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An issue I see with students a lot is abuse of the equals sign. For example, one problem asked "what is the degree of the polynomial: $\text{polynomial}$?", and I got answers like "$\text{polynomial}=3$". I tried to explain that no, that polynomial is NOT equal to 3, the DEGREE is equal to 3. They had a difficult time understanding what was wrong.

Another example, same student. The problem was something like "evaluate $5\times 2+5\times 7$". The student keenly asked "since there's a $5$ in both, can we divide by $5$?". The question of course is, "divide WHAT by $5$?". I could have said "we can let $x=\text{blah}$ etc." but that would be too confusing. I just said "since we don't have an equation, we can't divide, but what we can do is factor out a $5$ to get $5\times(2+7)$". They were satisfied with this.

I know it sounds pedantic, but conceptually equality is perhaps the most important concept in algebra to understand. Not to mention this can lead to real mistakes. For example I saw on a question like "expand $x(x+1)(x-1)$" the following answer:

$$x^2+x-x-1$$ $$=x^2-1$$ $$=x(x^2-1)$$ $$=x^3-x$$

They happened to get the right answer, but in doing things like that it's really easy to make mistakes. I tried to explain that lines 2 and 3 were clearly not equal, and again it just seemed so lost on them.

Why is this so difficult for students, and how do I combat it? What are they hung up on?

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Related: See my answer at http://matheducators.stackexchange.com/questions/1040/how-to-cure-students-from-the-idea-that-root-and-squaring-are-identity-operators/1058#1058, and the comments on that answer, for a further discussion of the "operator" conception of equality.

– mweiss – 2015-04-27T16:54:49.2801Regarding the last example - simply exchange x with some value (say, 5) and show them, that 24 does not equal to 120 :) – Spook – 2015-04-28T10:23:40.410

2My high-school math teacher used to explain equality with a pair of scales. Things are only equals if both sides "weigh" the same. You can only use an equality sign if the scales would be balanced. When classmates made the mistake your student makes, he would force them to draw a pair of scales and put the values on there, then asked them if their drawing made any sense. It never did :) – Kevin – 2015-04-28T11:18:15.720

3To complicate things, in computer science it's perfectly okay to say $x=x+1$. There "equals" means "assign the value of". – Aeryk – 2015-04-29T13:51:05.850

1I have tried to combat this with my elementary students who when rounding 28 to the nearest 10, will write 29=30. I point out that they are not equal and suggest using an arrow 29 => 30 to show that they are going to the next step. – Amy B – 2015-07-16T00:38:49.303

If I am doing a problem in that way (which would only be in one-on-one work, never with a whole class) I would either omit the sign entirely, or use an arrow instead. – Wildcard – 2015-10-27T05:59:49.637

3On the "how do I combat it" question: you have to grade on it. Every time an equals sign gets misused on a test, points come off. Every time a student writes a falsehood (that things are equal when they're not), points come off. State this expectation clearly up front. I think that proper writing of the language is

themost valuable thing we can share. If they don't get feedback with points on the line, they'll never attend to it (and some still won't). – Daniel R. Collins – 2015-11-29T19:49:16.450Also, I would advise

pleasenot to use arrows instead, for two reasons: 1. the student that try some math-heavy higher ed then get totally confused when we try to use "=>" precisely to denote logical implication; 2. it makes more sense to use a sentence explaining what is done or what happens. Student should not think maths is written in a sequence of symbols, it is written in a language. Sure, having them write sentence that make sense is hugely difficult, but for one main reason: it does not hide the depth of misunderstanding that happens also when it is covered by sloppy notation. – Benoît Kloeckner – 2017-04-03T14:54:57.323