Parentheses

Parentheses are usually absent from early school math. Later in the curriculum they are avoided when possible and treated as though their use is bad form. “Simplify”, for instance, frequently means “reformulate without parentheses”. There are many problems with this point of view.

Parenthesis encode valuable information

• can be expanded to get a parenthesis-free form but the factorization information is lost. If it is needed, e.g. for partial fractions, division, or finding roots, it will be hard to recover.
• The right-hand side of requires significantly fewer multiplications and is faster to evaluate either by hand or by computer.

Parentheses reduce errors

• The numerical product written without parentheses is easily mistaken for the number .
• Separating different mental activities, e.g. separating arithmetic or algebra from the translation step in word problems, or from the organizational step in polynomial multiplication, can significantly help students (see Separation of Tasks). Avoiding parentheses typically requires on-the-fly arithmetic or algebra that mixes tasks.

Parentheses aid understanding and abstraction

• When compound arithmetic problems, for instance coming from word problems, are written out before any operations are performed then the connection between the problem and the expression is clearer and the step to symbolic versions is simple. These full expressions usually require parentheses.
• There is a particular problem with calculators. It is not unusual for calculator-taught students to be able to do complex multi-step arithmetic problems on a calculator but be unable to write an expression for the calculation because they lack fluency with parentheses. Such calculator skills are not a step toward abstract or symbolic work and cannot be considered evidence of understanding.

Summary

• Parentheses happen, and can be useful in many ways.
• They should be introduced early in the curriculum and students should be encouraged to use them.