Workshop Overview: Knowing and Learning Mathematics for Teaching
The Mathematics Teacher Preparation Content Workshop, held on March 1921, 1999, at the National Academy of Sciences, was designed around two central questions:

What is the mathematical knowledge teachers need to know to teach well?

How can teachers develop the mathematical knowledge they need to teach well?
Mathematics teacher educators, mathematics education researchers, mathematicians, K12 school supervisors, and classroom teachers explored these two questions by considering actual tasks of teaching practice, such as remodeling problems, analyzing student work, or managing discussions. For the broader mathematics education community, the papers and reports collected in this proceedings are intended to inform and provoke discussion of these questions and the issues surrounding them.
The Workshop consisted of plenary sessions designed to set a framework for thinking about the questions, concurrent sessions based on specific tasks of teaching that illustrated how the questions might be enacted in the classroom, and panel sessions in which the panelists reflected on their experiences at the Workshop in the context of their own background. In addition, each participant was assigned to one often small discussion groups, with the membership of each group representative of the range of interests and professional responsibilities of the participants. Each group was assigned to respond to one of five overarching questions. The participants met with a designated leader in these small groups periodically throughout the Workshop to continue their thinking about their response to the assigned question in light of the plenary and concurrent sessions in which they had taken part.
Prior to the Workshop participants were given two tasks to do to lay the groundwork and provide a common platform from which to begin to address issues related to the content knowledge of teachers. They also read Chapter 1 and Chapter 4 from Liping Ma's (1999) Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States and considered explicit questions about each chapter.
The Workshop began on Friday evening with a reception and a general welcome from Rodger Bybee, Executive
Director of the Center for Science, Mathematics, and Engineering Education. Steering Committee chair Deborah Ball gave a welcome and a brief overview of the Workshop. Following the welcome, participants met in their small groups to discuss the preworkshop reading and to consider, for the first time, the overarching question assigned to their group. The opening session concluded with a panel consisting of Liping Ma, Mark Saul, and Genevieve Knight. Ma described what she meant by profound understanding of fundamental mathematics (PUFM). Fundamental mathematics is a foundation for later learning that is primary because it contains advanced mathematics topics in elementary form and is elementary because it is at the beginning of students ' learning. Profound understanding indicates a deep, vast, and thorough knowledge of the subject. According to Ma, teachers with PUFM are able to reveal and represent mathematical ideas in ways that are connected, that display multiple perspectives and awareness of basic ideas of mathematics, and that have longitudinal coherence.
Mark Saul addressed the question, “What is it that is essential to an elementary teacher's understanding of mathematics?” He offered a list not meant to be exhaustive but to stimulate thinking. Saul suggested there are a variety of ways to come by this knowledge, and it is our responsibility as a community to begin implementing some of these. He raised an important question of balance. People who work with teachers must themselves have a profound understanding of the mathematics they are teaching, but too much mathematics presented too quickly can result in disaster. Genevieve Knight described a 1959 text, the Fundamentals of Freshman Mathematics by Allendoerfer and Oakley, designed to give a modern treatment of topics needed to prepare students for calculus and emphasizing the authors' interpretation of essential ideas of fundamental mathematics. Knight questioned which teachers are those who must possess this fundamental knowledge and what is fundamental mathematics. Is it a list of topics, courses, experiences? She charged the group to think about how to create an explicit set of defining properties of what fundamental mathematics means and a welldesigned system to determine when a teacher or a student understands.
The first full day of the Workshop began with a presentation by Hyman Bass and Deborah Ball, focused on the task of establishing a classroom culture in which mathematics reasoning is both possible and called for. They posed the question: What lenses do elementary teachers need to see that they as teachers are involved in the development of children's capacity to construct proofs, to understand and follow mathematical proofs, to understand the need for justification, and to be able to distinguish valid justifications from invalid justifications? A videotape of Ball's thirdgrade classroom gave a concrete example of young children talking with each other about a mathematical idea and reasoning about its interpretation. Bass noted that his ongoing work analyzing the videotapes of Ball's classroom is essentially the same as the focus of the workshop: to look for the mathematics entailed in the core tasks of teaching. Because such tasks involve mathematical decisions, the focus is on discerning what that mathematics is and how to place it into a larger mathematical context. The two examples in the video were used to consider some core elements of teaching: What common knowledge do students have and how can a teacher use this common knowledge to help students understand how to justify a mathematical claim in a meaningful way?
Reacting to the video and the comments by Ball and Bass, Jim Lewis noted that establishing a classroom culture in which mathematical reasoning is called for, such as in the video, places a greater demand on the teacher than would a more traditional approach of demonstrating the process for solving a problem and having children practice. His concern with proper notation and language reinforces the challenges that the teacher faces on a daily basis, deciding when to introduce new mathematical notation and when to couch the discussion in the language of the children. Lewis suggests the mathematical knowledge required to teach in this manner is far superior to the mathematical knowledge that most students have when they are certified as ready to teach mathematics at the elementary school level.
The plenary session was followed by concurrent sessions on what mathematics teachers need to know to teach well. The participants were assigned to two of four concurrent sessions, each focused on the mathematics involved in a particular task of teaching. The session led by Erick Smith looked in depth at one classroom example and the mathematics the teacher needed to know to manage a classroom discussion around the student presentations. The participants in the session addressed the question: How do teachers make mathematical meaning as they develop an understanding of their students' mathematical thoughts; that is, what mathematical knowledge do teachers need to understand the mathematical work of their students?
