subtraction problems by a forward method that finds the unknown addend, thus avoiding the difficult and error-prone counting down methods (e.g., Baroody, 1984; Fuson, 1984, 1986b). So 14 − 8 = ? can be solved as 8 + ? = 14, and students can just count on from 8 up to 14 to find that 8 plus 6 more is 14.

Some first graders will also move on to Level 3 derived fact solution methods (see Box 5-11) such as doubles plus or minus one and the general method that works for all teen totals: the make-a-ten methods taught in East Asia (see Chapter 4 and, e.g., Geary et al., 1993; Murata, 2004). These make-a-ten methods are particularly useful in multidigit addition and subtraction, in which one decomposes a teen number into a ten to give to the next column while the leftover ones remain in their column. More children will be able to learn make-a-ten methods if they have learned the prerequisites for them in kindergarten or even in Grade 1.

The comparison situations compare a large quantity to a smaller quantity to find the difference. These are complex situations that are usually not solvable until Grade 1. The third quantity, the difference, is not physically present in the situation, and children must come to see the differences as the extra leftovers in the bigger quantity or the amount the smaller quantity needs to gain in order to be the same as the bigger quantity. The language involved in comparison situations is challenging, because English gives two kinds of information in the same sentence. Consider, for example, the sentence Emily has five more than Tommy. This says both that Emily has more than Tommy and that she has five more. Many children do not initially hear the five. They will need help and practice identifying and using the two kinds of information in this kind of sentence (see the research reviewed in Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b; Fuson, Carroll, and Landis, 1996).

Learning to mathematize and model addition and subtraction situations with objects, fingers, and drawings is the foundation for algebraic problem solving. More difficult versions of the problem situations can be given from Grade 1 on. For example, the start or change number can be the unknown in change plus problems: Joey drew 5 houses and then he drew some more. Now he has 9 houses. How many more houses did he draw? Children naturally model the situation and then reflect on their model (with objects, fingers, or a drawing) to solve it (see research summarized in Clements and Sarama, 2007; Fuson, 1992a, 1992b). From Grade 2 on they can also learn to represent the situation with a situation equation (e.g., 5 + ? = 9 as in the example above, or ? + 4 for an unknown start number) and then reflect on that to solve it. This process of mathematizing (including representing the situation) and then solving the situation representation is algebraic problem solving.