Download John Monaghan - University of Leeds

Eclectic theorisation: knowledge construction and practice
John Monaghan
School of Education, University of Leeds
Paper presented at the University of Leeds School of Education Research Conference, The
role of theory in education research, Hinsley Hall, Leeds, 14 May 2007
What I wrote
My main interests are tool use by teachers and by learners, practice and epistemic
matters – all with regard to mathematics. I also always try to get an holistic
perspective in my work.
Important theoretical influences are:
Soviets
Davydov on knowledge construction and Leont’ev on goals and actions.
Americans Wertsch on mediational means and Saxe on emergent goals.
French didactics On ‘instrumentation’ (turning an artefact into an instrument), the
institutional dimension of knowledge construction and tasks & techniques (epistemic
and pragmatic techniques).
My talk will briefly go over the above and then look at how I intend to employ these
theoretical ideas in a little project John Threlfall and I have just started on the use of
spreadsheets in KS2 and in KS3.
The discussion could focus on ideas I introduce or it could focus on the pros and cons
of ‘mixing theories’. Regarding ‘mixing theories’ – maybe it is a bad idea. I think I
do it because I’m basically a practical mathematics educator who wants an holistic
account of learning and teaching mathematics with ‘things’. I’d defend my position
by claiming that I’m not trying to get a ‘unified theory’ and recognise that the
theories I use have different historical roots.
Davydov
On the concrete and the abstract (one aspect of knowledge construction)
A lot of ‘abstraction’ in maths but what is abstraction?
Abstraction ain’t ‘pattern spotting’ but progresses from an initial entity and
ends with a consistent final form. He calls this progression the ‘ascent to
the concrete’, the germ of an abstraction develops into a form which can
be used to explain reality. It depends on the “disclosure of contradictions
between the aspects of a relationship that is established in an initial
abstraction”.
1
Relevant for ‘task design’, e.g. topic ‘multiplying by a number between 0
and 1’.
Learner probably thinks multiplying makes a number bigger – exploit this
contradiction.
Leont’ev, Saxe and Wertsch on goals and actions
To Leont’ev goals are the raison d’être of an activity but Saxe’s emergent
goals are little ‘must do’ things that come into being during activity and
usually pass away. Activity theory is generally viewed with regard to
activity systems á là Engeström - they are large scale systems. A school
qualifies as an activity system, a sequence of lessons does not. This remark
applies to Leont’ev’s version of activity theory too but Leont’ev
introduces: activity, actions, operations and goals can be viewed with
regard to actions. Tool use is essential to actions and operations.
Wertsch is another ‘micro activity theorist’. He says a lot about
mediational means (MM- tools, artefacts, conventions) and actions. 2 are:
1) Not so much humans as humans with MM
2) MM transform (mathematical) actions
Going back to ‘multiplying by a number between 0 and 1’ – quite different
depending on the tool you use to do it with.
French didactics (Artigue, Chevallard, Laborde, Lagrange, Rabardel,
Trouche – others)
Instrumented activity
Distinguishes between a tool, as a material object, and an instrument as a
psychological construct: “the instrument does not exist in itself, it becomes
an instrument when the subject has been able to appropriate it for himself
and has integrated it with his activity.” Resonates with Marxist views that
tools are simultaneously ideal and material.
‘Instrumental genesis’ – the process of turning an artefact into an
instrument. Different, in general, for learners and teachers.
Knowledge and institution
Knowledge lives in institutions (could read ‘communities of practice’).
Contrast knowledge generated by research mathematicians and the ‘same’
knowledge reproduced in schools! A lot of values come in, e.g. knowledge
2
to be taught, from people who work for, say, QCA. The teacher comes in –
contrast ‘knowledge to be taught’ with ‘taught knowledge’. The learner
comes in – contrast ‘knowledge taught’ with ‘knowledge learnt’. All this
makes one very careful about speaking of ‘knowledge’.
Task, technique, technology (talk) and theory
Tasks and techniques are problematised: tasks are artefacts that are
constructed (and reconstructed) in institutions; techniques are not simply
technical manipulations but are institutionally privileged to the extent that
only one, of many possible techniques, may be considered.
Two aspects of tasks and techniques: (i) nothing natural about specific
tasks or techniques (there are many things we can do and, invariably, many
ways to do them); (ii) social values are attributed to specific tasks and
techniques. We differentiate between pragmatic and epistemic values of
techniques. Pragmatic values concern the efficiency, or breadth of
application, of a technique. Epistemic values concern the role of
techniques in facilitating mathematical understanding.
Re ‘multiplying by a number between 0 and 1’ – I might introduce a
technique of multiplying by 0.3 as multiplying by 3 and dividing by 10
(and dividing by 10 as moving the decimal point). Done correctly this has
pragmatic value. Whether it has epistemic value depends!
Our little project
Teachers’ use of spreadsheets to enhance pupils’ mathematics subject
knowledge in KS2 & KS3
Aims:
1. To encourage cross-school and cross-phase teacher collaboration
with regard to the use of ICT in mathematics in KS2 and KS3.
2. To investigate teachers’ foci in the use of ICT in maths in KS2 & 3.
3. To investigate curricula opportunities and constraints with regard to
the use of ICT in maths in KS2 & 3.
4. To investigate issues concerning ICT-maths task design in KS2 & 3.
5. To investigate student engagement with maths in ICT-maths tasks in
KS2 & 3 and aspects of their understanding of maths.
3
6. To facilitate within-school continued professional development
through focused reflections on task design and lesson observations.
University team
Schools triad 1
Schools triad 2
John M, John T, Patricia George
secondary teacher, 2 primary teachers
secondary teacher, 2 primary teachers
Focus had to be on teachers but …
… our belief that studies of teachers’ use of ICT (or, indeed, any resource)
cannot be conducted without a simultaneous focus on pupils’ learning.
The emphasis on ‘mathematics subject knowledge’ reflects project
teachers’ concerns that proposed project activities must be related to
supporting subsequent learning within each school.
Two main work stages – May-July and October-November. Teachers (in
3s) will plan and then teach a sequence of lessons based around the use of
spreadsheets. We will monitor their task design and their perceptions of
curricula opportunities and constraints. Six pupils from each class will be
observed in detail. Screen capture software together with audio recordings
will be used to collect data. Lessons will be video-recorded. Lessons will
be mutually observed and a debrief will follow (audio-reorded).
OK, so how do theoretical ideas enter?
Well, a lot of ‘French’ foci:
 knowledge tasks – techniques, instrumental genesis
institution
How do teachers from each phase (school/institution) approach the design
of tasks? What knowledge and what instrumented techniques do: teachers
build into their tasks; do pupils realise? Monitoring the development of
teachers’ and pupils’ instrumental genesis. Do the pupils actually do/learn
what the teachers expected? (is maths knowledge enhanced?)
Re Saxe, Wertsch and Davydov
Monitoring teachers’ emergent goals in lessons – what, why and how do
they lead pupils to knowledge?
Focus on the what and why of pupils’actions and what knowledge they
lead to:
4
 pragmatic and epistemic values of techniques
o and where, if anywhere, they lead to
 their epistemic actions and do they lead to abstractions
(and if so, do these abstractions reside in this tool use)
NB the underlined bits may be beyond what this project can do
5