**Suggested Citation:**"Concluding Remarks." National Research Council. 2001.

*Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop*. Washington, DC: The National Academies Press. doi: 10.17226/10050.

**Concluding Remarks**

*Deborah Loewenberg Ball*

Over the last two days we analyzed teacher practice for the mathematics entailed by that practice and opportunities teachers had to learn mathematics from the actual work of teaching. This consideration of what teachers need to know to teach well and how they come to learn what they need to know raises many issues. Much of what teachers need to know is under-specified. What kind of knowledge do teachers need beyond what students need? On one hand, we have been talking about the “packages” of knowledge that organize mathematics that is important for teaching. On the other, we have also seen the importance of being able to “unpack” mathematical knowledge in order to listen to students, choose tasks, and otherwise use mathematics in teaching. How can we design opportunities for learning that enable teachers to understand the nature of learning mathematics, to appreciate their students ' struggle to learn, and to recognize the mathematics in what their students are saying? How can we provide opportunities for teachers to learn that will give teachers a sense of the mathematics they choose to pursue and an understanding of why they made those choices? How can we as a community of mathematics educators frame a structure for teacher development that will begin to address these issues?

The Workshop was not structured to provide answers, but rather to set the stage for the next steps. What can we take from this Workshop? The experience was designed to be an intellectual resource for your own thinking shaped by the notion from Ma's work of a profound understanding of fundamental mathematics—that teachers need to know what they are teaching in a deep and substantial way. The artifacts presented to stimulate your thinking were based on using mathematics in practice, centered around some of the core tasks of teaching. Opportunities to learn mathematics can be found by using sites of practice or outside of practice. In the first instance, it is easy to get lost in the doing and not pay attention to the actual mathematics involved. In the second, it is necessary to mediate among course materials, texts, what knowledge is actually needed and learned in the process, and the actual use of that knowledge in the classroom. The bridge between these ways of coming to know raises new questions: What is understanding of mathematics for teaching? How can we develop this understanding in the profes-

**Suggested Citation:**"Concluding Remarks." National Research Council. 2001.

*Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop*. Washington, DC: The National Academies Press. doi: 10.17226/10050.

sional education of teachers?

As mathematics educators, we need to pursue the important questions and begin to develop the ideas. In your own work, try out some activity or approach centered around the kind of thinking that framed the Workshop. Document what happens and probe the effects to see what underlying factors may be present. Think carefully about the nature of the evidence you present. Write about your experiences and present your ideas to your colleagues in a variety of forums. As a community, we can begin to collect these experiences and evidence, testing our theory against the reality of teachers and students, and begin to move forward in a real analysis of what mathematics teachers need to teach well and how they come to learn the mathematics they need to know.