To prevent slipping when walking on ice, one should take small...

Let the leg make an angle $\theta$ with the vertical while walking.

The normal force $N$ and the frictional force $f$ provided by the ground are related by $f = N \tan\theta$.

To avoid slipping, the required frictional force must be less than or equal to the limiting friction, i.e., $f \le \mu N$.

Since the coefficient of friction $\mu$ for ice is very small, the required friction $f$ must be small.

Taking shorter steps makes the angle $\theta$ smaller, which in turn reduces the required frictional force $f$.

Therefore, shorter steps ensure smaller friction is needed to walk without slipping.