calculators and computers), issues in research (ethical practice, impact of research on practice and the role of teachers as researchers) and professional practice (teacher education).

One feature of research on learning mathematics at the tertiary level and its ­translation into curriculum is that it often focuses on the ways that lecturers understand teaching and learning, and the nature of mathematics itself, such as ideas about the precision and rigour of mathematics, the cumulative nature of the subject, and the importance of mathematical skills (see, for example, Thomas and Holton 2003; Burton 2004; Nardi 2008). This is based on the view that lecturers are best placed to make changes to the learning environment, and the underlying assumption that changes and developments in teaching practice will result in changes – hopefully improvements – in learning. Lecturers of mathematics believe, not unreasonably, that they know what is critical for students to learn from their own experience of being learners and mathematicians. With this belief it would seem that developmental efforts could focus on current and early-career lecturers’ conceptions of mathematics. A curriculum in mathematics, particularly in specialist mathematics degrees, is developed primarily from this knowledge base of the lecturers, combined with the strategic requirements of the university and the demands of relevant industries (Bowden and Marton 1998); the ways that students understand learning in the discipline are often assumed.

Nardi has carried out more than a decade of research on tertiary mathematics education that included interviews with students, mathematics lecturers and mathematics educators. She has summarised her results (Nardi 2008) about learning mathematics in the form of a series of dialogues between a university mathematician and a researcher in mathematics education; although the student does not participate in the dialogues, her voice is heard implicitly, and samples of her work are discussed. Nardi’s mathematician shows some of the characteristic attitudes of his colleagues, but he seems genuinely interested in the insights into learning that he is offered in the form of samples of student work and discussions with the mathematics educator. He talks of the importance of conception of mathematics and its relation to learning mathematics:

Getting to see how each one of us understands the subject is useful: and communicating our own misconceptions amongst each other as mathematicians before having a go at innocent students! (p. 263) … A lot of the problems you have to deal with when you meet our students is that they have a very singular view of mathematics, a rather poor view of mathematics. (p. 262) … May I say that it is in these discussions exactly that these sessions have proved enormously valuable already. There are things I will teach differently. There are things that I feel like I understand better of mathematics students than I did before. (p. 260)

We discussed in the previous chapter Burton’s (2004) study of university research mathematicians’ ideas about mathematics and coming to know ­mathematics. Based on the 70 interviews that she carried out, she concluded that “the mathematicians’ experiences, as learners, are relevant to less sophisticated learners in schools, and in universities” (p. 178), and used this position to argue for “a pedagogical approach to mathematics that treats learners as researchers” (p. 183). Burton’s findings support a view that, in at least some important respects, ­mathematicians’ experience researching in mathematics parallels students’ learning of mathematics. Houston (1997) considered the way of life of professional mathematicians, and argued that there were three corresponding ways in which students learn mathematics. The first step is learning the methods or tools of the discipline; these tools are used to understand models of the world that have been created by others; finally, this understanding enables the student to engage in the creative activity of building models, so modelling – and mathematics – becomes a way of life. Houston’s three steps are reflected in our identification of three levels of conceptions of learning mathematics.

Looking specifically from the students’ viewpoint, we return to the study ­carried out by Crawford et al. (1994). They investigated students’ conceptions of learning mathematics, asking them to respond to the survey question: How do you usually go about learning some maths? They identified five conceptions of learning mathematics: the narrowest was “learning by rote memorization with an intention to reproduce knowledge and procedures”, and the broadest was “learning with the intention of gaining a relational understanding of the theory and looking for situations where the theory will apply.” They distinguished between learning for reproduction (the first two categories) and learning for understanding (the other three) – the classical distinction between surface and deep approach to learning. Students who aimed to learn for reproduction were likely to have a fragmented conception of mathematics, while those who aimed to learn for meaning were likely to hold a cohesive conception. Meyer and Parsons (1996) reported on a quantitative investigation of student learning in mathematics using a questionnaire developed from an investigation of students’ qualitative approaches to their study of mathematics. They identified two major factors: an association of desirable features (such as strategic problem solving, deep approach to learning, incorporation of group work and explaining to others, intrinsic motivation and confidence) and another association of undesirable features (such as a memorising approach, single-strategy problem solving, insecurity and fear of failure). However, relatively little has been investigated from the viewpoint of students who plan to be professionals in the mathematical sciences. The studies mentioned in this paragraph were carried out in large first-year mathematics classes that ­contained few students planning to specialise in mathematics.

