What are some promising ways to help teachers not only develop mathematical understanding but learn to *use* mathematical insight and knowledge in the context of practice? What are the key features of what makes an approach promising? Are there ways to engage preservice teachers in learning mathematics through the tasks they will actually do in practice?

Leader: *Mark Klespis;* Members: *Tom Carpenter, Liping Ma, Patricia Reisdorf, Skip Fennell, Paul Kuerbis,* *Judy Merlau, Ginger Warfield*

We interpreted this question to mean we needed to examine ways in which preservice teachers can synthesize the content knowledge they receive and then apply this knowledge in practice. Prospective teachers must come to appreciate how basic mathematical ideas are introduced, developed, and retained by students. The interplay among a teacher 's content knowledge, understanding of that content, and awareness of strategies for presenting it in practice is a complex endeavor. At each level—preservice, student teaching, and inservice—there is a need for teachers to talk with others so they can tap into the shared knowledge of all practitioners in the field.

*Preservice teachers can synthesize mathematical content and pedagogical* *content knowledge through core conversations with other members of* *the mathematics education community.* Teachers share their mathematical knowledge in conversations with their students. At the same time, students share their knowledge with their peers. The group agreed it was essential that students and their instructors actively converse about mathematics. This idea gave rise to the phrase “core conversations ” to describe instances where students and teachers engaged in an in-depth discussion of one or more aspects of a piece of mathematical knowledge. In content and methods classes, these core conversations can be enhanced in many ways. One way is to have students view video clips of elementary school students learning mathematics with a goal of analyzing the mathematical knowledge needed to teach effectively. A second suggestion was to have students participate in a mathematics forum where different solution paths to a problem could be discussed.

One emphasis at this Workshop was to ask people to think of a list of core mathematical ideas that preservice teachers should know. Such a list is necessary, but

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Question #5
DISCUSSION GROUP #5
What are some promising ways to help teachers not only develop mathematical understanding but learn to use mathematical insight and knowledge in the context of practice? What are the key features of what makes an approach promising? Are there ways to engage preservice teachers in learning mathematics through the tasks they will actually do in practice?
Leader: Mark Klespis; Members: Tom Carpenter, Liping Ma, Patricia Reisdorf, Skip Fennell, Paul Kuerbis, Judy Merlau, Ginger Warfield
ASSUMPTIONS MADE IN FRAMING THE DISCUSSION
We interpreted this question to mean we needed to examine ways in which preservice teachers can synthesize the content knowledge they receive and then apply this knowledge in practice. Prospective teachers must come to appreciate how basic mathematical ideas are introduced, developed, and retained by students. The interplay among a teacher 's content knowledge, understanding of that content, and awareness of strategies for presenting it in practice is a complex endeavor. At each level—preservice, student teaching, and inservice—there is a need for teachers to talk with others so they can tap into the shared knowledge of all practitioners in the field.
SUMMARY OF THE MAIN POINTS OF DISCUSSION
Preservice teachers can synthesize mathematical content and pedagogical content knowledge through core conversations with other members of the mathematics education community. Teachers share their mathematical knowledge in conversations with their students. At the same time, students share their knowledge with their peers. The group agreed it was essential that students and their instructors actively converse about mathematics. This idea gave rise to the phrase “core conversations ” to describe instances where students and teachers engaged in an in-depth discussion of one or more aspects of a piece of mathematical knowledge. In content and methods classes, these core conversations can be enhanced in many ways. One way is to have students view video clips of elementary school students learning mathematics with a goal of analyzing the mathematical knowledge needed to teach effectively. A second suggestion was to have students participate in a mathematics forum where different solution paths to a problem could be discussed.
One emphasis at this Workshop was to ask people to think of a list of core mathematical ideas that preservice teachers should know. Such a list is necessary, but

