How to reduce INDEPENDENT SET to INDEPENDENT SET SIZE?
There's a couple of issues to address, first is what definition of INDEPENDENT SET you are using, the &quot;standard&quot; definition is

INDEPENDENT SET (Decision)
Input: A graph $G$ and a number $k \in \mathbb{N}$.
Question: Does $G$ have an independent set of size at least $k$?

This is the decision version of the problem, and it should be pretty obvious how to use INDEPENDENT SET VALUE to solve it.
If you however mean the search (or optimization) version of the problem:

INDEPENDENT SET (Search / almost Optimization)
Input: A graph $G$ and a number $k \in \mathbb{N}$.
Output: An independent set $I\subseteq V$ where $|I| \geq k$.

If we're looking at the optimization version, then we have the optimization criterion &quot;maximise $k$&quot;.
In these cases we can still use INDEPENDENT SET VALUE to solve the problem in polynomial time. What we want is an algorithm that makes a polynomial number of calls to our INDEPENDENT SET VALUE sub-routine, and at each step decides what to do with some element of the graph.
Further hints in spoilers (don't cheat! ;) ):

 Say you have a graph $G$ where the size of the largest independent set is $\alpha(G)$ Consider what happens if you delete a vertex from $G$ to get $G'$, then run the INDEPENDENT SET VALUE algorithm on $G'$ - what possible values can you get for $\alpha(G')$, and what does that mean for the vertex you deleted?