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 well-designed 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 third-grade 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?