I came across some interesting mistakes in many area of mathematics with my students and do not let me also to tell you for university students level, I would like to know How do i deal with students who make these mistakes :


1) For ‎$x,y >0 , \log (x+y)=\log x+\log y ‎$

2) For ‎$z\in {C} ,\cos²z+\sin²z \neq 1 ‎$

3) ‎$1 ‎$ is a prime number because ‎$1 ‎$ divides ‎$1 ‎$ and itself

4) For ‎$x, y \in {R},\sqrt{(x²+y²)}=x+y ‎$

5) For ‎ ‎$g(x)\neq 0, \displaystyle \int_{a}^{b}\frac{f(x)}{g(x)}dx=\frac {\int_{a}^{b} f(x)}{\int_{a}^{b}g(x)}dx ‎$

6) ‎${0}^{\infty}‎$ is indeterminate case .

Some Teachers in mathematics believe that the claim 6 is true .

My question here is : How do i deal with students who make these mistakes?

Edit: I edited the question to be clear and according to the below comments

Thank you for any help !!!

  • 2
    $\begingroup$ As a passing comment, I would suggest wording these a little more carefully for students. "For $x,y \ldots$" is ambiguous --- do you mean "for some $x,y$" or "for all $x,y$"? I can guess which you mean, but seeing this makes me want to ask: Is there a good way to get teachers to pay more attention, and say what they mean rather than mean what they say? $\endgroup$ Dec 27 '16 at 16:48
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    $\begingroup$ I thought your question is good, but it was hard to read because of many spelling and grammar mistakes. I tried to edit the grammar and spelling so that it would be easier for people to understand. I didn't understand , "do not let me also to tell you for university students level". Can you explain that?? $\endgroup$
    – Amy B
    Dec 27 '16 at 19:30
  • 6
    $\begingroup$ (Although I was not the down-voter...) I see an issue with this question as being overly broad. In particular, to believe that there is "a good teaching method" to avoid all of those mistakes. For example, your first listed mistake alone is discussed at length in an earlier post, MESE 926. There is not a panacea for all the ills that you describe; so, I am not sure how to respond to all of these questions... maybe focus on one. $\endgroup$ Dec 27 '16 at 19:31
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    $\begingroup$ I voted to close mainly for two reasons. First, as @BenjaminDickman implied above, the question is too broad. Second, and I think, more importantly, it is wrong to ask how to "avoid these mistakes". We might ask how to "deal with these mistakes". That is quite another question, yet too broad. $\endgroup$ Dec 28 '16 at 12:32
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    $\begingroup$ Not only too broad, but such different kinds of issues. Numbers 1, 4, and 5 are easily dealt with by trying an example. (And maybe #6.) How to get students to do that is a bigger question. #3 is by definition, though. $\endgroup$
    – Sue VanHattum
    Dec 28 '16 at 14:38