Virginia Bastable led a session focused on investigating students ' mathematical thinking. The session was divided into two components: reviewing a written case study to share observations about the mathematical thinking of the students and reflecting on the experience to make more general statements about the mathematical knowledge needed by teachers to understand this mathematical thinking. Olga Torres engaged participants in remodeling a mathematical task, either to make it simpler or more difficult. In the process, the participants considered issues related to teachers' understanding of mathematical language, the underlying mathematical ideas, and ways of reasoning, and the prior knowledge of both teacher and student. The session led by Michaele Chappell centered on the teachers' mathematical knowledge, skills, and dispositions that matter in examining and analyzing student work. The basis for the discussion was student work on a seventhgrade problem dealing with proportionality.
The day ended with a panel session on the kinds of mathematical knowledge that matter in teaching. Alan Tucker observed that the typical three or six credits in a teacher preparation program seem insufficient for a preservice teacher to pick up all of the knowledge needed to do the tasks featured in the workshop. He suggested that learning to teach must be a process that is continually evolving through the life of the teacher. To make this experience successful, teachers need a foundation for their learning: critical thinking skills that enable them to reason from basic mathematics principles. Tucker concluded by observing that the level of common mathematical knowledge among the public seems to be actually much higher today than in the past and pointed out that the exciting part of being a teacher is to take advantage of and build on what people already know.
Based on the experiences of the day, Deborah Schifter added to Mark Saul's list of what is essential to an elementary teacher's understanding of mathematics.
She suggested that teachers need to be connected with the notion that mathematics makes sense, must be able to work with uncomfortable feelings to come to the place where mathematics does make sense, and must be curious about how mathematics works. She observed that teachers must learn to see the mathematics in the situations they encounter. Gladys Whitehead pointed out that as faculty in community colleges work with elementary majors, they struggle with two concerns: providing teachers with an adequate content base and adequate teaching strategies. Her question was: How do we capture our own experience and pass it on to new teachers? She closed by noting that postsecondary teachers need inservice development and that working with school systems and their teachers can help those in postsecondary revamp their programs for teachers.
The second full day of the Workshop was framed around the two questions: How might prospective elementary teachers be helped to develop the kind of mathematical knowledge they need to teach well? What are alternative and promising approaches to the mathematics education of beginning teachers? Participants were again assigned to two of four concurrent sessions. Using student curriculum materials to help teachers learn mathematics was the focus of the session offered by ShinYing Lee and Marco Ramirez. Lee focused on Japanese teachers ' manuals as a significant source of information that contributes to teachers' knowledge of mathematics. The manuals provide different aspects of mathematical knowledge and tie the content knowledge to an understanding of how students learn and the common difficulties students have. Ramirez took teachers through a lesson from Investigations in Number, Data, and Space, a relatively new U.S. elementary curriculum created by TERC. The materials focus primarily on the role of the teacher and include only a small set of student activities for each concept. The teachers ' manuals suggest questions to ask and point out problems students may encounter and various strategies students may use for a given task.
Carne Barnett discussed how studying case materials enables teachers to learn mathematics. A major goal of the session was to illustrate how a deliberately facilitated case discussion can help teachers acquire an advanced and flexible knowledge of the mathematics content they teach. A second goal was to demonstrate how an analysis of the mathematics in a classroom situation prepares teachers to make informed and strategic teaching decisions. Jill Lester, Virginia Bastable, and Deborah Schifter illustrated the development of mathematical content knowledge through a discussion of programs and practices from Developing Mathematical Ideas, a mathematics inservice program for teachers. Participants worked through a case on number and discussed how this provided the opportunity for preservice and inservice teachers to learn mathematics. Bradford Findell and Deborah Ball offered an analysis of video and its role as a delivery mechanism for helping teachers learn mathematics. The goal of the session was to consider how video can be used to provide opportunities for teachers to engage in mathematics content. The session was organized around two video clips: a 5minute teacherdirected class discussion taken from VideoCases for Mathematics Professional Development project and the beginning of a 30minute lecture prepared as part of a 51video course, Gateways to Arithmetic by Herb Gross, for prospective elementary teachers.
Participants discussed the advantages and disadvantages of each case as an opportunity to learn mathematics.
The closing panel discussed how teachers come to learn the mathematics they need to know to teach well. Richard Askey used an analysis of problems presented in textbooks to illustrate his view of important mathematics teachers should know and be able to do. He discussed the mathematics in different texts and how the need for deep knowledge of mathematics is reflected in teachers' manuals. Carol Midgett described the learning of mathematics from her own experience as a teacher, attributing much of her growth in understanding mathematics to professional activities outside of the classroom. She indicated that the Workshop was important because it provided the opportunity for people from all levels of educational practice to work together in their common struggle to deal with the complexity of teaching and what mathematical knowledge it takes to teach well. Alice Gill reemphasized that teachers need to know mathematics and to know it well. She urged the Workshop participants to keep in mind not only the mathematics but also where the teachers are and what additional supports they may need. James Lightbourne closed the panel with reflections on the importance of helping teachers come to learn mathematics in the ways described during the Workshop. He mentioned several National Science Foundation programs as other possibilities for promising ways to develop this teacher knowledge. Joan FerriniMundy, the panel moderator, suggested potential sites of practice other than those featured at the Workshop that could be vehicles for learning mathematics and offered an initial set of reasons why sites of practice are useful as ways to develop teacher knowledge.
Deborah Ball closed the Workshop by observing that it was not designed to provide answers to the many questions about teacher content knowledge but to serve as an intellectual resource for the participants to use in framing their own work. She encouraged the participants to try some activities based on the thinking of the Workshop, centered on core tasks of teaching, document the results, and share them with their colleagues. In this way, she suggested, the field can begin to move forward in a real analysis of what mathematics teachers need to teach well and how they come to learn that mathematics.