Approaches to studying and conceptions of mathematics among undergraduate mathematics students

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Approaches to studying and conceptions of
mathematics among undergraduate mathematics
students

• Judith MacBean

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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

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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

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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

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Data collected

• Other
• Exam results
• Departmental documentation
• Departmental website
• Meetings with Departmental tutor (discussion of
  student progression from year to year)

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The cohort Originally 142 students (123
completed initial questionnaire)
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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)

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Conceptions of Learning and of Mathematics

• Brief potted history
• Säljö (1979)
• Marton et al (1993)
• Crawford et al (1994)
• Crawford et al (1998a)

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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?

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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.

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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?

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Harry high exam results
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Sarah high exam results
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Adam low exam results
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Charlotte low exam results
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Interview data

• Harry, when asked what he liked about maths
  Ive asked myself this question a few times.
  Its a difficult question to answer, isnt it?
  Obviously, one of the reasons was that Im pretty
  good at it. But I was quite bad at it, actually,
  before I got interested in it. Once youre
  interested in something, I suppose you become
  better at it. With Maths, I dont really see it
  in terms of symbols and, and calculations and
  numbers, I see it in terms of ideas. And I think
  its kind of the purer subjects, in my opinion, .
  . . ideas. And although theres a proof its
  not about the little tricks and everything, with
  the algebra and everything, its about the idea,
  where did it come from. You know, its just
  always that sort of flash of inspiration, thats
  the aspect that Ive always liked. And you
  really prove something, if you like, youre
  completing a jigsaw puzzle or games or whatever,
  its just like that. Its so vast, theres so
  many different things, so much. Youll never
  know all of it.

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Interview data

• Harry, when asked how well he thought he was
  doing Its difficult to say. I normally dont
  judge my progress so much on what my marks are,
  because theyre 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 doesnt change
  who you are, it doesnt change how good you are,
  it just means you worked hard. So thats not
  what Im interested in. What it means is to be
  able to understand what were doing. And so far
  in Analysis I feel like I understand things.

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Interview data

• Sarah, at end of first year Well, I kind of
  knew what to expect from the papers, like
  Analysis, youve just got to learn the proofs . .
  ., its just going to do the proofs on there, so
  that one was OK.,
• Sarah, at end of second year I think its all
  about memory. I think exams, . . ., like,
  analysis, the whole thing is just memory, you
  could not I dont understand it, but if Ive
  done well in it, its just memorising it
• Sarah, at end of third year But, like, now I
  know that you dont always have to understand
  everything to do well in the exams. Like, maybe
  the first year I didnt know, but, now, I think,
  now, like, Im more relaxed, like if I dont
  understand something Im not really panicking and
  trying to understand it. I know you can just
  use your memory to memorise it, you dont really
  have to understand it, but thats just in some
  subjects.

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Interview data

• Sarah, when asked how much she enjoyed maths, at
  end of second year I just do it, I just dont
  enjoy it, I just do it. I wouldnt say I go into
  a lecture and I really find it interesting. I
  just can do it.
• I dont 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 its like when you have to keep
  memorising all these things, it just takes the
  enjoyment out of it.

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Interview data

• Adam Im here because I enjoy mathematics and I
  want to understand it, not because I want to pass
  an examination and so, for me, its 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, Im not really interested in that, for
  me, its more looking at it and getting to grips
  with it.

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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 youve got a pretty good
  memory, then you write the proofs out over and
  over again, then you can do it. And I think
  thats a real shame because its such a really
  nice subject, not that Id want to study it much
  further than this, but it is great that
  everything is really proved and set down in
  stone.

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Interview data

• Charlotte, asked why she chose to study
  mathsBecause 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 dont think
  mine does.
• However Yeah, at A Level I really loved it but
  since Ive been here, its like theyve taken
  something I love and crushed it and now I hate it
  and I never want to use it again.

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Interview data

• Charlotte I think probably in the second year.
  Because I didnt have very many friends, because
  a lot of them had just gone. I hadnt got many
  friends, I couldnt do the work, there werent
  that many people to ask anymore. So, I found that
  sometimes it was very difficult to get the work
  done, I couldnt do it myself. It was at that
  point where I started to think, it wasnt really
  what I wanted to do anymore. And from there it
  just went downhill. Its just, every time you
  get homework, you get in and you just cant do
  it, it gets you down after a while. I think
  thats what happened in the second year.

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Interview data

• I didnt do enough work, especially in Analysis,
  because Analysis, the actual exam is rote
  learning. Its 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, theyve 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 doesnt help you through the test,
  but its fun to do, I suppose. James, straight
  A grade student

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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?

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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?