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.
Includes the parts and tools to replace the touchscreen on your iPhone 3GS. Front panel does not include ear piece speaker and home button. Replace your cracked, broken, or shattered iPhone 3GS display. Compatible with all iPhone 3GS models. — 3 available at 14.95 each.