Christine Funk, yes it is entirely possible that the digitizer circuit is interrupted where your glass is broken. Replace the digitizer, or my personal; favorite, replace the assembly. It's more expensive but also quicker and less chances for failure. you can get the assembly from here or the replacement from here. The guides for your repair can be found right here. Hope this helps, good luck.
This kit includes all the parts and tools to replace the touchscreen on your iPhone 3GS. Does not include... — 38 available at 39.95 each.