# Looking at Change in Number Sense

One of the issues raised by parents is how to look at change in children's understanding of mathematics. In short, as a parent, how do I know if my child is really learning anything worthwhile about mathematics? Although there is no simple answer to this question, we can offer some guidelines for tracking change. In this newsletter, we will focus on arithmetic (addition, subtraction, multiplication, division).

### Taking a Long View

Most programs of assessment try to measure change in short periods of time, a few minutes, a lesson, a few days. CGI takes a different view-changes in children's understanding of mathematics occur gradually over a long period of time - years not days. Generally, we expect children's problem solving strategies to grow in two ways.

 (l) First, children will progress from reliance on directly modeling the action in arithmetic word problems (or other situations) to some form of counting or use of number facts.

So, over time, children who first directly modeled the action in a joining problem (like 5 + 7 = ?) by using counters, like blocks or teddy bears, will use a counting strategy, like counting on from 7. . .8(1), 9(2), 10(3), 11(4), 12(5), or a fact strategy, like 5 + 5 = 10 and 2 more is 12.

 (2) Second, children will develop more flexible strategies. They will be able to solve problems in more than one way.

For example, a child who first solves a joining problem like (7 + ? = 12) by counting on from 7 to 12 (8(1), 9(2), 10(3), 11(4), 12(5) should come to see that the problem can also be solved by counting down from 12 - 11(1), 10(2), 9(3), 8(4), 7(5). In other words, the child can see the problem both as joining objects and as separating objects.

### Using Problem Solving to Promote Understanding

CGI helps children build understanding by solving problems. Problems differ in the amount of planning ahead that they require. Problems involving joining and separating with the result unknown (7 + 8 = ? or 12 - 5 = ?) can be modeled one step a time. Multiplication problems involving groups also can be solved this way (Bart has 4 boxes of pencils. There are 6 pencils in each box.) How many pencils does Bart have altogether?). So too can some types of division problems (How many tens in 70?) These types of problems are good problems for children who rely primarily on direct modeling. With larger numbers, these problems are also good ways of helping children develop understanding of place value, especially when children invent their own ways of solving the problem.

To help children go beyond direct modeling, problems that involve some planning are helpful. For example, Join Change Unknown (12 + ? = 21) and Compare (Gregory has 11 books. James has 4 books. Gregory has how many more books than James?) problems help children think about the whole problem and about part-whole relations. Problems that involve even more planning include the join and separate problems with the start unknown (? + 7 = 12 or ? - 21 = 44). Multiplication and division problems with "derived" quantities (quantities that relate two or more things like miles per hour or quantities that do not have discrete, "touchable" units, such as pounds) are also often difficult. These types of problems usually cannot be solved by direct modelers, so children's ability to solve them (even if they use counters) is a mark of progress.

Teachers use different types of problems in the classroom to help children learn new ways of thinking about relationships among numbers. Your child's portfolio contains a sample of the types of problems that your child has solved and how your child has solved them.