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Student worksheets in the early grades, meant to build both math skills and the ability to communicate mathematically, often lead to a specific misconception.  Here is an example:

3 + 5 = ___

This type of question accurately shows that we are looking for the sum of 3 and 5.  The misconception that develops, however, is about this symbol: =.  From completing problems like this one, students from a young age see "=" as meaning "the answer is coming next."  But the equals symbol has a very different meaning than this: it means "the two sides are the same."  This is not a small difference in meaning.

I see the misconception show up most clearly in something I think of as a math run-on sentence.  A student working through a string of calculations will often write something like this:

2 + 10 = 12 + 15 = 27

Read it the way a mathematician would, and the notation is actually making a false claim: that 2 + 10 is the same as 12 + 15 — in other words, that 12 is the same as 27.  In our base 10 system, we know that 12 ≠ 27.  This student's arithmetic is correct; the mistake is in the punctuation. They've used the equals sign to mean "and then I did the next thing."  But in mathematics the symbol means something very specific, and this isn't it.  Whatever sits on the left side of an equals sign needs to have the exact same value as whatever sits on its right. This is the only definition of equal.

While worksheets and the way we teach kids equation notation contribute to this misconception, calculators reinforce it: you type in your equation and then press "=", the answer appears.  For years, nearly everything a child sees tells them that = is what produces an answer, not a claim about balance.

For plain arithmetic, that shortcut mostly works, which is exactly why it survives so long.  The trouble arrives in algebra.  An equation like x + 5 = 12 is not asking a student to "do" something to the left side and record an answer — there is nothing to compute.  It is a statement of balance: some number, plus five, has the same value as twelve.  Every technique for solving equations, from keeping both sides balanced to substitution, depends on reading = as "is the same as."  A child who still thinks of it as "the answer comes next" hasn't just picked up a bad habit, they're missing a foundational part of algebra.

The good news is that the fix is simple, and it can start long before algebra. The most powerful thing we can do is treat the equals sign like a balance scale — the old-fashioned kind with two pans that have to hold the same weight.  At home, you can play with this using nothing but conversation.  Try asking your child whether a statement is true or false: Is 7 = 7? (Yes.)  Is 6 + 2 = 5 + 3? (Yes — both sides are 8, even though they look different.)  Is 4 + 3 = 8? (No.)  You can also flip the familiar format around: instead of 3 + 5 = ___, ask ___ = 3 + 5, or 8 = 5 + ___.  These take two seconds, but they send a clear message: the equals sign is about sameness, and it reads in both directions.

And if you happen to spot one of those run-on sentences on your child's homework — 2 + 10 = 12 + 15 = 27 — you don't need to treat it as a careless slip. It's a window into how they're thinking. A gentle "I know what you mean, but let's check whether each equals sign is actually true" turns it into a conversation about what the symbol really says.  Small as it is, that little sign carries one of the biggest ideas in mathematics: that two things, written differently, can be exactly the same.