I notice that the given equation has a symmetric form: $m+\frac{1}{m}=8$. This makes me think of squaring both sides to see if I can get a similar expression for $m^{2}+\frac{1}{m^{2}}$. Squaring both sides, I get $(m+\frac{1}{m})^{2}=64$. Expanding the left side, I get $m^{2}+\frac{1}{m^{2}}+2m\cdot\frac{1}{m}=64$. Simplifying the middle term, I get $m^{2}+\frac{1}{m^{2}}+2=64$. Subtracting 2 from both sides, I get $m^{2}+\frac{1}{m^{2}}=62$. This is almost what I want, except for the extra 4 in the problem. So I just add 4 to both sides and get $m^{2}+\frac{1}{m^{2}}+4=66$. 
If $m+\frac{1}{m}=8$, then what is the value of $m^{2}+\frac{1}{m^{2}}+4$?