Just a quick note/summary for any who came here via Google looking for the answer to the general question asked in the title (as I was). Any of the following should work for getting access to shell variables inside quotes:
echo "$VARIABLE"
echo "${VARIABLE}"
Use of single quotes is the main issue. According to the Bash Reference Manual:
Enclosing characters in single quotes (
') preserves the literal value of each character within the quotes. A single quote may not occur between single quotes, even when preceded by a backslash.
[…]
Enclosing characters in double quotes (") preserves the literal value of all characters within the quotes, with the exception of$,`,\, and, when history expansion is enabled,!. The characters$and ` retain their special meaning within double quotes (see Shell Expansions). The backslash retains its special meaning only when followed by one of the following characters:$,`,",\, or newline. Within double quotes, backslashes that are followed by one of these characters are removed. Backslashes preceding characters without a special meaning are left unmodified. A double quote may be quoted within double quotes by preceding it with a backslash. If enabled, history expansion will be performed unless an!appearing in double quotes is escaped using a backslash. The backslash preceding the!is not removed.
The special parameters*and@have special meaning when in double quotes (see Shell Parameter Expansion).
In the specific case asked in the question, $COLUMNS is a special variable which has nonstandard properties (see lhunath’s answer above).