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we need to ascertain the core conversations surrounding preservice teacher issues as well.
Several questions arose: Do we practice what we preach here? How often do we meet with colleagues as we have done at this Workshop to discuss issues related to teaching at our own campus or school? If we believe these conversations have value for our students, we need to have them ourselves. Core conversations help all of us refine and improve our teaching practices. The group felt strongly that these conversations need to begin early. This led to our second central assertion.
The mathematics education community's resources must be available to preservice teachers now—starting in their college classroom. Prospective teachers need to be aware of the multiplicity of mathematics education resources early in their preparation—resources beyond their college instructors. One immediate suggestion was to make sure students know at least one good mathematics Web site. Another suggestion in a similar vein was the development of a virtual online community. A key feature of this community would be to facilitate communication among all the groups involved in mathematics teacher preparation.
Some examples of mathematics education networks include the following:
Texas Statewide Systemic Initiative
ToToM (Teachers of Teachers of Mathematics)
Mathematics Forum at Swarthmore
Operation Pipeline
National Science Foundation's best practices—a report on curriculum reform projects
America Counts
The Eisenhower National Clearing-house Web site
Liping Ma described at length the working conditions of Chinese teachers. A great deal of their day is spent discussing the next day's lesson with other teachers on their team. This team can consist of teachers working at the same grade level, but it can have teachers from the grades immediately above or below the grade levels being taught. This structure is not common in current U.S. schools. The team approach to teaching, however, is an excellent program to adapt to preservice programs, especially in light of the group's discussion of core conversations about mathematical content and pedagogy. A cohort of students could discuss fundamental mathematical concepts throughout the semester in a mathematics content class and then attempt to incorporate the mathematical insight in the context of teaching practice—in either a content or methods course.
Early apprenticeships in mathematics teaching must be required in the teacher preparation program. One way to make sure preservice teachers feel connected to the larger mathematics education community is to require field experiences early in their college careers—experiences that go beyond the level of classroom observation and that engage prospective teachers in some actual acts of teaching. Such experiences will help students see themselves as teachers-in-training rather than simply college students. It is also important for students to see the issues with which inservice teachers need to deal daily.
Field experiences must flow continuously from the preservice level, to “internship,” to master teacher level. As one grows in mathematical and pedagogical knowledge, a person's core conversations move outward. That is, talking begins within groups in a preservice mathematics classroom, expands to mentor teachers

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during the first year of teaching, and then to colleagues at all levels as one enters the profession.
There must be an agreement on the fundamental mathematics needed for teaching. There must be a consensus on what core mathematics is essential for teaching. Such a consensus was not reached in our group. However, if the U.S. curriculum is a “mile wide and an inch deep,” as the Third International Mathematics and Science Study researchers suggest, what implications does this have for our preservice curriculum? The Texas Statewide Systemic Initiative's Guidelines for the Mathematical Preparation of Prospective Elementary Teachers (Molina, Hull, & Schielack, 1997) is one attempt to provide guidance to colleges and universities in this area.
ISSUES/COMMENTS
It is easy to say we need to stress the what and how of mathematics. But how do we do this in practice? Multiple representations of concepts? Using student work to develop understanding? Group members saw evidence during the Workshop that an in-depth exploration of a small domain of mathematical knowledge allowed us to unpack the layers of mathematics needed to fully understand (and teach!) a concept. As important as it is to discuss the core knowledge of mathematics needed to teach, we realized it was equally important for students to be aware of core conversations that arise from these experiences. Then students and instructors can lay bare their understandings of mathematics so that each group can continue to grow professionally. The challenge we face is to get students to “unpack” their knowledge and, most importantly, induce them to look to the larger mathematics education community for help.
REFERENCE
Molina, D. D., Hull, S. H., & Schielack, J. F. ( 1997). Guidelines for the mathematical preparation of prospective elementary teachers. Austin, TX: Texas Statewide Systemic Initiative for Science and Mathematics Education .

