What does the equals sign really mean?
Every number sentence with an equals sign (for example, 2 + 3 = 5) is saying that the amount on the left is the same as the amount on the right. The language in these games helps learners by modeling this concept over and over.
To succeed in mathematics, children need to learn the correct meaning
In the sentence “Two plus three is the same as five”, or the sentence “Two plus three is the same amount as five”, the wording appropriately causes 2 + 3 and 5 to sound like equivalent amounts. Hearing sentences like this helps children understand that the equals sign means that whatever is on the left side is the same amount as whatever is on the right side. The children who grasp this concept are the ones who go on to become high achievers in mathematics.
Children who do not grasp this concept are at an enormous disadvantage in a mathematics classroom. The idea that the equals sign means equivalence is essential for success in mathematics. Without this knowledge, children who want to produce correct answers have no alternative other than executing procedures without a full understanding of what they are doing.
A potential pitfall: Developing wrong ideas about the equals sign
After enough experiences doing a certain procedure every time they see an equals sign, children may reach the incorrect conclusion that the equals sign is a signal to do a procedure (Ginsburg, 1977, p. 85). While playing the game “Count by twos from an arbitrary even number”, one learner gave a clear demonstration of confusion in this area. He drew a 12 card, so he was supposed to say “12” and then count up from there to 20 by twos: “14, 16, 18, 20.” Here is what he actually said: “12 equals 14, 16, 18, 20.”
Of course, no adult had ever told this child that 12 equals 14. He used the word that way because he was working under the incorrect assumption that the equals sign is a “do something signal” (Behr, 1980).
This child’s assumption, although wrong, was understandable. Most children come into contact with the equals sign long before it is formally presented in their school curriculum. So it is very easy for children to develop mathematically wrong ideas about the symbol they are seeing. Ideally, we can step in and help them learn the right ideas.
How you can help
The games in the Mathematical Symbols section teach explicitly that the equals sign means equivalence. That concept is reinforced throughout the games.
Another good way to help learners focus on the meaning of the equals sign is to help them compare the equals sign with the inequality sign, ≠ (also called the “not-equals” sign). The game “Inequality sign” can be used to introduce the not-equals sign.
Once learners know what the not-equals sign means, you might enjoy using these ways of discussing how to use it:
- You can write a false number sentence that has an equals sign, such as 3 = 4 (“three is the same as four”), and say “This number sentence is false, because three isn’t the same as four. How could we change it to make it true?” One way is to turn = into ≠ by adding a slash, so that the sentence becomes 3 ≠ 4. Another way is to add 1, turning the sentence into 1 + 3 = 4. A third way to make this sentence true is to erase the 4 and substitute another 3, turning the sentence into 3 = 3. Some learners will think the sentence 3 = 3 looks strange, because it has an equals sign but no plus sign. However, it just means “three is the same as three”, and this statement is perfectly true. The statement “3 = 3” is not something that people usually take the time to say, because it is not very interesting. But it is still true, and it is OK to say it. The activity “Is this number sentence true or false?” helps learners talk about these ideas.
- You can write several number sentences and ask whether each one is true or false. For example, you can write 2 + 2 ≠ 4 and ask “Is this true or false?” It is false, because two plus two is the same as four. The activity “Is this number sentence true or false? You decide” asks learners to decide whether different number sentences are true or false.
- Ask learners to create their own true and false number sentences, and to show how the false sentences could be made true.
Troubleshooting
You may have the concern that seeing an instructor write down something false — even if the instructor clearly says what is happening — could confuse or distress learners.
However, children old enough to be learning the meaning of the equals sign already know that people sometimes make false statements. Generally, children find purposefully written false number sentences to be hilariously funny, not confusing. If you do find that a learner seems distressed, you can always rewrite the number sentence so that it is true. You can tell that learner “You’re right, true number sentences are the ones we want to use in mathematics. That was an example to show you what a false number sentence might look like, because false number sentences are the kind we want to avoid.”
Some learners may object to the idea that “equals” means “is the same as”. These learners are probably thinking along these lines: A group of five all in one place is not the same thing as a group of two in one place and a group of three somewhere far away.
This point of view has validity, and these learners will appreciate it if you give them credit for making a good observation. You can explain that “is the same as” is a shorter way to say “is the same amount as”. The five in that group of five truly is the same amount as the two in the group of two plus the three in the group of three.
In fact, you could say “is the same amount as” any time you talk about an equals sign. It takes time to say all the words in this phrase, but if it helps your learners understand the concept, it is worth it.
Next section: Children can learn mathematics by doing something fun
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References
Behr, Merlin, Stanley Erlwanger, and Eugene Nichols. 1980. “How Children View the Equals Sign.” Mathematics Teaching 92: 13-16.
Ginsburg, Herbert P. 1977. Children’s Arithmetic: The Learning Process. Oxford: D. Van Nostrand.
August 17, 2020