Our Study of Students’ Conceptions of Learning Mathematics

The empirical basis for our investigation of students’ ideas about learning ­mathematics consists of the transcripts of a series of interviews that we carried out with 22 undergraduate students majoring in the mathematical sciences. In terms of their views about learning mathematics, we asked them the key questions: What do you aim to achieve when you are learning in mathematics?, How do you go about learning mathematics? and What do you think you want to take with you from your learning of maths? These were followed by further probing questions that depended on the student’s responses – general questions such as Can you give me an example of that? and specific questions such as What part does logic play in mathematics? and How does doing the exercises help your learning?

As a first step in the analysis, all members of the research team read through the interview transcripts several times to get an overall idea of their content. This was followed by several days’ exploration of various aspects of the transcripts by the whole team – here we focus on the aspects pertaining to learning mathematics. Although our initial approach was based on phenomenography, and the results are reported in the form of a classical phenomenographic outcome space, we extended the approach in several ways. Early in the analysis, we identified a framework of ‘intention, approach, outcome’ (IAO) that could be used to analyse the information concerning learning mathematics. This was not only suggested by the group of transcripts themselves, but also receives theoretical support from previous work (see for instance Dahlgren 1997, and other articles in Marton et al. 1997). We took ‘intention’ to mean statements where students referred to future plans or aims; ‘approach’ statements included general descriptions of their methods of learning or specific details of what they did as learners; and ‘outcome’ statements were clearly oriented towards skills, both procedural and conceptual, or attitudes that they had developed. To some extent, our three main questions could be interpreted as directing students towards IAO, but our aim was simply to encourage students to talk about all relevant aspects of their learning of mathematics. Moreover, we (and other researchers) have used similar questions in previous studies of learning and the IAO framework has not emerged (for example, Petocz and Reid 2001).

Having decided on this analytic approach, we utilised the qualitative research package NVivo (QSR International 2007) to go through each transcript and code any statement about learning mathematics under one of these three aspects. This was done by two of the team independently (Anna Reid and Peter Petocz) and any differences were then discussed and resolved. Statements about ‘intention’, for example, were often made in response to the question What do you aim to achieve in learning mathematics?, but this was by no means the case all the time. NVivo made it easy for us to extract and investigate all the statements for ‘intention’, ‘approach’ or ‘outcome’ separately, coding them as ‘free nodes’ without any pre-determined structure, appropriate for such an emergent analysis. This then allowed us to identify qualitatively different conceptions for each of the IAO aspects, and to place them in a tentative hierarchy. The parallel nature of the outcome spaces, particularly for the ‘intention’ and ‘outcome’ aspects, became quite clear. Each conception was examined separately and tested against students’ transcripts to determine its place in the hierarchy; the student quotes given in the following section are all taken from this level of the coding. The analysis showed that the conceptions for each IAO aspect could be grouped into three hierarchical orientations which we have labelled ‘techniques’, ‘subject’ and ‘life’.

Individual participants’ statements were then used to categorise their conceptions of each aspect of IAO; this was particularly easy since the NVivo codings were immediately accessible. The first two transcripts that we examined showed a consistency across IAO, suggesting that we investigate the consistency across aspects for each participant. Such a classification of individual transcripts takes the analysis beyond a standard phenomenographic one; however, our aim was not to categorise individual students, but rather to use such classification as a method to extended the analysis. We have used this approach before in a comparison of individual students’ conceptions of statistics and learning statistics (Reid and Petocz 2002) and an investigation of the relationship between their ideas of learning statistics and their views of their teacher’s role (Petocz and Reid 2003).

