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 viewchanges 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 partwhole 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?
Portfolios
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 childrenproblems 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.