jmc

Without loss of generality, let $x_1$ be the smaller root. In the quadratic $a{x}^2+bx+c$, the roots sum to $\frac{-b}{a}$ and multiply to $\frac{c}{a}$. Therefore, $x_1{x}_2=\frac{24}{1}=24$ and $x_1+x_2=m$. Since $x_1$ and $x_2$ must be integers, there are only 4 positive integer pairs of $(x_1,x_2)$ such that the two multiply to 24 -- $(1,24),(2,12),(3,8),(4,6)$ -- and the 4 corresponding negations of those values. Note that for each of these $(x_1,x_2)$, each $m=x_1+x_2$ is distinct. Because $x_1+x_2=x_2+x_1$, the value of $m$ does not change if the order of the roots are reversed, so there are only $4+4=\boxed{8}$ possible values of $m$.