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Question #5
DISCUSSION GROUP #6
What are some promising ways to help teachers not only develop mathematical understanding but learn to use mathematical insight and knowledge in the context of practice? What are the key features of what makes an approach promising? Are there ways to engage preservice teachers in learning mathematics through the tasks they will actually do in practice?
Leader: Dale Oliver; Members: Virginia Bastable, Marilyn Hala, Marco Ramirez, Gerry Rossi, Jack Schiller, Terry Woodin
ASSUMPTIONS MADE IN FRAMING THE DISCUSSION
The phrase, “learn to use mathematical insight and knowledge in the context of practice,” was the focal point of the group discussion. What is the nature of content knowledge that makes a difference in the classroom, that allows teachers to recognize and act on teachable moments in mathematics? How can mathematics teacher educators facilitate the transition from content knowledge to the practice of teaching?
SUMMARY OF THE MAIN POINTS OF DISCUSSION
The following summary is a reflection of the general consensus that emerged in the group through the four small group meetings within the conference. The summary begins with a brief description of two qualities of mathematical knowledge that are particularly important for teachers to effectively apply that knowledge to the practice of teaching. Following the description are three assertions about promising practices in teacher preparation that support the development of these qualities.
Teachers who use mathematical insight and knowledge in the context of practice draw from a coherent view of the development of mathematical ideas in elementary school children. One aspect of a deep and profound understanding of mathematics in the elementary curriculum is the understanding of how these ideas grow over time. At a minimum, a teacher needs a coherent view of this process within the one-year curriculum of the level at which they teach. This understanding of how mathematical ideas grow over time includes knowledge of effective strategies for working with children and a sense of meaningful assessment.
Ideally, teachers have a coherent view of the development of mathematical ideas encompassing the entire K-8 mathematics curriculum. In a sense, the teacher who is most ready to apply mathematical knowledge of a content area to the practice of teaching mathematics knows the K-8 mathematics curriculum of that content

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area as a whole. While this is a lofty and perhaps unachievable goal during undergraduate teacher preparation, such a goal is not out of reach for a specific content package (place value, for example) that can then be extended to other content packages as teachers continue to extend their mathematical understanding in the context of practice.
Teachers who use mathematical insight and knowledge in the context of practice value the process of mathematical inquiry for themselves and for their students. One piece of evidence that a teacher is making a useful transition from content knowledge to the practice of teaching is when the teacher takes advantage of “teachable moments ” in the classroom. (A teachable moment is that time when a student or a group of students asks a question or makes a conjecture from their own thinking and is poised to expand their understanding of the mathematics from that thinking.) The mindset (beliefs) of a teacher toward mathematics has great influence over the use of mathematical insight and knowledge in teaching mathematics. Teachers who believe that mathematics is no more than a set of rules, a collection of techniques, or a list of vocabulary terms are less likely to see mathematical inquiry as relevant to the teaching process and, therefore, less likely to take advantage of teachable moments that fall outside of these definitions. However, if the process of inquiry (exploration, conjecture, and proof) is part of what a teacher believes is mathematically relevant and important, then such a teacher is more willing to wrestle with mathematics ideas on their own and thus better prepared to build on student insights for fostering student learning. Moreover, such a teacher will view their own development of mathematical knowledge as an ongoing process that can be achieved in various ways, including through their work in the classroom.
Consider the topic area of place value. This phrase encapsulates many mathematical ideas. A teacher who practices active inquiry into mathematics is more willing to ask questions like “What ideas are here? How are they articulated? How can these ideas be unraveled to reveal possible pathways for leading student inquiry?” This teacher is also more likely to view such questions as being important
Studying elementary mathematics from an advanced viewpoint demands intellectual intensity. Teaching elementary mathematics is an intellectually intense and challenging endeavor. To do so well requires that teachers acquire a deep understanding of elementary mathematics, in itself a substantial intellectual challenge (Ma, 1999). Hence, prospective teachers need intensive and focused opportunities in which they can acquire a deep understanding of, appreciation for, and confidence in elementary mathematics. On one hand, it is not enough for departments of mathematics to offer survey courses in elementary mathematics that do not reflect the required intensity or depth of inquiry that is required. On the other hand, departments of mathematics are urged to recognize that such intense intellectual activity focused on elementary curriculum or on elementary mathematics is valuable and worthy of undergraduate mathematics credit.
There are two important forms that mathematics department offerings can take: those that are curriculum-centered (and thus address a coherent view of the development of mathematical ideas in K-8 curriculum), and those that are foundational (and thus address the nature of mathematics as an inquiry-based discipline). In either case, the pedagogical aspects of teaching mathematical content

