Step 1: List out the numbers and sort them.
We have nine numbers: $1, 2, 0, -1, -3, 1, 2, 0, 1$. Instead of a single sweep, let's group and count how many times each value shows up: $-3$ appears once, $-1$ appears once, $0$ appears twice, $1$ appears three times, and $2$ appears twice. Adding these counts, $1+1+2+3+2=9$, which matches the total number of entries, so the count is correct.
Step 2: Build the sorted order from the counts.
Listing each value the number of times it occurs, from smallest to largest, gives $-3, -1, 0, 0, 1, 1, 1, 2, 2$.
Step 3: Cross off from both ends.
For an odd count of $9$ terms, we can find the median by removing the smallest and the largest value together, again and again, until one value remains. Removing $-3$ and $2$ leaves $-1, 0, 0, 1, 1, 1, 2$ (seven terms). Removing $-1$ and $2$ leaves $0, 0, 1, 1, 1$ (five terms). Removing $0$ and $1$ leaves $0, 1, 1$ (three terms). Removing $0$ and $1$ leaves just $1$ (one term).
Step 4: State the result.
The single value left over after this pairing process is the median.
\[ \boxed{1} \]
