The overall goal of this 45 year project is to create tools that
will help learners (at all levels) make greater sense of complex phenomena
and to study how learners come to understand complexity. Complexity is the
study of systems in which phenomena or global behaviors arise from the
interactions of simpler parts. Many everyday phenomena exhibit complex
behavior: the growth of a snowflake crystal, the perimeter pattern of a
maple leaf, the dynamics of the Dow Jones or of a fourth grade classroom.
These are all systems which can be modeled as composed of many distributed
but interacting parts. They all exhibit nonlinear or emergent qualities
which place them beyond the scope of current K12 mathematics curricula.
Complexity is a theme that cuts across traditional discipline boundaries.
Yet it is rarely found as an explicit theme in K16 curricula. Indeed,
many studies have shown that, in both the public at large and in science
classrooms, "good thinking" about systems of interacting agents is not
easily found.
This is a time that is seeing a ground swell of interest in the
sciences of complexity. It is also a time of increasing globalization of
economies and increasing awareness of the interconnectedness of both
natural and human systems. This has resulted in a strong need, in both the
public at large and in the scientific community, for increased abilities
in systems (or ecological) thinking .
This project grows out of and responds to that need. The concrete
goals of the project are to:
 Create (and enhance existing) socalled "objectbased parallel
modeling languages" (OBPML) (sometimes also known as agentbased modeling
languages) that can be used by learners to create rich and detailed models
of large systems of interacting agents and objects. The languages should
strive to obey the maxim "low threshold, no ceiling"  that is, they
should be simple enough that middle schoolers can use them to create
models, yet powerful enough that research scientists can do their research
using these languages.
 Create a library of socalled "extensible models" (i.e., domain
specific models created using these toolkits) for learners to explore and
extend. Since complexity is a theme which cuts across traditional content
domains, the models developed will include content material from across
the natural and social sciences as well as from the discipline of
mathematics .
 Characterize the developmental path of learning to think about
complex systems. How does "complex" thinking develop and mature? How do
learners shift from thinking in terms of hierarchical and deterministic
control to probabilistic and decentralized approaches? The data for this
developmental theory will be gathered through conducting indepth learning
interviews with users of the OBPML and the extensible models.
All of these goals will be pursued with the framework of
"Connected Mathematics". Traditional mathematics education has proceeded
from a view that mathematics is "given" rather than constructed and is to
be transmitted to learners primarily through formalism. As a result,
mathematics is usually taught in isolation from other domains and the role
of technology in mathematics education has primarily been to better
inculcate or animate the existing formalisms. In contrast, the theory of
Connected Mathematics sees the fundamental activity of mathematics as that
of making and designing new mathematical representations and connecting
these representations to each other and to other domains. The vision of
mathematics as being made and not simply received leads naturally to a
role for technology. Technology is not there simply to animate received
truth, it is an expressive medium Ð a medium for the making of new
mathematics. It follows that we can make better use of computational
technologies than simply running blackbox simulations Ð we can make
mathematics by constructing computational embodiments of mathematical
models. The true power of the computer will be seen not only in better
teaching of the old topics but in transforming ideas about what can be
learned.
As the project unfolds, we expect to develop:
 New modeling environments which move beyond current OBPML  that
integrate multiple levels and modes of representation
 A library of extensible models  a valuable resource shared through
publication on the world wide web
 A body of case studies of learners building computational models of
complex phenomena
 A cognitive science theory of how learners come to make sense of
complex phenomena and of the role of modeling in facilitating that
sensemaking.
