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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.

What About Number Facts

Children need to invent their own ways of solving problems (see the newsletter on place value). Over time, they will develop recall of number relationships, first about doubles (4 + 4 = 8), or counting by 2's or 5's, and then by recalling number facts that are convenient for solving the problems they experience. Although number facts are often thought of as the building blocks of arithmetic, CGI takes the opposite approach -- by deriving number facts as they solve problems, children develop an understanding of number. Parents can help their children see relationships and patterns, so that number facts aren't merely recalled. The best way to learn about number facts is by solving problems. However, once your child knows some facts, you can help your child see relationships. For instance, instead of just using flash cards to learn that 6 + 7 is 13, you can ask: how many different ways can we make 13? Are there more, less, or the same number of ways of making 14? Instead of memorizing that 6 x 7 = 42, you can ask if 6 x 7 is the same as 7 x 6? Why? Instead of simply remembering the 3 "times" table, you can ask about the pattern of differences (3 x 1 = 3, 3 x 2 = 6, 3 x 3 = ? What's the relationship?). For more elaborate thinking, you might want to ponder with your child patterns like: 2 x 9 = 18, 3 x 9 = 27 (Notice that 1 + 8 = 9 and 2 + 7 = 9?) or what happens when a number is multiplied by itself? What's the pattern?


A portfolio is a collection of your child's work in mathematics. Looking at the portfolio will help you determine the types of word problems that your child has solved: joining and separating problems, compare problems, multiplication and division problems. Recall that more difficult problems are those that cannot be easily modeled by children-problems involving unknown starting values (? + 7 = 23, ? - 6 = 13, etc.), or multiplication/division problems involving rates (Ellen walked 15 miles. It took her 5 hours. If she walked at the same speed the whole way, how far did she walk in one hour?) or some other derived relationship (A hamster weighs 3 times as much as a gerbil. The gerbil weighs 9 ounces. How much does the hamster weigh?). Your child's teacher can help clarify some of the important differences among the problems.

Your child's solutions to the problems are written down so that you can see how he or she solved each problem. As you look at your child's portfolio, it is important to notice the type of solution strategy and the number of different solution strategies (Does your child use direct modeling to solve every type of problem, or is there evidence of using some form of counting or recall strategies for easier problems?), and the way that your child communicates his or her ideas.