The Meaning of the Equals Sign - More About

An understanding of the meaning of the equals sign is essential if students are to grasp algebraic equations. The language of arithmetic focuses on answers whereas the language of algebra focuses on relationships. While they are learning arithmetic, students come to interpret the equals sign as meaning 'work out the answer'. For example, when they see the questions below, they immediately write the answers after =.

5 + 11 =
16 - 8 =
26 + 10 =
3 × (14 - 5) =

They interpret the left-hand side as the question and the right-hand side as the answer. This interpretation can be a major obstacle to algebraic thinking in later years.

Classroom instances

This interpretation of the = sign as an instruction to work out a calculation leads to misuses that are frequently observed in the classroom. This is observed around the world. Kieran (1992) gave the following example from Canada. In response to this problem

Jay had $2. When he visited his grandmother she gave him another $5. He then spent $3. How much did Jay have now?

it is common to see students write

2 + 5 = 7 – 3 = 4

By this, they mean that 2 + 5 = 7 and then from this answer they subtracted 3 to get the final answer of 4. Again, students are treating the equals sign as an indication of the result of a calculation. They do not see that they are writing false statements; in this case, saying that 2 + 5 = 4.

Students should set out their working for multi-step problems in a logical sequence of number sentences where the equals sign shows that the number expressions on each side are equal. For example,

2 + 5 = 7
7 – 3 = 4

Students also tend to write down a calculation as they would perform it on their calculator. For example, if asked to calculate 3 × (14 + 36), students frequently write 14 + 36 = 50 × 3 = 150.

This is a false mathematical statement, because it implies that 14 + 36 = 150.

Misuse of the equals sign should not be accepted because of the danger to subsequent work on algebraic equations. 'Getting the right answer' is not enough!

Causes

Students misuse the equals sign for two main reasons.

• Firstly, it is common school practice to use it as an indicator of where to put the answer in early arithmetic lessons.
• Secondly, many calculators use a button labelled = in this way. Some calculators now helpfully label the "work it out now" button as EXE (short for execute).

Students will also see = used in signs and slogans to indicate a loose association (e.g. CONFIDENCE = SUCCESS), rather than precise meaning of equals. If students use it this way for fun (e.g. on a poster) make sure that they understand what is wrong with this in the mathematical sense.

On a spreadsheet, = is used in a special way to indicate a formula.

Students will inevitably meet these uses of = in the environment. This is partially helpful and partially unhelpful for learning mathematics; in either case it is unavoidable. Regular discussions about the meaning of equals provide a good opportunity to reinforce the correct 'is equal to'.

 

References

Kieran, C. (1992). The Learning and Teaching of School Algebra, In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390-419). New York: Macmillan.

MacGregor, M., & Stacey, K. (1999). A flying start to algebra. Teaching Children Mathematics. 6(2), 78-85.

Stacey, K and MacGregor, M (1997) Building Foundations for Algebra, Mathematics Teaching in the Middle School. 2 (4, February).