All things are not created equal…including the understanding of the equals sign! Did you know that Americans have a very poor understanding of the equals sign, especially when compared to students in other countries?
“Ninety-eight percent of the Chinese sample correctly answered 4 items indicating conceptions of equality and provided conceptually accurate explanations. In contrast, only 28% of the U.S. sample performed at this level.” (Li, Ding, Capraro, & Capraro 2008)
Students on our continent tend to see the equals sign as meaning ‘the answer to’. In other words, the answer to 2+3 is 5.
What’s wrong with that you ask? The true meaning of the equals sign is as a symbol that indicates equivalence; it does not mean ‘an answer’. It means that one side is equivalent to the other. The equals sign acts as a balance in that the two sides must balance; it represents sameness as in the same amount, but not necessarily using the same things.
Still don’t see the difference? You’re not alone! Most of our students don’t understand the difference either. When students think that the equals sign means ‘the answer’, this is what happens:
Want another example? How many of you have written a problem like this on the black-board: 10 = 6+4 and had your kids tell you “that’s wrong!” because you wrote it backwards? If so…you have some work to do!
But don’t fret! You can quickly transform your students’ thinking by incorporating a few new habits into your routine.
Rory quickly and easily conceptualized the true meaning of the equals sign using this home-made balance scale. I was surprised at how accurately it worked too! It was also a great model to demonstrate equivalence such as: 4+4 is the same as 5+3; and Rory was very able to prove their equivalence to me, by rearranging the towers of 4 +4 to look like 5+3, with unifix cubes. In later lessons, I would use the balance scale to ensure concrete understanding of the other relationship symbols, ‘less than’ and ‘greater than’, as well.
Here’s what you can do:
- Most importantly: Use your words carefully! When reading equations, reinforce the idea of equivalence by reading the equal sign as ‘is the same as’ or ‘is equivalent to’. Don’t ask for the answer to a number sentence, ask for what it is equivalent to.
- Write number sentences backwards and forwards. In other words, alternate which side has the operation to be performed. For example, write 9 = 5+4; don’t always show just 5+4=9.
- Give questions that ask children to find equivalent expressions, not just questions with one number answers. For example, 5 + 4 = ? + 1 instead of 5+4 = ?.
- Reinforce the commutative property: 3 + 2 = 2 + 3.
- Cuisinairre rods! These are great for finding and showing equivalent representations of expressions.
- Use a balance scale! The more concrete and real you can make the understanding, the better. And not just for early grades, upper levels appreciate the visual as well.
- Give experiences with true AND false number sentences.
- Watch for textbooks and worksheets that don’t promote this way of thinking. The study mentioned earlier, blames a lot of the misinterpretation of the equals sign on North American textbooks.
Building the understanding of the equals sign, as a relationship symbol, starts in kindergarten and therefore the earlier you can promote the proper understanding of it, the better. The equal sign is the primary symbol used to understand relationships in our number system. Understanding its meaning promotes algebraic reasoning and gives students access to powerful relationships for working with numbers. With a thorough comprehension of the equal sign, other representations, such as the symbols for ‘less than’ and ‘greater than’, make more sense. And that’s our goal! To make more sense!