Children around the world pass through three levels of increasing sophistication in methods of single-digit addition and subtraction. The first level is direct modeling by counting all of the objects at each step (counting all or taking away). Students can be helped to move rapidly from this first level to counting on, in which counting begins with one addend. For example, 8 + 6 is not solved by counting from 1 to 14 (counting all), but by counting on 6 from 8: counting 8, 9, 10, 11, 12, 13, 14 while keeping track of the 6 counted on. For subtraction, Children’s Math Worlds does what is common in many countries: it helps students see subtraction as involving a mystery addend. Students then solve a subtraction problem by counting on from the known addend to the known total. Earlier we saw how Maria solved 14 - 6 by counting up from 6 to 14, raising 8 fingers while doing so to find that 6 plus 8 more is 14. Many students in the United States instead follow a learning path that moves from drawing little sticks or circles for all of the objects and crossing some out (e.g., drawing 14 sticks, crossing out 6, and counting the rest) to counting down (14, 13, 12, 11, 10, 9, 8, 7, 6). But counting down is difficult and error prone. When first or second graders are helped to move to a different learning path that solves subtraction problems by forward methods, such as counting on or adding on over 10 (see below), subtraction becomes as easy as addition. For many students, this is very empowering. The third level of single-digit addition and subtraction is exemplified by Peter in the vignette in Box 5-2. At this level, students can chunk |

The approaches in the three chapters that follow identify the central conceptual structures in several areas of mathematics. The areas of focus—whole number, rational number, and functions—were identified by Case and his colleagues as requiring major conceptual shifts. In the first, students are required to master the concept of *quantity*; in the second, the concept of *proportion* and relative number; and in the third, the concept of *dependence* in quantitative relationships. Each of these understandings requires that a supporting set of concepts and procedural abilities be put in place. The extensive research done by Griffin and Case on whole number, by Case and Moss on rational number, and by Case and Kalchman on functions provides a strong foundation for identifying the major conceptual challenges students