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How Children Solve Addition and Subtraction Problems

Children do not usually begin solving addition and subtraction problems by memorizing equations. Instead, they often start with direct modeling, which means they represent the objects and actions in the story. They might use counters, fingers, drawings, or other objects to show what is happening step by step.

In direct modeling, children follow the structure of the story, not just the operation an adult might use to solve it. For example, if a problem describes objects being joined together, a child may physically build both groups and then count them all. If a problem describes objects being separated, the child may start with the full group and remove some objects. Strategies such as joining all, joining to, separating from, separating to, matching, and trial and error show how children make sense of the action or relationship in the problem.

As children’s number sense develops, they often move from direct modeling to more efficient counting strategies. These strategies still connect to the story, but the child no longer needs to model every object. For example, a child might count on from one number, count on from the larger number, count up to find a missing amount, or count down to subtract. These strategies are important stepping stones between acting out a problem and using more formal arithmetic.

As future teachers, it is important to recognize these strategies as meaningful mathematical thinking. A child who uses counters or counts on fingers is not “just guessing” or “doing it the baby way.” They are building connections between quantities, actions, counting, and operations.

 

Student Learning Goals

By the end of this topic, students should be able to:

  • Describe direct modeling and explain why it is important.

  • Identify common direct modeling strategies for addition and subtraction problems.

  • Identify counting strategies children use as they become more efficient.

  • Explain how a child’s strategy connects to the structure of a word problem.

  • Compare direct modeling strategies and counting strategies.

  • Recognize children’s developing mathematical thinking instead of focusing only on speed or memorized facts.

Key Vocabulary

  • Direct Modeling - Representing the objects and actions in a story problem.

  • Joining All - Making both groups and counting everything together.

  • Joining To - Starting with one group and adding more to it.

  • Separating From - Starting with the whole group and taking some away.

  • Separating To - Removing objects until a certain amount remains.

  • Matching - Pairing objects from two groups to compare them.

  • Trial and Error - Trying possible amounts until one fits the problem.

  • Count On - Starting from a known number and counting forward.

  • Count On From Larger - Starting with the larger number and counting forward by the smaller number.

  • Count Up - Counting forward from one number to another to find the difference.

  • Count Down - Counting backward to subtract.

  • Count Down To - Counting backward until reaching a target number.

Common Student Misunderstandings

  • If a student is thinking direct modeling is not “real math,” then the student may not yet recognize modeling as meaningful problem-solving.

  • If a student is assuming they should immediately write equations, then the student may be overlooking the developmental role of concrete strategies.

  • If a student thinks counting on fingers is always a weakness, then the student may need to distinguish between early counting strategies and inefficient overreliance.

  • If a student treats all subtraction strategies as the same then the student may need to compare count down, count up, and separate-from strategies.

Example Strategies

  • Joining All - Builds both groups, then counts all objects.

  • Joining To - Starts with one group and adds more.

  • Separating From - Starts with the whole and removes some.

  • Separating To - Removes objects until a target amount is left.

  • Matching - Lines up two groups to compare.

  • Count On - Starts from one number and counts forward.

  • Count On From Larger - Starts from the larger addend.

  • Count Up - Counts from the smaller number to the larger number.

  • Count Down - Counts backward by the amount being subtracted.

  • Count Down To - Counts backward until reaching a target.

Teacher Connection

When teachers understand children’s strategies, they can respond more thoughtfully. Instead of simply marking an answer right or wrong, a teacher can ask:

  • How did you think about the problem?

  • What did you count first?

  • What did each counter or finger represent?

  • Could you show your strategy another way?

  • Is there a way to solve it with less counting?

These questions help children explain their reasoning and gradually move toward more efficient strategies.

Quick Reflection Question

Why is it useful for a teacher to know whether a child solved a problem by joining all, counting on, or counting up? What might each strategy tell the teacher about the child’s understanding?