**November 20-21, 2008**

Our goal from these two symposia is to focus on what are some of the main foundational issues in mathematics education today and where do we see the field moving in addressing these issues. The result of focusing on this goal will hopefully yield a resource that will be of use to the field in one of many ways. Such may include:

*A white paper/list of recommendations that the Kaput Center will publish on its website and distribute to alert researchers and practitioners of our discussions

*A scholarly article/monograph or special issue in a well-read journal/venue

*New projects

*Archive of existing resources (papers, datasets) or on-going work that researchers and practitioners can access to help establish partnerships and new projects

It is not expected that the Kaput Center will be a lead in all of the final products of this meeting. But, it will aim to keep track of subsequent activities particularly in respect of the original funding (NSF).

In reviewing our original work and attempting to consolidate these with subsequent feedback, I present the following five themes for our preliminary discussion in conjunction with the plenary speakers.

**Ubiquity of Technology**

*Role of new dynamic technologies in reformulating mathematical knowledge for education and vocational use

*Transparency of technology

*In what form does an affordable technology exist (i.e.. where and for whom)?

*What are the new "design principles" for our educational and research practices?

*How can we explore new ways of mathematical thinking vs use technology to prop up existing practices?

*What is the role of a commercial company such as Verizon in providing the right kind of wireless infrastructure to help make our theoretical visions for accessible education become a widespread reality?

**Data Visualization**

*How can we help children make sense of large datasets and data of the world and society around them as well as adults making sense of natural and alternative ways of formulating ideas?

*How do we make sense of understanding the dynamics of complex systems? And more so, how do we understand what and how to focus on?

*There are many ways to visualize complex datasets but we need to know what constitutes an effective set of visual tools and how we apply them to educational environments.

*How do we explain them with different epistemologies? i.e. can young children today see a patterns in relevant data and make claims about truth.

*Fundamental: What do our young children believe in when they look at data in the daily papers? Who is their expert?

**Students as co-participants in building predictive and explanatory models**

*How do learners identify themselves with the mathematical "stuff" they operate with, and on, in digital environments?

*How do students identify with their work and in what ways?

*In essence, how can we develop from where students are, in thinking through mathematical ideas to, something that they identify with?

**Characterize particular ways of thinking in mathematics as a human activity**

*What are the next research questions to address advances in sociocultural studies and the evolution of tools and symbolic thinking?

**Psychologising, socializing, and formalizing**

I see a new theme that extends our March 2008 discussions and integrates thoughts from colleagues that could not make the original discussions.

*John Mason presented the following idea:*

Consider the integration of 'psychologising the subject matter' (as Dewey put it), with 'socialising (enculturating) learners into mathematical practices', with the changing 'absolutes' (as Gattegno put it) of children as they pass from early childhood through adolescents to late teens.

*In addition, Ubiratan D'Ambrosio wrote to me reflecting on our earlier themes:*

Is mathematics accessible and necessary for every human being? Rather than asking "particular ways of thinking in mathematics as a human activity" I would ask "mathematics as a particular way of thinking as a human activity...the obstacle in Math Education has to do with language."

**Overview of the symposium**

The symposium is organized across two days. Video clips of the discussions can be found below.

## Day 1 Video Clips

Part 1 | Part 2 | Part 3 |

Part 4 | Part 5 | Part 6 |

Part 7 | Part 8 | Part 9 |

Part 10 | Part 11 | Part 12 |

## Day 2 Video Clips

Part 1 | Part 2 | Part 3 |

Part 4 | Part 5 | Part 6 |

Part 7 | Part 8 | Part 9 |

Part 10 | Part 11 | Part 12 |

Part 13 |

**Associates of the Kaput Center wish to thank the National Science Foundation for partly funding this symposium.**