Teaching math in the elementary grades is justified by practical applicationseveryday mathematics. But math is far more important. It is a way of thinking that is different, beautiful, and profound. Unfortunately, many children go through school mathematics without a glimmer of this way of math. The key to the way of math is insistence on the why? in every step. But arithmeticthe main subject matter of elementary mathrequires non-trivial steps. It is unreasonable to expect young children to grasp these by means of abstract reasoning. But physical models, or manipulatives, can help. We argue that if one wishes to create a manipulative that would make arithmetic intuitively obvious to children, then one is driven to a particular physical model of the base-10 number system. We describe this model, and the dramatic impact that it has had on children. BACKGROUND INFORMATION: Elon Kohlberg is on the faculty at Harvard Business School, but he also works with elementary school teachers on how to teach arithmetic. He has designed manipulatives to help children understand why arithmetic works the way it does. For example, they make clear what place value means and what it means to carry. These manipulatives represent arithmetical operations more faithfully and elegantly that those traditionally used, and have been used successfully in the classroom. Thus his ideas may be of interest to mathematicians, mathematics educators, and teachers.