DIFFERENT VERSIONS OF THE SAME TASK: CONTINUOUS BEING AND DISCRETE ACTION

1
DIFFERENT VERSIONS OF THE SAME TASK CONTINUOUS
BEING AND DISCRETE ACTION

• Anne Watson
• University of Oxford
• MADIF 2008

2
Interest in task design

• Realisation of importance of task in teaching and
  learning (Sierpinska, 2004 Burkhardt and
  Schoenfeld, 2003)
• Extended, multi-stage, authentic, assessment (van
  den Heuvel-Panhuizen Wittmann 1998)
• Day-to-day tasks in classrooms (Swan, 2006
  Mason and Johnston-Wilder, 2006 Karp, 2007).
  Variation and object of learning (Runesson 1999
  Emanuelsson 2001)
• Affordances of mathematical activity doing and
  generalising (Watson 2004)

3
Not teacher-proof!!

• Tasks adopted from materials differences quality
  of activity (Stein, Grover and Henningsen, 1996)
• Need to understand different mathematical
  qualities (PME RF 2008)

4
Ways to analyse differences in lessons

• Different questions and prompts for different
  mathematical thinking
• Variation of examples and exercises for different
  objects of learning (e.g. Runesson)
• Teachers role in interactive sequences
• Different epistemological aspects of mathematics
  emphasised (e.g. METE project)

5
Mathematical quality

• measured by people with strong mathematical
  qualifications (e.g. TIMSS 7-nation video study,
  2003)
• descriptions of mathematical activity can be
  fractal and meshed together (Davis Sumara,
  2006)

6
My approach

• What is available mathematically in the public
  activity of the classroom?
• What is the class supposed to be doing right
  now?
• What are they supposed to be thinking about?
• What is being said and done, and by whom, that is
  shaping and is shaped by the activity?
• (what is the object of learning as perceived by
  the learner)

7
Comparing different teachers, different classes,
same task

• Parallel classes
• Shared purpose to understand and learn how to
  construct some loci
• Task A making loci by following instructions in
  open space (e.g. find a place to stand so that
  you are the same distance from these two
  points)
• Task B compass and straight-edge constructions
• Teachers chose order of tasks, language, how
  links are made, whether all or some involved in
  physical task, whether rulers are allowed

8
Similarities

• asking, prompting, telling, showing, giving
  reasons
• referring students to other students work
• explaining choices and actions
• working out how to do the constructions,
• variation offered was similar within each locus
• choice of loci was shared
• teachers stated intentions
• all teachers praised accuracy
• written work similar range of rough sketches and
  neat constructions

9
Unhelpful dichotomies

• open/closed
• teacher-centred/learner-centred
• traditional/reform
• individual/collaborative
• procedural/conceptual
• etc.

10
Differences

• order of tasks
• different sub-tasks
• different things said at different points in
  activity
• what was said to whole group or small
  group/individuals
• different order of loci
• different emphases
• different tools at different times
• different conceptualisations afforded

11
Lesson X

• compasses used as a tool for reproducing equal
  lengths
• what can we use to get equal lengths?
• what do compasses do for us?
• why would I use the compasses?
• same distance equidistant
• compare use of compasses in two constructions
• relationships as caused by equal lengths
• physical acting out using a different
  representation to rehearse newly-understood
  relationships (nothing said here about equal
  lengths)

12
Lesson Y

• find individual points that fulfil conditions
  collective task
• rough diagrams and reasoning, join up points
• use compasses and straight edges as tools to join
  up points, how not why
• locus as obey rule
• physical activity use of same language as in
  construction task obey rule, same distance

13
Lesson Z

• visualisation in context
• physical activity same distance
• reproduce this when drawing
• small groups compasses for equal lengths
• public statements about links between situations
  what was the same?

14
Comparing lessons

• No implied hierarchy of lessons
• All lessons included
• relationships between variables
• properties
• reasoning about properties
• relationships among properties
• Different relationships emphasised
• Different tool use

15
Discrete action

• When broken down into separate actions there were
  no significant differences in lessons, yet each
  lesson was distinctive
• Cannot describe mathematics teaching just as a
  collection of actions, utterances, tasks,
  examples, student groupings, epistemological
  aspects

16
Continuous being

• the teacher is guided by her personal perception
  of the general relationships embedding the task
• different language, different sequencing,
  different emphases, different comparisons,
  different connections are made by different
  teachers acting out their own continuous sense of
  working with loci
• these are not necessarily right or wrong, just
  different

17
Continuous being

• there is coherence in the relationships among
  tasks, language, emphases, prompts and other
  components.
• these are manifested as continuous threads of
  mathematical awareness

18
Implications

• teaching mathematics is the more-or-less fluent
  expression of an understanding of a mathematical
  context
• mathematics teaching is a way of being
  mathematical
• education of mathematics teachers could be
  approached as a continuous complex mathematical
  experience

19
Thanks to

• Ella Fitzgerald and Sy Oliver
• It aint what you do its the way that you do
  it
• and teachers at Oxford Community School