Mathematics Education Summer SchoolNαύπλιο, Greece

21 – 27 August 2003

MATHEMATICIANS AS EDUCATIONAL CO-RESEARCHERS:Reflections on Learning Mathematics

at University Level

Elena Nardi

Session Summary

 

In recent years research into the teaching and learning of mathematics at the undergraduate level has begun to acquire increasing significance within the field of mathematics education. Beyond research that examines student difficulties with regard to specific mathematical concepts (such as function or limit) or forms of mathematical reasoning (such as proof), studies now consider pedagogical beliefs and practices at the undergraduate level. In the three 90-minute sessions we will

- discuss a series of studies in this area currently taking place at the University of East Anglia in the UK (Session 1)

and examine data from these studies including:

- mathematics undergraduates’ written responses to problems from their Year 1/2 courses in Analysis, Linear Algebra and Group Theory (Session 2); and,

- extracts from group interviews with mathematicians who lecture the courses and assess the students’ work in which the mathematicians reflect on their students’ mathematical thinking and related pedagogical issues (Session 3) 

Proposed outline of Session 1

Welcome and introductionsSession summary introduction

Study introductionDiscussion of example from study

Setting up for Sessions 2 and 3

MATHEMATICIANS AS EDUCATIONAL CO-RESEARCHERS

a study conducted by Elena Nardi and Paola Iannone, University of East Anglia

and managed by Chris Sangwin, University of Birmingham

Responding to an increasingly urgent need for collaboration between mathematicians and mathematics educators, the study engages mathematicians as educational co-researchers in a series of themed Focus Group interviews where a pre-distributed sample of mathematical problems, typical written student responses, observation protocols, interview transcripts and outlines of relevant bibliography is used as a trigger for reflection upon and exploration of pedagogical issues.

An introduction to the study

Rationale

Aims and methodology

Preliminary map of themes

Example

Implications

Rationalerapid changes within teaching mathematics at university level

fewer and fewer students opt for exclusively mathematical studies

recruitment of good graduates to mathematics teaching at a low

gap between secondary and tertiary mathematics teaching approaches

student alienation from traditionalism of university-level teaching

university accountability regarding quality of teaching

modifications of the tertiary syllabus in the 90s and topic-centred studies

reform should be focusing on teaching

underlying principles

practices

students' experiences and needs

…and on the fragile relationship between

mathematicians and mathematics educators…

the need to engage mathematicians in self-reflective processes

Aims and methodology: the use of Focus Group Interviews

groups of mathematicians from seven institutions in the UK

main body of data: team of five mathematicians based in UEA

15-month clinical partnership funded by LTSN

forum of collaboration: Focus Group Interviews

 

Focus Groups allow

observation of ‘collective human interaction’ (Madriz 2001)

 

‘…the researcher usually dominates the whole research process, from the selection of the topic to the choice of the method and the questions asked, to the imposition of her own framework on the research findings. Focus group minimises the control that the researcher has during the data gathering process by decreasing the power of the researcher over the research participants. The collective nature of the group interview empowers the participants and validates their voices and experiences’. (p838)

Methodology: Data CollectionThe six themes

11 Cycles of Data Collection, six in UEA, five elsewhere in the UK

audio recorded discussions of a dataset, one on each of six themes

 

1. Formal Mathematical Reasoning I:

Students’ Perceptions of Proof and Its Necessity

2. Mathematical Objects I:

The Concept of Limit Across Mathematical Contexts

3. Mediating Mathematical Meaning:

Symbols and Graphs

4. Mathematical Objects II:

The Concept of Function Across Mathematical Topics

5. Formal Mathematical Reasoning II:

Students’ Enactment of Proving Techniques and Construction of Mathematical Arguments

6. A Meta-Cycle: Collaborative Generation of Research Findings in Mathematics Education

Methodology: Data CollectionThe Dataset

a Dataset consists of:

a timetable for the half day meeting

a short literature review

samples of data on the theme from previous studies

students’ written work

interview transcripts

observation protocols

list of issues to consider

 

group members prepare for the audio recorded discussion in advance

Methodology: Data Analysis

full transcription of each digital recording

200 minutes long, a Verbatim Transcript of 30,000 words

transcript roughly structured according to the structure of the Data Set

researcher intervention: minimal co-ordination and consolidation

an almost natural emergence of 80 Episodes, our analytical units

an Episode: self-contained piece of conversation with a particular focus

80 Episodes from Cycles 1 – 6 recordings, 80 Stories

a Story: a narrative account

which summarises content and highlights conceptual significance

first and second level analytical triangulation

 

 

Preliminary map of themes

On students’ attempts to adopt ‘genre speech’

On pedagogical insight: tutors as initiations in ‘genre speech’

On the impact of school mathematics

on students’ perceptions and attitudes

On one’s own mathematical thinking

and the culture of professional mathematics

On the relationship, and its potential,

between mathematicians and mathematics educators

(25, 25, 4, 20, 6)

A sample of data from Cycle 1… and the fourth preliminary theme

 

The context: a Linear Algebra question which involves proving certain properties of the adjoint (or adjugate) of an nxn matrix, thus extensive use of det(M), the determinant of a matrix.

 

15 minutes into the discussion, foci have included:

question setter’s intentions

typical tendencies in student responses

speculations on student perceptions of determinants

 

Or…

‘what the students actually feel when they do these things. And when you see a determinant how do you… how is one supposed to relate to it?’

 

Example of one discourse in the data: Concept Image Construction

‘a bit of garbage that is sort of coming your way?’

‘something to be worked out’

‘I cannot handle a thing that is very complicated’

‘it removes the true power of the adjoint’

the role of the ‘guidance given to the student’

the need to try to ‘understand and appreciate the student’s landscape’

for students: ‘a number to be worked out’?

the need to provide ‘some structure’

the need to ‘share landscapes’

students’ resistance to cross-topical images

the limitations of a ‘compartmentalised view of mathematics’

the need to build subtle instrumental images:

det(M) as ‘something which just saves you writing down a large number of elements’

but …. ‘it is not easy to make it good’

image construction as a personal venture

suspicion towards ‘forced networking of all mathematics’ where ‘everything relates to everything’

you ‘need to have your own, tailor-made brain version of what the thing is’

is there ‘just one’ such ‘network’ anyway?

 

Some observationsconcept image spaces as

dynamic loci of human cognition

a creative fluctuation between

epistemological, psychological and pedagogical analysis

a vindication for

non-deficit models of research on teaching

 

a sense of ownership can exceed the impact of

externally imposed pedagogical prescriptions

In place of a conclusioneach Episode re-embedded in original aim:

identify patterns in attitudes, beliefs and practices

 

two discreet but distinct roles of the mathematics educator:

strategic question posing

view consolidation

 

group enthusiasm helping conversations

escalate beyond the remit of pre-determined themes

 

the Dataset: not a straightjacket

but a solid basis for discussion

 

participants, by constantly re-shaping the focus of the discussion, determine content of data and eventual focus of the research…They are

thus becoming co-researchers…

 

a topical and much needed pedagogical enterprise