School District of Holmen Title I Math
Volume 3
Addition and Subtraction Problems
Part One
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-resultunknown. 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 gave
him 9 more shells.
How many shells
does Byron have
now?
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.
School District of Holmen Title I Math
Volume 3
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."
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 joinresult-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.
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, children 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.
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