The checkerboard is referred to a lot when Holden is thinking of his old friend Jane Gallagher. The checkerboard is symbolic of Holden's obsessiveness with Jane. Holden is always asking people and wondering if Jane still keeps her kings in the back row. In checkers, if you keep your kings in the back row, it's like a security. You can't lose if you play that way. This is why Holden is so curious about Jane and if she still keeps her kings in the back row. What he really wants to know is if Jane has lost her security and let someone besides Holden in. This is why when Stradlater returns from a date with Jane, Holden repeatedly asks Stradlater, "Did you ask her if she still keeps all her kings in the back row?" Holden is actually asking if Jane let down her guard and let go of her innocence to Stradlater. Holden ponders a lot about if Jane keeps her kings in the back row over the period of the book. This just shows that Holden is totally obsessed with Jane's innocence and if she still has any innocence left.
School is also a symbol that tells us about Holden. It actually tells us about what will probably happen to Holden in his life. School is a structured system with many steps that must be completed in order to pass. It is a lot like life where there are many obstacles and challenges one must overcome in order to succeed. Holden had already failed a number of schools before he failed Pencey. Holden doesn't even care if he fails school because his parents don't care. Holden tells people about how he failed so many schools like it's something to be proud of. He