Approaches to studying and conceptions of mathematics among undergraduate mathematics students

1. Approaches to studying and conceptions of mathematics among undergraduate mathematics students Judith MacBean

2. The PhD Study • Longitudinal study following a year group of students through three years of single or joint honours Maths BSc/MSci • Looking at: • experiences of learning mathematics • approaches to studying, and conceptions of mathematics • exam performance • influences during time at university • attitude towards mathematics, and change over time

3. Data collected • Questionnaires: • Background information, prior experience of maths and expectations of university life • Approaches to studying, and conceptions of, maths (completed twice) • Progress in 2nd year • Brief review of the 3 years

4. Data collected • Interviews – semi structured • A total of eight students were interviewed up to six times during their degree • A further four students took part in one off interviews – 2 in their 1st yr, 2 in their 3rd yr

5. Data collected • Other: • Exam results • Departmental documentation • Departmental website • Meetings with Departmental tutor (discussion of student progression from year to year)

6. The cohort: Originally 142 students (123 completed initial questionnaire)

7. Approaches to Studying • Brief “potted” history – Approaches to studying (deep/surface or meaning/reproducing): • Marton and Säljö (1976) • Biggs (1979), Entwistle & Ramsden (1983) • Richardson (1990) • Thomas and Bain (1982, 1984), Scouller (1998) • Webb (1997) • Ramsden (1997), Entwistle (1998a, 1998b), Prosser & Trigwell (1999) • Coffield et al (2004) • Entwistle et al – shift towards qualitative (2001-2005)

8. Conceptions of Learning and of Mathematics • Brief “potted” history: • Säljö (1979) • Marton et al (1993) • Crawford et al (1994) • Crawford et al (1998a)

9. Research questions • How do these students’ approaches to studying mathematics relate to their conceptions of mathematics? • Did their approaches to studying mathematics and conceptions of mathematics change over the three year of their degree? And, if so, how? • How do their approaches to studying mathematics, and conceptions of mathematics, relate to their exam results? • How do contextual factors impact on these students’ approaches to studying mathematics and conceptions of mathematics?

10. Previous research results • Positive correlation between a meaning approach to studying and a cohesive conception of maths. Positive correlation between a reproducing approach to studying and a fragmented conception of maths e.g. Crawford et al, 1998a • Positive correlation between high exam results and meaning / cohesive. Positive correlation between low exam results and reproducing / fragmented e.g. Crawford et al, 1998a, Entwistle, 2000 • However Mji (2003) found no such correlation with exam results amongst a cohort of mathematics students at four universities and one teacher college in South Africa.

11. My results • Significant positive correlations between meaning / cohesive and reproducing / fragmented – both times questionnaire administered • No change in students’ approaches or conceptions found, over time. • No correlation found with exam results (other than small, but significant negative correlation between mid-sessional 1st yr test results and fragmented conceptions i.e. the worse a student did in the tests the more likely they had a fragmented conception of maths) • Why?

12. Harry – high exam results

13. Sarah – high exam results

14. Adam – low exam results

15. Charlotte – low exam results

16. Interview data • Harry, when asked what he liked about maths: “I’ve asked myself this question a few times. It’s a difficult question to answer, isn’t it? Obviously, one of the reasons was that I’m pretty good at it. But I was quite bad at it, actually, before I got interested in it. Once you’re interested in something, I suppose you become better at it. With Maths, I don’t really see it in terms of symbols and, and calculations and numbers, I see it in terms of ideas. And I think it’s kind of the purer subjects, in my opinion, . . . ideas. And although there’s a proof – it’s not about the little tricks and everything, with the algebra and everything, it’s about the idea, where did it come from. You know, it’s just always that sort of flash of inspiration, that’s the aspect that I’ve always liked. And you really prove something, if you like, you’re completing a jigsaw puzzle or games or whatever, it’s just like that. It’s so vast, there’s so many different things, so much. You’ll never know all of it. “

17. Interview data • Harry, when asked how well he thought he was doing: “It’s difficult to say. I normally don’t judge my progress so much on what my marks are, because they’re not really a good indication of how good you are, and you know that as the course goes on the harder it gets. You know, if you work a little bit and then you get a six, you work really hard, you get ten, but it doesn’t change who you are, it doesn’t change how good you are, it just means you worked hard. So that’s not what I’m interested in. What it means is to be able to understand what we’re doing. And so far in Analysis I feel like I understand things.”

