1. TSG: 34 Task design and analysis: a response to the first four papers
Anne Watson
University of Oxford
ICME 2008

2. The intention of task design is to influence changes in the mathematical activity of learners
To act differently in future mathematical situations
To draw on different resources in future
use tools/representations differently
extend and/or enrich personal example spaces, concept images, mathematical repertoire…

3. How representation mediates learning
Representation focuses attention on visual similarities and differences – using the syntax of graphs to make new distinctions in meanings (Zaslavsky)
Representations need interpretation - create dialogue between representation and meaning (Swan)
Representation to provide a visual object to talk about (all)
Representation is syntactic, semantic, and enactive – moves enact grammar and meaning and afford reversal (Ainley)
Representation models a mathematical structure – transformations in one can be mapped onto the other so each is a model-of and model-for (Schmittau)

4. Shifts/extensions/enrichments of experience afforded
Multiplicative to exponential (Schmittau)
Discrete to continuous (Schmittau)
The utilitarian power of symbolic generalisation (Ainley)
Inversion (Ainley/ Swan)
Clarification/definition of distinctions between mathematical objects (Swan/Zaslavsky)

5. How are these enrichments achieved?
All tasks draw on students’ perceptions and past experience, and offer extensions
All tasks afford conjectures and feedback on effects
All tasks constrain choices to fit models, images, alternative representations, of conventional mathematics
All tasks exploit students’ search for familiarity, similarity/difference, and effect

6. Beyond purpose, context, curriculum, norms, and what is to be done:
How example choice and sequencing expose mathematical structure and potential confusions
How representations structure the affordances and constraints of the task
How variation, equivalence & difference are organised
How learners’ propensities to seek similarity/difference and familiarity are used
How learners’ choices and responses are exploited
How feedback and conflict are managed
How extension and enrichment of PES is afforded
What shifts of perception/critical ideas are afforded
What is there to argue about and how does this argument take place?
What is the payoff?