Some students have trouble with this experiment. Some students never get the temperature to stay constant at boiling, while other students can't get the temperature to stay level while the ice is melting. The most common problem is that the the student heats the beaker too quickly. If heat is supplied faster than your stirring can distribute it, then you will have a problem keeping the temperature constant during phase changes. Another common problem is that our thermometers are not very precise. Mercury thermometers are more precise and would not show this rise in temperature during melting.
If you really want to do the experiment over again, try to reduce the heat. You can do this by partially covering the flame source or moving the beaker away from the flame. You can also crush the ice. That helps the distribution of heat.
However, if you know what is supposed to happen and understand that, then just move on. You shouldn't be concerned enough about it to repeat the experiment, especially if you saw the temperature stay constant through one phase change