## External Consultants

**Deborah Loewenberg Ball** is Dean of the School of Education and Professor of Education at the University of Michigan. Ball is internationally known for her research on elementary mathematics instruction; she was a member of the President's National Mathematics Advisory Panel. Prof. Ball is currently studying the nature of the mathematical knowledge needed for teaching elementary mathematics.

**Liping Ma** is a deeply original scholar whose ideas have shifted paradigns in the field of mathematics education. Her 1999 book Knowing and Teaching Elementary Mathematics and her concept of "profound understanding of elementary mathematics" have had a major impact on the direction of mathematics education in the U.S.

**Akihiko Takahashi** is Associate Professor of Education at DePaul University. He taught elementary school in Japan for more than ten years and served as a designer of the Japanese national curriculum, and later received his PhD in mathematics education from the University of Illinois. He has worked extensively on elementary curricula development and on teacher preparation.

**Hung-Hsi Wu** is Professor of Mathematics at the University of California at Berkeley. He has written numerous articles and gives frequent invited talks on curricular issues in K-12 education and on teacher preparation and professional development. He runs summer workshops for teachers and was a member of the President's National Mathematics Advisory Panel.