My Worst Lesson Ever

My worst lesson ever happened during my student teaching days.  I'm not sure if it was my worst lesson ever, but it is in the one that stands out in my memory.  
I remember going to a baby or bridal shower where each woman was given a clothespin.  If you saw another person with her legs crossed, you got to take all of her clothespins.  The person with the most clothespins at the end of the shower won a prize.

I thought this would be a great idea to use in the classroom.

My plan:

Give each student a clothespin and a problem as they walked in the room.
They were to sit in their seat and complete the problem.  Each student had a different problem.
Then, in pairs, the students would attempt the other persons problem.
The clothespins were only exchanged if one student was right and the other was wrong.
The student with the most clothespins won bragging rights.

What really happened:

I gave each student a clothespin and a problem as they walked in the room.
Many students didn't know how to do the problems I gave them, so in an effort to save time (because I wanted this completed in one period) I gave each student the correct answer to their problem, but not how to solve it.
The students sat with each other and got more and more miserable as the period went on.
No clothespins were "won" and the students and I left that classroom feeling like losers.

What went wrong:

I didn't allow enough time for this activity.  Don't all new teachers do this?  At least one day should have been dedicated to making sure each student knew how to do their problem.

I made the students compete against each other on a topic they were still learning.  I am now aware that it is more productive to have students work together cooperatively while learning something.

Students who already had a low level of confidence in math just had it drop another notch.  Who wants to walk around the room and admit they have no clothespins?

What incentive is there to do a problem against a person who has no clothespins?  There's nothing to win so why bother?

If the activity were to continue, it would turn out being the better students participating until only one person had all the clothespins.  The students who truly need the practice would have given up long ago.

Can this be salvaged?:

My thought is no.  Forget the clothespins and competition and try this activity.  Problem Experts.