 Children solve addition and subtraction problems in many ways. As adults we are likely to think first about the operation (+ or -) used to solve a problem. But children think mainly about two things. First they think about the action or relationship in the problem and then they think about what is unknown. In the next few weeks we will be talking about different kinds of problems. We will talk about join, separate, and compare problems.

#### Action in word problems

One type of action is joining -putting things together. Depending upon what is unknown, there are three different types of joining problems. This week we will be covering the first kind, join-result-unknown. In such problems a number of items are joined by a number of other items. The child must figure out the unknown, i.e. how many items are the result of this joining. For example, Byron had 7 shells. His friend Melissa gave him 9 more shells. How many shells does Byron have now?

#### Strategies for Solving Join-Result Unknown Problems

Adults are likely to solve addition or subtraction problems by recalling facts. Children solve problem in many ways, called strategies, before they get to the point where they recall the facts. It is important that children have the opportunity to solve problems in many ways. Solving problems in more than one way helps children learn more about numbers. Beginning math instruction by teaching children their "facts" can short circuit this development. Mathematics is more than just getting answers.

## Types of Joining Problems

 Result Unknown Change Unknown Start Unknown Byron has 7 shells.Then Melissa gavehim 9 more shells.How many shellsdoes Byron havenow?

The best way to learn how children figure out problems is to ask them. Knowing about children's typical growth in strategy use helps adults understand what to expect from children. There are three main ways that children solve joining problems:physical (direct) modeling, counting, or using facts.

1. Physical modeling: The first strategy children develop is to use objects or their fingers to put the two groups together. Then they count them all starting with one. For example, to find out how many shells Byron has children put 7 shells (or use pennies or some other counters) and the 9 shells together and count them all "1,2,3,4,5...16." 2. Counting strategies: At other times children will use a counting strategy. Counting strategies use numbers to represent quantities. Usually children begin to use counting strategies to solve join-result-unknown problems by starting with the number that came first in the problem and then they will count on. For example, because they know Byron had 7 shells, they would count, "7 (pause) 8, 9, 10, 11, 12, 13, 14, 15, 16." This is called counting on. This is more mentally demanding than physical modeling because children must keep in mind that 7 represents 7 shells, and that 8 represents the first shell Melissa gave him, 9 represents the second shell Melissa gave him and so on up to 16. A more advanced counting strategy is counting on from the larger number in the problem. In this case, the child would start with 9 (the largest number in the problem) and count on "10, 11 ...16". This is a major development in children's understanding of number because at first children do not realize that 7 and 9 more are the same amount as 9 and 7 more. While this may not make much of a difference for a problem like this one, it does make a difference for problems like: Byron had 2 candies and Melissa gave him 8 more. It's a lot easier to count on from 8 than from 2.

3. Using facts: Children also learn to use facts to solve join-result-unknown problems. Children memorize certain facts more quickly than others. Often they learn the "doubles" first. They will use this knowledge to figure out the answers. This is called deriving facts. Using the shell problem as an example, a child who knows that 7+7 is 14 may reason that 9 is 2 more than 7 so the answer must be 2 more than 14, which is 16. This strategy is based on understanding relations between numbers; and so it's a significant growth in understanding numbers. Finally, chidren do remember their number facts and can generate answers by recalling the fact, 7+9=16. When this recall is based on children's experiences with number and with their emerging development in problem solving, then children understand where their answers come from. It is more important that students understand how they got an answer than how quickly they find the answer. 