Conceptions of Learning Mathematics

Students’ conceptions of learning mathematics can be considered from three aspects, intention, approach and outcome (IAO) and can be organised into an outcome space for each aspect. The conceptions can be grouped into three broad orientations for each aspect that we have called ‘techniques’, ‘subject’ and ‘life’. Table 3.1 displays the outcome space for students’ conceptions of learning mathematics and shows that the conceptions forming each of the three aspects are broadly comparable, the intention and outcome showing almost parallel conceptions.

Table 3.1 Students’ conceptions of learning mathematics
Full size table

In common with other phenomenographic outcome spaces, these conceptions are hierarchical and inclusive (Marton and Booth 1997). The narrowest level focuses on the extrinsic and technical attributes, the middle level is essentially concerned with the subject of mathematics itself, and the broadest level looks beyond the mathematics to its place in students’ lives. Those students who describe the narrower, more limiting views of learning mathematics seem unable to appreciate features of the broader, more expansive views. However, those students who describe the more holistic views are aware of the narrower views, and are able to integrate characteristics of the whole range of conceptions to further their own understanding of learning mathematics. It is for this reason that we as educators value the broader, more holistic conceptions.

The conceptions are now described and their aspects are illustrated with succinct quotes from the students’ transcripts, each labelled with the student’s pseudonym. Each individual quote is not necessarily indicative of the meaning of the category, but merely supportive, and the richness of each category is defined by the whole set of transcripts. Quotes from a particular student may appear in several conceptions, and at more than one level, illustrating the hierarchy discussed previously.

(1) Techniques Orientation

These conceptions are concerned with the extrinsic and practical features of learning mathematics, and where mathematics itself is explicitly involved the focus is on the tools and component skills rather than the mathematics itself. In terms of intention, Brad’s quote puts forward the idea of passing the subject or course, often mentioned in a jocular way by students. Dave is aiming for a job, and Grant for a better job – this is an important feature for some students and is in the background for others:

Brad: [What do you think it’s important for you to learn about mathematics?] Is ‘whatever I think’s going to be on the final exam’ the wrong answer or …?

Dave: [What do you aim to achieve through learning mathematics?] I, well I, a job, fairly straightforwardly, yeah.

Grant: I basically have had a bit of a soul-searching time over the past few months, because like I just thought, oh yeah, I’ll just sort of get this degree and be more employable and earn lots of money kind of thing.

Marios and Candy talk explicitly about acquiring the tools and skills of mathematics, but their intention for learning is a fragmented one:

Marios: I see it as it’s probably going to be a tool that I use, so I’m going to be doing something else and all of a sudden I’m going to need this tool and ‘oh okay, yes I’ve learnt that, so I’m going to be able to use that tool to solve whatever problem I have in front of me.’ So that’s how I see I’m probably going to use the maths that I’m studying now, yep.

Candy: [What do you think are the important things that you need to learn about maths while you are here?] Basic things like probably like calculating, the fundamentals of maths are really important, yes, but I think a lot of the theory that comes with mathematics, most students seriously don’t understand it, and I can say I don’t understand it either.

In terms of the way they actually go about learning, students focus on course requirements and expectations. This conception describes an approach to learning mathematics that emphasises doing the set work and completing the subject or course requirements. This approach is put forward by Candy, and then more fluently described in Dave’s extract:

Candy: Let’s just say I do what is required in the course, I probably don’t do as much study as I should. It’s a matter, I think a lot of people just do enough to be able to pass and understand the basics of it, that is basically how I go about it.

Dave: And most of, when you say, you know, ‘how have you focussed on learning maths?’, most of it has simply been getting the basic work done, getting the assignments in and just hoping that you have enough time to revise for the exams. And it’s just been, I’ve felt on the back foot throughout, I haven’t had that much trouble getting assignments in on time, and I reckon I probably revise a little bit more than average for most of the exams. But, so relatively speaking my performance has been above average, my marks are quite high, but from a personal basis I do feel I could have done better.