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cannot be ignored. In fact, pedagogic and content issues are intertwined in the pursuit of a deep understanding of elementary mathematics, and thus education departments must also guard against offering methods courses without the requisite intellectual intensity or connection to content issues.
Explicit connections between mathematics and classroom practice strengthen content and pedagogical knowledge. For the most part, teachers have a hard time unraveling packages of mathematical ideas into component parts and supporting ideas that inform their teaching. An approach that seems promising for improving a teacher's facility with mathematics is to design opportunities for learning mathematics including explicit connections to the elementary curriculum. For example, if it is important that elementary teachers learn about calculus, then explicit connections between the concepts of calculus and the content in the K-8 curriculum need to be a substantial part of the experience of learning calculus. Mathematics teacher educators are encouraged to study and articulate the connections that are most helpful for teachers.
Alternatively, any new opportunities for learning about the teaching of mathematics at the elementary level need to highlight the process of mathematical inquiry and connect to significant mathematical ideas. Mathematics teacher educators are encouraged to design experiences for prospective teachers and teachers to study and analyze “teachable moments” in the classroom (some where teachers capitalize on such moments and others where teachers miss teachable moments). There are two means for studying classroom application in this way that seem promising: the study of written cases of an episode or string of episodes in the classroom and the analysis of classroom videotapes.
The combination of connections, from mathematical ideas to curriculum and from teaching practice in the classroom to mathematical ideas, works together to establish a professional habit of mind that includes asking questions such as, “Why would I ask a student to do this mathematical task? What is the mathematical purpose of the task? What other mathematical or related areas connect to this mathematical activity?” Such questions can play an important role in the intellectual development of prospective teachers.
Teaching is an intellectual activity that requires time and an expectation of inquiry. How are teachers motivated to study the connections mentioned under the previous assertion? How can overburdened and undersupported teachers find time to look at even the next page of the mathematics text and ask themselves, “Why am I having my students do this? What is the purpose?” If teachers are to make real gains in the application of mathematical knowledge to the practice of teaching, then the culture of teaching must begin to change. The new culture must support time and the expectation that a significant use of that time will be to study mathematics and the elementary mathematics curriculum.
The process of changing the culture can start in the undergraduate preparation of teachers by including coursework in the use of resources (print and electronic materials, professional organizations and conferences, mentors, and peers) to pursue questions that arise in classroom contexts. This kind of inquiry into mathematics and the related pedagogical issues can be an integral part of being an elementary educator if the process is modeled carefully and consistently in teacher preparation programs.

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ISSUES
Throughout our discussion, the issues that surfaced most often were connected to a common belief in the broader educational community that teaching elementary mathematics is elementary. How can mathematics and education faculty be engaged in reexamining the intellectual challenges in knowing elementary mathematics deeply? How can such faculty be motivated to rethink the connection between knowing mathematics and teaching mathematics? How will mathematics departments be motivated to offer curriculum-based courses for teachers? How will teachers and the school districts in which they work be motivated to value learning more about the mathematics that they teach?
The Workshop was a good start toward establishing the intellectual intensity of mathematics teacher education: that knowing and teaching elementary mathematics is anything but elementary. There is much work left to do.

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