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=Blue Skies Maths Curriculum=

History & Aims
An opportunity arose to gather several colleagues to address afresh the question of what a mathematics curriculum could look like that took into account the ubiquitous presence of and familiarity with the Web and social networking. We met on Oct 5th 2010 to initiate discussion of possibilities.

1) What could mathematics itself look like as a human endeavour if it is assumed that learners have access to powerful tools via the internet?
===2) What are the pros, cons and interesting features of the use of applets (for rehearsal, for specific topics or concepts) and of generative software (dynamic geometry, dynamic function plotting and manipulating, computer algebra systems)?===

3) How might social networking be used productively without intruding on learners' own spaces and yet without denying this form of human interaction in mathematics?
For example, mathematics in primary and secondary is principally focused on getting the answer to calculations (through rehearsing procedures). Mathematicians do not see mathematics as calculation, so why does this view pervade school mathematics? Mathematics could be seen as a constructive enterprise, as a way to reveal underlying structural relationships and so model phenomena in the material, imagined and virtual worlds of human experience. As in the USMES project, tools and their conceptual underpinnings could arise as and when needed to make sense mathematically of a variety of phenomena.

Core Issues
These arose during our face2face discussions: **Learning by Rote –– Doing Calculations –– Learning without Purpose** There was a tendency in discussion to slide between these three aspects. Purposeful learning is a pedagogical-psychological-social rather than a cognitive or curricular issue. It is connected with the next three:

Phenomenal mathematics (surprise & expectation; need to make sense)
All mathematical concepts and techniques in school and early university can be initiated and motivated by trying to make sense of phenomena of different kinds

Making Mathematical Sense & Making Sense Mathematically
**Importance of Ways of Working on … in a conjecturing atmosphere**

Curriculum specification:
In stead of seeing curriculum as uni-dimensional, it could be seen as a collection of acetates: • Curriculum is usually specified by topic, which leads to assessment of routine proceudres on routine tasks; • Curriculum could by mathematical use of human powers (imagining & expressing, specialising & generalising, conjecturing & convincing, interpreting & (re)presenting, organising & classifying, ...) and mathematical themes (invariance in the midst of change, freedom & constraint, doing & undoing, ...); • Curriculum could be specified by license to use unaided (each of) a set of powerful mathematical tools (Spreadsheet, CAS, DGS, SPSS, ...) through having, as in a driver's license, shown appropriate use of that tool in several demanding situations; •Curriculum could be specified in terms of a collection of Habits of Mind (Cuoco, Goldenberg & );

The curriculum as manifested is then developed by teachers working collaboratively to 'shine light through the various acetate sheets'.

Role of Theorems & Proofs re Modelling
If modelling dominates the curriculum

Keeping teachers on board
Any proposed changes, however blue or cloudy skied, must involve the entire community of people associated with mathematics teaching and learning, at all ages and in all phases

Teacher as Curator –– Searching for some tool/phenomenon –– criteria of quality
Teachers making mathematically productive use of the resources available on the internet become curators of a library of applets and M-E software (manipulatively expressive: see Andy diSessa)

Invoking human powers (including puzzling things out)
Challenging learners to make sense of phenomena