The outcomes described by students seem to be broadly parallel to the intentions. First, there are the extrinsic outcomes of qualifications and jobs, important for some and in the background for others. Candy explains this viewpoint:

Candy: [What do you aim to achieve while you are learning mathematics?] Well, my degree, hopefully. Some, it’s more, to be quite honest, I think most uni, most people who go to university to get university degrees today just are into it for the recognition of having a degree, and to be quite honest when it comes into the workplace, whatever, just say I have a mathematics degree right now, I’m a graduate with a mathematics degree, I can go into some sort of workplace or work area that’s just totally unrelated at the moment. It’s just, I think to a lot of people, it’s just having a degree means you have some sort of level of achievement or intelligence that employers will recognise.

When mathematics is explicitly mentioned, the outcome focuses on acquiring the tools and skills of mathematics, rather than on the mathematics itself. As an example, Candy sees the skill of using a statistical package as an outcome of her mathematical learning:

Candy: [What is it that you want to hold on to and remember?] I want to be able to remember how, how things are done, so just say I’m doing, I’m doing the statistics right now, yeah I take away from the course, so that’s actually very practical because it uses a lot of computer packages and they teach how to calculate the statistics using these packages, so that is something I can actually take away with me and be able to use.

(2) Subject Orientation

These conceptions are concerned with learning the actual subject of mathematics itself. The intention is to understand all aspects of the mathematics being studied, the practical and theoretical, the pure and the applied. In the two following quotes Heather focuses on understanding the theory while Ian is more interested in the application of mathematics to his area of finance:

Heather: [What do you aim to achieve when you are learning in mathematics?] A deeper understanding of the use of formulas and techniques of mathematics, so not just rote ­learning a whole lot of formulas and saying ‘okay, this is such and such theorem and this is what it means’, but more ‘how has he come to this conclusion and why?’/…/So you’ve got to understand the actual techniques, not just how to apply them. That’s usually my aim because I don’t really care about the applications much, as understanding, because I figure that if you understand then you can easily apply it, usually.

Ian: My objective is to learn what areas, and in finance and yeah I can learn more about them and at higher levels using the mathematics and understanding the mathematics of how I guess financial markets and that sort of thing, and prices in financial markets and basically just the dynamics and pricing of assets in the financial markets I’m primarily interested in.

As part of this conception, while retaining the focus on the mathematics itself, some students talk about an intention to use their mathematical skills and abilities to help other people in a variety of situations. Paulo and Ashleigh both express this notion in general or specific ways:

Paulo: Yeah, so in some ways you can help the world when it has a problem and you have this background in applied maths and you can say, ‘oh, I can help’. That’s good.

Ashleigh: [What do you think it will be like to work as a qualified mathematician?] It will be good for me because, you know, I would know everything, and I think it’s different because not many people know it, so I would be able to help them with a lot of things, like mathematical things, yeah./…/For example if I was to work as an analyst, with using stats packages and things, I would be able to create for the company, different strategies and improvements.

With this conception, students describe an approach to learning mathematics that focuses on various aspects of the mathematics itself. This includes selecting, connecting and applying particular aspects of mathematics, learning from theory or from examples and applications. There is an underlying idea of the cohesive nature of the mathematics being studied (as opposed to the fragmented ideas in the ‘techniques’ conception). Elly and Sujinta explain the view:

Elly: [What kind of things would you say you focus on when you are learning in maths?] I guess I try and understand what I’m learning first and then try to put it into practice, that’s the way I go about it. [What do you mean by understand?] If I’m given a theory I’m not just going to use it blindly, I’m going to try not to. I’m going to try and see where the theory comes from, what it relates to and then be able to use it, yeah.

Sujinta: So I’ll look at the theory and sort of understand that and then actually see numbers changing, so if you are using a different, well, oh okay, if you are using one method perhaps, seeing the starting numbers and seeing all that at the start and then seeing someone work through, seeing how the numbers change as you work through the process and getting to the final number, and then I can, if I don’t understand anything in the theory, then what I can do is check that back with the numbers and play with the numbers and go ‘oh okay, this number came from this one divided by that one’ or something, and look at the theory and then I understand it more there because I’m getting the practical side of it.