18. Interview data • Sarah, at end of first year: “Well, I kind of knew what to expect from the papers, like Analysis, you’ve just got to learn the proofs . . ., it’s just going to do the proofs on there, so that one was OK.”, • Sarah, at end of second year: “I think it’s all about memory. I think exams, . . ., like, analysis, the whole thing is just memory, you could not – I don’t understand it, but if I’ve done well in it, it’s just memorising it” • Sarah, at end of third year: “But, like, now I know that you don’t always have to understand everything to do well in the exams. Like, maybe the first year I didn’t know, but, now, I think, now, like, I’m more relaxed, like if I don’t understand something I’m not really panicking and trying to understand it. I know – you can just use your memory to memorise it, you don’t really have to understand it, but that’s just in some subjects.”

19. Interview data • Sarah, when asked how much she enjoyed maths, at end of second year: “I just do it, I just don’t enjoy it, I just do it. I wouldn’t say I go into a lecture and I really find it interesting. I just can do it.” • “I don’t think I ever got enthusiastic, even at school, I think I just did it because I could do it. And some physics I did find quite interesting, but it’s like when you have to keep memorising all these things, it just takes the enjoyment out of it.”

20. Interview data • Adam: “I’m here because I enjoy mathematics and I want to understand it, not because I want to pass an examination and so, for me, it’s more about that I get to grips with it. OK, this examination is a benchmark, have you had time enough to assimilate this information? Can you put it down within two hours? Well, to be honest, I’m not really interested in that, for me, it’s more looking at it and getting to grips with it.”

21. Interview data • Adam: “I achieved my highest mark in the first year, only because it was just state and prove. I said to my girlfriend, “I bet you could actually pass analysis without knowing what the hell was going on.” If you’ve got a pretty good memory, then you write the proofs out over and over again, then you can do it. And I think that’s a real shame because it’s such a really nice subject, not that I’d want to study it much further than this, but it is great that everything is really proved and set down in stone.”

22. Interview data • Charlotte, asked why she chose to study maths:“Because I loved it at A level. Yeah, it was really good. I mean, A level, they said that they really preparing us to come to uni but then, when I got here, it was just so … nothing like what I expected. You can only do it, with your brain, like, in a certain way. I don’t think mine does.” • However… “Yeah, at A Level I really loved it but since I’ve been here, it’s like they’ve taken something I love and crushed it and now I hate it and I never want to use it again.”

23. Interview data • Charlotte: “I think probably in the second year. Because I didn’t have very many friends, because a lot of them had just gone. I hadn’t got many friends, I couldn’t do the work, there weren’t that many people to ask anymore. So, I found that sometimes it was very difficult to get the work done, I couldn’t do it myself. It was at that point where I started to think, it wasn’t really what I wanted to do anymore. And from there it just went downhill. It’s just, every time you get homework, you get in and you just can’t do it, it gets you down after a while. I think that’s what happened in the second year.”

24. Interview data • “I didn’t do enough work, especially in Analysis, because Analysis, the actual exam is rote learning. It’s very odd because the whole course is one of the most… it challenges you more than the other courses, the first term, it makes you think more and, yet, when it comes round to the exam, they’ve just got prove - . . ., prove this and this and this, rather than – I mean, it makes one wonder why they have, on the homework sheets, all these things that you spend ages banging your head against tables for, to come up with solutions to these questions when, at the end of the day, it doesn’t help you through the test, but it’s fun to do, I suppose.” James, straight A grade student

25. Tentative Conclusions • Mid-sessional tests – formative, but provide students with first insight into… • “Rules” of assessment – mismatch between coursework and exams? • Memorisation – “memorising with understanding” Marton et al (1996,1997) or “rote memorisation”. Is there something specific to the study of maths, or is this only to do with the students’ perceptions of assessment?

26. So? • Is this just another way of looking at the data? While embracers probably correspond to those with meaning approaches and cohesive conceptions, are pragmatists, instrumentalists and hangers-in more likely to be reproducers with fragmented conceptions?