I think the part that you're describing is the scape of a hardneck garlic. If what you bought in the store was one of the silverskins or an artichoke variety, you will get few, if any, scapes since those types don't typically produce them.

Normally, if you've planted garlic from a local farmer's market or other place that has locally grown garlic (i.e. a hardneck for zone 6), you'd definitely want to cut the scape once it appears so that the plant directs its energy into producing a larger bulb for the summer harvest.

we let a few scapes form bulblets then use those in stir fry also. We also use the scapes laid on top of a pot roast, they get soft and yummy.

Thank you for all your responses. I don't know the variety of the garlic bulbs I used. I guess I will just be patient and wait.

I have a question too. Do you cut them all the way down or do you leave some of the scape?

Cut off where it comes out of the stalk.