ROBERT SPEISER is a mathematician and a mathematics educator. His research in both fields has gained international recognition. He is Editor of the Journal of Mathematical Behavior, and the chair-elect of the North American Chapter of the International Group for Psychology of Mathematics Education. Speiser's research in mathematics education centers on the development of understanding.CHUCK WALTER'S academic career has focused on doing mathematics and learning to so mathematics at all levels. His research interests in mathematics education center on the growth and development of understanding, especially in school classrooms. He is assistant editor of the Journal of Mathematical Behavior.

• Preface Introduction Frames for counting, numbers and arithmetic Five women build a number system Analysis Discussion Appendix A: Mathematics 305, Fall 1995 Appendix B: Mathematics 305, Fall 1997 Notes References

ROBERT SPEISER is a mathematician and a mathematics educator. His research in both fields has gained international recognition. He is Editor of the Journal of Mathematical Behavior, and the chair-elect of the North American Chapter of the International Group for Psychology of Mathematics Education. Speiser's research in mathematics education centers on the development of understanding.CHUCK WALTER'S academic career has focused on doing mathematics and learning to so mathematics at all levels. His research interests in mathematics education center on the growth and development of understanding, especially in school classrooms. He is assistant editor of the Journal of Mathematical Behavior.

• Preface Introduction Frames for counting, numbers and arithmetic Five women build a number system Analysis Discussion Appendix A: Mathematics 305, Fall 1995 Appendix B: Mathematics 305, Fall 1997 Notes References

ROBERT SPEISER is a mathematician and a mathematics educator. His research in both fields has gained international recognition. He is Editor of the Journal of Mathematical Behavior, and the chair-elect of the North American Chapter of the International Group for Psychology of Mathematics Education. Speiser's research in mathematics education centers on the development of understanding.CHUCK WALTER'S academic career has focused on doing mathematics and learning to so mathematics at all levels. His research interests in mathematics education center on the growth and development of understanding, especially in school classrooms. He is assistant editor of the Journal of Mathematical Behavior.

• Preface Introduction Frames for counting, numbers and arithmetic Five women build a number system Analysis Discussion Appendix A: Mathematics 305, Fall 1995 Appendix B: Mathematics 305, Fall 1997 Notes References