The parallel outcome is an understanding of the practice, theory and applications of mathematics. In this conception, the outcomes are described in terms of the aspects of mathematics that students are able to use. Ian refers to financial mathematics while Yumi talks about the use of her statistical learning:

Ian: [What do you want to take with you from your learning of mathematics?] Oh, I just, well for me in particular I guess the most important for me is a deeper or more technical or involved understanding of financial markets and the dynamics of that sort of field.

Yumi: [What do you think you will be taking with you from your learning of maths into work?] Oh, just the ability that I can apply things, especially with the, I think statistics is really, really helpful and that’s really important I think in the workforce as well to have that sort of knowledge, how to use it, because you are always looking to improve things in the company and if you have got historical data, then you can perhaps analyse it to see what areas are, do you know what I mean?

Some students are aware of an outcome that allows them to help other people using their mathematical skills and abilities. Joseph has found that he is able to help school students with a subject that they find difficult, while Andy has a more general idea of being a mathematical ‘trouble-shooter’:

Joseph: [What is it that you liked about teaching mathematics?] Oh, because like most of the students are in trouble with mathematics, you know, so if I explain to the students about the concepts and then they are happy, very, they are happy straight away and then I am happy teaching them. So I want to get people to get more happy, you know, I want to help as much as I can do to people.

Andy: [What do you think that work will involve, the work that you are going to do?] Well, I think the ability to, well being able to give advice to certain people as to how to go about solving a problem that they might have, a bit like a trouble-shooter maybe, or actually doing a lot of the analysis myself, like one or the other or both I think.

(3) Life Orientation

In these conceptions, students go beyond the actual discipline to focus on the role of learning mathematics in their personal and professional lives, and the way it changes their view of the world around them. Students may talk explicitly about their intention to develop a mathematical way of thinking and looking at the world – Richard and Hsu-Ming explain this view:

Richard: My personal aim in learning in mathematics is to strive for clarity of thought more than anything, and by saying that, I guess I lean towards the more abstract models because it requires a greater effort intellectually to grasp what it is that is being proposed.

Hsu-Ming: Maybe I want to have an understanding of the world around me and I believe that with mathematics, with the principles of mathematics, I feel as though I will be more able to, I guess yeah, more able to do that, to understand, with yeah, with the basic concepts, just that, that line of thought will be able to help me understand the world around me better.

Some students talk about being intrigued by mathematics and its place in the world, and describe an intention to learn mathematics to broaden their mind and satisfy their intellectual curiosity. Eddie describes his feelings of wonder about learning mathematics:

Eddie: There are some things that you can study that are, that are intellectually so beautiful it’s gratifying to study it for its own sake, I mean when you can see it you think ‘oh yes, this, this has to be right’. And it’s just such a beautiful and elegant chain of reasoning like I, I sometimes hear people talking about murder mysteries or something or where the inspector, Inspector Poirot, you know, sort of engages in this, or Sherlock Holmes or whatever, engages in this beautiful piece of deductive logic to determine who did something. Well that’s sort of about one percent of what you can be talking about with, with some mathematics.

The corresponding approach to learning mathematics involves going beyond formal studies of the subject and using that as a way to understand mathematics itself. Grant tries to explain this approach and its motivation of satisfying his intellectual curiosity. Hsu-Ming explicitly utilises his broad life experience in his learning project:

Grant: I just find maths sort of fascinating as a subject, so it’s just a matter of sort of finding out, you know, because I have a lot of sort of questions that I sort of, I’m wondering is it possible to do this or, you know, how, or what’s the basis behind that or whatever, and it’s just interesting finding out, you know, how these various theories have come about to sort of allow you to do things when you kind of wondered if it was possible to get the answer to certain questions and, you know, then finding out that you can. So yeah, it was just a matter of sort of satisfy my own curiosity I guess.

