# Background

Mathematics Teaching and Learning

While the mathematics teacher education sites address different issues
of teaching and learning, they share a set of principles about what it
means to teach and learn mathematics.

Our shared principles are represented well in Adding it Up (NRC,
2001). We agree that all students need opportunities to develop both
concepts and skills, to develop flexibility in their abilities to
engage with mathematical ideas and to engage in what some may call
higher order or critical thinking. Developing these ideas is about
rigor in both depth and breadth and is ultimately how we conceive of
understanding. We agree that developing mathematical understanding
involves developing knowledge and skill as well as a disposition that
orients one as someone capable of doing mathematics well.

## Preparing Teachers

While the mathematics teacher education sites address different
issues of teaching and learning, they share a set of principles about
what it means to teach and learn mathematics.

Our shared principles are represented well in Adding it Up (NRC,
2001). We agree that all students need opportunities to develop both
concepts and skills, to develop flexibility in their abilities to
engage with mathematical ideas and to engage in what some may call
higher order or critical thinking. Developing these ideas is about
rigor in both depth and breadth and is ultimately how we conceive of
understanding. We agree that developing mathematical understanding
involves developing knowledge and skill as well as a disposition that
orients one as someone capable of doing mathematics well.

## Shared Perspectives

While there is growing consensus about the perspective we share
about the teaching and learning of mathematics, we also know that it is
not a perspective that is widely implemented in schools and classrooms.
We know from studies like the Third International Mathematics and
Science Study (TIMMS) that in the U.S. most mathematics instruction is
not consistent with current reform ideas (Hiebert et al., 2003; Hiebert
& Stigler, 2000; Stigler & Hiebert, 1997, 1999). Most U.S.
mathematics classrooms maintain a focus on answers rather than
strategy. The teacher assumes responsibility for solving the problem
while student participation often involves providing the next step in a
procedure. TIMMS also reports that U.S. students had little opportunity
to discuss connections among mathematical ideas and reason about
mathematical concepts. Further we know that often “reform” becomes
about using manipulatives, putting students in cooperative groups, and
asking higher order questions in ways that in and of themselves do not
lead to classrooms that support the development of mathematical
understanding (Stigler & Hiebert, 1997, 1999; Webb et at al.,
2004). How these approaches play out within the classroom context
matters. How the teachers and students engage with the higher order
questions or the manipulatives also matters. So, our job as mathematics
teacher educators is to find ways to support teacher education students
to learn the practices in deep and connected ways that become
generative, mathematically meaningful, and make sense for each child in
the classroom.

In addition to these principles related to the teaching and learning
of mathematics, as teacher educators we have come, through this work,
to a better understanding of learning from practice in ways that become
generative. A principle driving all of this work is that teacher
education students need an opportunity to figure out what it means to
learn about practice from practice. To see that teaching practices are
not separate from the principles we have laid out and that they are not
separate from each other. We want to provide students with the tools,
knowledge, skills and dispositions they need to learn as they teach and
to do so in a way that is generative. The websites allowed us to focus
on learning from practice and we could use that learning to structure
opportunities to learn from the practice of teaching.