Children typically see arithmetic problems written in either a left-to-right format with the equal sign and answer blank at the end (e.g., 2 + 3 = __), or a traditional vertical format with the arithmetic fact written on top of a horizontal line. Neither of these formats highlights the interchangeable nature of the two sides of an equation. Research shows that children construct a better understanding of mathematical equivalence when they get to practice arithmetic problems presented in a variety of different problem formats.

Regardless of the problem format, arithmetic problems may activate unhelpful, operational ways of thinking about the equal sign (e.g., that it’s a signal to “calculate" or "put the answer"). For this reason, it’s beneficial to teach children the relational meaning of the equal sign outside of an arithmetic context first (e.g., 12 = 12) before moving on to including the equal sign in a variety of different problem formats.

Children encounter physical equivalence relations in everyday life, such as sharing stickers or balancing a scale. We can use these well-known concrete contexts as a springboard for helping children construct a formal understanding of mathematical equivalence. We can do this by asking children to solve problems in the familiar concrete contexts first, before fading away the concrete details, linking first to two-dimensional pictures, and then finally to the abstract, formal mathematics symbols.

We can help children achieve deep understanding of mathematical equivalence by asking them to compare and explain different problem formats (e.g., 7 = 5 + 2 and 5 + 2 = 7) and problem-solving strategies (e.g., explaining why a specific correct strategy is correct and why a specific incorrect strategy is incorrect). When we ask children to engage in comparison and explanation, we show them that we value attentive and mindful problem solving. We also provide them with an opportunity to reflect on and justify generalizations about the underlying structure of arithmetic.