Multiple Meanings of the = & - Signs

In the previous chapter we saw that symbols are also part of the language of mathematics. Symbols can also have different interpretations depending on the intent with which they are used.

Two of the most common symbols used in primary school, and which have multiple meanings, are the equal sign “=” and the minus sign “-“.

The Equal Sign “=”

The equal sign “=” is use in 3 different interpretations.

Operational Interpretation of =

A common example is the “=” sign called “equals as in 4 + 9 =?

Here the = sign is interpreted as “what is the result of this operation?” or “what do 4 & 9 make when added”.

This can be called the operational interpretation of =.

In the primary classes, = sign is used almost exclusively in this sense.

So much so that students never realise that there is another sense in which the = sign is used, which is called the relational sense.

Relational Interpretation of =

Take this problem 3 + 4 =? + 5

Many students get confused with such problems as they cannot interpret it in the normal way.

Here = has been used in the sense “Is the same as”. This is a broader interpretation of =.

Once this interpretation has been understood, the solution to the problem is clear.

With this interpretation the following variations also start making sense

5 = 5

5 + a = a + 5

If a = b and b = c then a = c

Here the focus is on the equivalence relation.

The 2nd variation is particularly useful in simplifying algebraic equations.

The 3rd variation is a statement in logic.

Substitutive Interpretation of =

It can also lead to the understanding that if x=3, then wherever x occurs, it can be substituted by 3.

Relational & substitutive interpretations of = are very important while learning algebra and coding languages.

The Minus “-“Sign

The minus sign is used in 2 interpretations.

Operational Interpretation of “-“

When we say 9 – 5, we mean that a quantity of 5 has to be taken away/ separated from 9. Here it represents the subtraction operation.

This is the most common interpretation.

Qualitative Interpretation of “-“

This interpretation is introduced in the topic of integers. Here -2 is called “negative 2”.

The “-“ sign qualifies the number 2. We can think of it like an adjective or an adverb.

It could mean a movement of 2 units in a direction opposite to what was indicated by +2.

For example, if climbing up 5 steps on the stairs can be called +5, then -5 would mean climbing down 5 steps.

A lot of confusion is caused among students as the two different interpretations of “-“ are not clarified to them.

Hence in primary school, though the focus is predominantly on the operational interpretation, students should also be introduced to the other interpretations through simple examples like one given above.