Hsu-Ming: Well I’m a mature age student, I had to do something between leaving high school and I had a few years until I was back in the TAFE college and then quite a few years and now I’m here, so I know from those times that I’m not in an educational facility, that nothing stops me from learning and we have to learn every day. Again, it’s just that knowledge base, this gives me a broader knowledge base to start from, and that’s, and my starting point is much broader now, yeah my understanding of things, or if I wish to learn something, and I’ll have the confidence to go out and do it too, I guess.

In terms of outcomes, students in this conception are aware of having acquired a mathematical philosophy, or approach to life, or way of thinking. Richard, Dave and Hsu-Ming express this awareness of the outcomes of their mathematics learning.

Richard: My learning of maths, I think that is fairly straightforward, when I say that, in my mind what I mean is that my learning of maths gives me, hopefully, a robust framework, clarity of thought in which to apply that framework to future problems.

Dave: But overall, as I said, that doesn’t detract from the style of thinking that maths gives you, which I think is brilliant.

Hsu-Ming: Now, I wanted to develop my mathematical skills to, as I said, that would certainly enhance that field, with the under…, the knowledge that I’m gaining. However, I’m finding that, back to the original goal, I’m finding it, the knowledge that I’m gaining, more adaptive to the broader spectrum of the world rather than just in a particular situation, so in that sense, that’s how it’s diverging. That field is specific whereas I’m finding more general, generalisation occurring.

And some students, such as Julia, describe the personal growth and intellectual satisfaction, the awareness of the broad role of mathematics in the world, that they have acquired in the course of their learning:

Julia: I think it’s, it’s mind dazzling how much maths accounts for. I mean, to the naked eye maths is one, two, three plus four, have to do this subject and let’s get out of school, but if you sit there and look a little bit further, you find maths in all sorts of areas and that, I think, is wonderful. It’s like you’ve learnt this, you’ve got this knowledge that can be applied to so many different areas, that’s got to count for something. It’s a powerful thing actually, it’s the same theory that engineers use, it’s the same theory that financial managers and leaders use, it’s exquisite. /…/ Yeah, it explains so much in life, technology. It’s amazing how many things are explained by maths and applied, I mean I don’t even know all the applications and I’ve done three years, maths has got a whole lot of applications in our life, in our world, which you wouldn’t be aware of until you scratch the surface and try to find out. /…/ So the more you learn, the more you find out there is to learn.

Discussion

Early results from the NVivo coding encouraged us to investigate the consistency between IAO in individual transcripts. Since IAO are three aspects of learning, it would seem reasonable that a particular student’s conception of their own intention, approach and outcome would display the same general orientation. So if a student viewed mathematics learning in terms of ‘techniques’, then their conceptions of IAO would show consistency across the technical orientation. This is the case with Candy, for example, and we have given quotes from her under all three aspects. Importantly, no sections of her transcript were coded at any of the broader conceptions. On the other hand, if a student viewed mathematics as an integral part of their life orientation, then their conceptions of IAO would show consistency across the life orientation: Hsu-Ming presents an example with quotes under all three aspects. However, due to the hierarchical nature of outcome spaces, we might also expect to see evidence of the narrower conceptions at various points in her transcript, and indeed this was the case.

The majority of our respondents showed this sort of consistency in their transcripts: 14 out of 22 were easily in this group, and six of the others could have been included but for a single section, often in a summary of their views, where they expressed broader conceptions of one of the aspects. We believe that the discussion in the interview had the effect of encouraging them towards articulating broader conceptions, and we have experienced this feature in previous interview studies. We also noticed that two mature-aged students expressed narrower intentions but broader outcomes. In each case, it seemed that they had started their study of mathematics with very pragmatic intentions (getting a better job), and were somewhat surprised at the broader outcomes (a mathematical way of thinking). Dave’s quotes in the previous section illustrate this situation – he noticed the ‘brilliant’ style of thinking that he developed through his study of mathematics. Another respondent, from a non-Australian background, showed an interesting transcript that combined the narrowest conceptions focusing on the pragmatic concerns of job, status and money with the broadest conceptions exploring the philosophical and intellectual excitement of learning mathematics, but with nothing in between these extremes.

