General misconception about $\sqrt x$ I noticed a large portion of general public (who knows what square root is) has a different concept regarding the surd of a positive number, $\sqrt\cdot$, or the principal square root function.

It seems to me a lot of people would say, for example, $\sqrt 4 = \pm 2$, instead of $\sqrt 4 = 2$. People even would correct a statement of the latter form to one with a $\pm$ sign. Some also claim that, since $2^2 = 4$ and $(-2)^2 = 4$, $\sqrt 4 = \pm 2$. Some people continue to quote other "evidences" like the $y=x^2$ graph. While most people understand there are two square roots for a positive number, some seem to have confused this with the surd notation.

From an educational viewpoint, what might be lacking when teaching students about surd forms? Is a lack of understanding to functions a reason for this misconception?



Now I have noticed another recent question that hinted that poster was confused. Following @AndréNicolas's comment below, might these confusion really come from two different communities using the same symbol?
They probably just think that $\sqrt{x}$ means any number whose square is $x$, and don't know that the definition is just the positive root. I don't really think it's anything more than that.
In my opinion it's lack of logical thinking that generates such confusion. 

Also mixing natural language with mathematics gives problems: 


  $Q_1$: What are the squares roots of $4$?
  $A_1$: They are $2$ and $-2$.


Proper interpretation of $Q_1$: find the extension of the set $\{x\in \Bbb R\colon x^2=4\}$.


  $Q_2$: What is the square root of $4$, i.e., $\sqrt 4$?
  $A_2$: The square root of $4$ is $2$.


Proper interpretation of $Q_2$, once it has been established the truth of statement $(\forall x\in \Bbb R^+)(\exists !y\in \Bbb R^+)(y^2=x)$, what is the only positive number such that its square is $4$?



I find that most students don't have enough logic in them to make the above translations and since mathematicians insist on abusing natural language, confusion is bound to rise.

Once students known that $(\forall x\in \Bbb R^+)(\exists !y\in \Bbb R^+)(y^2=x)$ and that given a positive real number $x$, $\sqrt x=y$, there is absolutely no danger of mistakes. It comes directly from the definition.