All Equals Are Not Equal

Understanding key mathematical concepts early on helps learners make smoother transitions from basic arithmetic to more advanced thinking later in their educational journey. One concept that research has shown to be surprisingly tricky — yet important — is the meaning of the equal sign.

At first glance, the equal sign may seem simple: it appears in equations like 3 + 4 = 7, and students often learn “do the operation, then write the answer.” But research spanning decades shows that many learners interpret the equal sign as an instruction to calculate or find an answer (“do something”), rather than as a symbol indicating a relationship between two expressions.

This misconception — thinking of “=” as a prompt to compute rather than a marker of equivalence — can persist beyond early arithmetic and make it more difficult to reason about expressions and relationships in later mathematical contexts.

For example:

  • In studies dating back to the 1980s and continuing through more recent work, researchers have documented that a large proportion of learners interpret the equal sign operationally (as “compute the answer”), even when presented with non-standard equations like 8 + 4 = __ + 5.
  • This operational interpretation can lead to incorrect answers or avoidance of forms of equations that don’t fit a simple “operations → result” format.
  • A relational understanding of the equal sign — recognizing it means “the same as” or “balanced with” — has been linked to better performance on tasks that require strategic thinking about mathematical structure.

In other words, grasping the equal sign as a statement of equivalence rather than just a cue for calculation appears to be a key cognitive step that supports later mathematical reasoning.

More recent research further refines this idea. Some studies suggest that there may be multiple facets to a deep understanding of equivalence, including the ability to think about expressions as interchangeable or substitutable when they are equal — a subtle but powerful idea in mathematical reasoning.