The one distinct difference between students’ generally parallel discussions of intention and outcome is concerned with a group of professional skills and dispositions. These include personal qualities such as hard work, persistence and patience, technical abilities such as computer skills, social skills such as teamwork, and intellectual skills of decision making and problem solving (which comes close to ‘mathematical way of thinking’, but isn’t the same). These were not mentioned by any of our respondents in terms of intention, but were discussed by several of them as outcomes of their learning. It seems that students were not expecting to develop these professional skills as part of their mathematics learning, but some of them were surprised to see how they had in fact acquired them. Here are some examples from the transcripts:

Ashleigh: I think like so far it’s helped me a lot, like yes to help, it’s like I’ve actually learnt over the years like doing maths, how to, you know, get myself involved, more involved in like teamwork, yeah I’ve noticed, especially with stats, so, it’s interesting.

Monique: Thinking back over the subjects that I have done, they’re, they’re contributing to the kind of work that I want to, so not only the subjects, the things that I’ve learnt in the subjects, but also the skills that I’ve learned while doing the subject. And not only the skills but also the qualities, like being patient and having to persevere in studying and doing projects and I think that will be useful too when I go to work force.

Yumi: Into work, yeah, so I’ve said, applying my knowledge to certain problems and discipline. I think just the skills you learn in general just by going to university. How to deal with people as well, how to use, you know, a computer and processes and packages and things like that.

Gabrielle: Seeing patterns in things and analysing things, analysing data and breaking it down and looking at problem solving obviously as well. So not only in a mathematical context, but I think it’s given me analytical, not powers but reasoning in other areas as well, yeah and problem solving.

We will discuss this outcome of students’ learning in mathematics Chap. 5, when we report on recent graduates’ comments on their previous mathematics learning. It seems that when students start a degree in mathematics, such professional skills do not generally form part of their expectations. As their course progresses, they become aware of learning a broader range of skills and dispositions than they had previously expected.

Implications for Teaching and Learning

Our findings concerning students’ conceptions of learning mathematics have some immediate implications for mathematics pedagogy, and in this section we will discuss some of these. Our aim is to indicate how the research results that we have obtained can be utilised in the process of teaching and learning mathematics. The first point follows from our discussion of conceptions of mathematics in Chap. 2. Students are usually unaware of the range of variation in thinking about their learning, and tend to assume that their fellow students share their own views. Yet students sitting in the same class can have very different ideas about the nature of learning in mathematics and these ideas contribute to the approach that they adopt in any learning situation. Introducing them to the full range of conceptions seems to be an effective initial step in helping them develop broader views. Students could be asked to think about how they view learning in mathematics, and a short description of the results in Table 3.1 could be presented, maybe when their first assessment task is introduced or as part of an activity in an early tutorial class. Telling students about the range of variation in ideas about learning mathematics will not, of itself, broaden their conceptions, but it will make it easier for students to think more deeply about their own views (Reid and Petocz 2003 gives a case study of such an approach in the context of a course in regression analysis).

When students are aware of the range of conceptions of learning mathematics, they will be more likely to discuss their learning with their colleagues, especially when opportunities are made for them in the form of group work in classes or ­laboratories, and group assessment. Students can have quite sophisticated views of their own learning, and can be strong advocates for a deeper approach. Julia’s quote expresses the situation better than many teachers could:

Julia: There’s monkey learning and there’s proper learning. Monkey learning is finding out what you need to learn for the exam to get through, proper learning is finding out what’s behind the numbers that you are writing down so that you know for yourself. There are people that do very well in a subject because they learn what they need to know for the exam, but you ask them three or four weeks later and they couldn’t tell you. There are people that won’t do that well in their marks, but you ask them three years down the track and they will be able to explain to you how that matrix works or whatever you are talking about. There’s always a difference. And it takes a lot more time to learn the background than the ‘what you need to know’.

