Can you find the winning strategy for this simple game?

There are two players, A and B. A has a 6 by 6 grid on a piece of paper and B has a row of 6 boxes.
A goes first and fills in the boxes in the first row with X or O.
Then B fills in the first box of the row with either X or O.
A continues to fill in a row at a time with X and O and B responds by filling the next box in the row with X or O.

B’s objective is to duplicate one of the A’s rows and A objective is to prevent this.

1. What is a winning strategy for B?
2. If B makes the wrong first move, how can A assure a win?

I lifted this problem from the book Heart of Mathematics. The first part of the problem is a good introduction to working with infinity, which is one of the areas covered in the book. I found out about the book from a liberal arts math course I heard about. From the Amazon reviews, it appears that there are a number of such courses that use the book. I got a copy of it from the library, because I have an interest in math education and because I thought I might fill in a few gaps in my own math education. So far I have mixed opinions on the book.