Another important strategy is to arrange learning situations that encourage ­students towards the broadest conceptions of mathematics learning and away from the narrower conceptions. For instance, a class environment that presents a course in terms of a sequence of definitions, theorems and proofs, and rewards students in examinations for rote learning them, encourages students to focus only on passing the course and acquiring the appropriate techniques. In such a situation, even those students who are aware of broader conceptions of learning will be encouraged to work using the more limited ones. On the other hand, laboratory work and assignments that ask students to analyse the solutions to a differential equation, or to carry out an analysis of a set of statistical data, and then explain the meaning of the analyses to the people involved (clients, colleagues, readers of a professional publication) immediately expand students’ focus. So too do assignments or projects that allow students to select material based on their own interests and explicitly ask them to think about and discuss their own learning (see Viskic and Petocz 2006, for example). Another option for teachers is to develop learning materials that try to engage students at a broader level with an expanding notion of learning mathematics. One example is the book Reading Statistics (Wood and Petocz 2003), which asks students to read and engage with research articles in a variety of areas of application, and to communicate the statistical meaning in a range of professional situations. Using such materials and pedagogy, we can set up learning situations that afford scope for students to become aware of the broader role that mathematics can have in their studies and their professional lives. Our experience is that many students will take up these opportunities.

Summary and Looking Forward

We continued the description of the first stage of our research project by reporting on students’ conceptions of learning mathematics. Early in our analysis, we identified the IAO – intention, approach and outcome – framework, and used this to guide our analysis. We found that students viewed learning mathematics in three quite different ways. At the narrowest level, which we have termed ‘techniques’, they focused on the extrinsic and atomistic aspects, including obtaining a pass or a qualification, and learning isolated technical skills. At the next ‘subject’ level, they broadened their focus to the subject of mathematics itself. At the broadest and most inclusive ‘life’ level, they appreciated the contribution of mathematics learning to their personal and professional lives. These three levels of conception were identified in statements about intention, approach and outcome, with a particularly strong parallel between intention and outcome, though with some unexpected results in terms of professional skills for some students. Our identification of the IAO aspects of learning builds on the work of previous researchers, particularly the notion of surface and deep approaches to learning (Marton and Säljö 1976a) and the 3P ­(presage-process-product) model (Biggs 1999). Our ‘techniques’ and ‘subject’ ­orientations are loosely linked with surface and deep approaches to learning, but our broadest ‘life’ orientation seems to extend the notion of deep approach to learning. Biggs implicitly acknowledges the importance of intention and outcome in students’ views of learning: we have provided a description of the qualitative differences in these components, as well as the process or approach component, specifically in the discipline of mathematics.

In discussing the pedagogical implications of these results, we again pointed out the importance of helping students to become aware of the full range of views about learning mathematics, and of using appropriate pedagogy and learning materials to encourage students towards the broadest and most inclusive ‘life’ conception. There is much evidence to indicate that this is contrary to the common practice in many school classes and even university lectures. Wiliam (2003, p. 475) states this clearly: “If one observes practice in mathematics classrooms all over the world or looks at textbooks, the predominant activity seems to be the repetition of mathematical techniques through exercises.” Classes of this type direct students strongly towards the narrowest conception of learning mathematics and of mathematics itself. Students are likely to see little relevance of learning mathematics to their own studies, professions and life situations. With a clearer understanding of students’ ideas of mathematics and learning mathematics, we are in a better position to develop a pedagogy of mathematics that will encourage students towards the broadest views of their subject and its uses in their professional lives.

In the next chapter, we begin the process of extending our results, obtained from a small group of mathematics undergraduates, to a larger group of students studying mathematics in a wider variety of contexts and in a number of countries with different educational systems. Having obtained information from in-depth interviews, we are now in a position to make use of it to design and carry out a project to investigate the views of a larger group of students. For obvious practical reasons, we have to collect their views using a less labour-intensive approach, replacing interviews with open-ended survey questions.