Pictured above, from Shoe Box Math Learning Centers: What's In a Shape? (Creating closed shapes by gluing toothpicks to paper), Race to 100 (An adding game), a domino adding game, and Number Collage (creating rubbings using crayons and sandpaper numbers). You don't need a book, though. Pattern blocks with some crayons and printable pattern block grid paper is a popular shoebox on our shelf. We've also got a box with paper, pencil, ruler (for a number line), and dice for creating addition or subtraction problems. Cuisenaire rods in a tub, one-inch colored blocks and some pre-drawn patterns build (you can just draw the patterns yourself with a few colored markers), a ruler and a measuring chart (or just a ruler!), a tub of Math-U-See blocks, a balance scale with items to weigh, counting bears and little logic puzzles to match, macaroni and glue with predrawn pie templates....it just takes a little time and some imagination. Google can be a great help too, if you're stumped for ideas.
A math shoebox can be as structured or as open-ended as you wish. The idea is for the child to have a math activity box that can be explored independently, though it may take some pre-teaching or exploration together initially. Which isn't to say that it must be explored independently, as it's a lot of fun to sit down with the gents and try new things or gently guide some of the exploration. But to be a successful exploration box for our purposes the child should be able to pull the game off the shelf and explore in a way that's meaningful for him or her, which ideally leads back to a conversation or two about math concepts.
I thought the sandpaper numbers might be too simple for kindergarten gent, but it is something he could explore on his own or with his younger brothers. He occupied himself independently for a while. He was intrigued by the multiplication sign. "What's 9 x 9, Mom?" Which led to a mess of math blocks, more multiplication, and a lovely conversation about zeroes and multiplication. By the time we'd finished, the file folder was abandoned for a project involving writing 1 followed 50 zeroes then multiplying the entire thing by 9.
And for some overall math fun, check out this collection of links:
Links to math games, activity ideas, puzzles, articles, learning and teaching aids, freebies, math in daily life, "unschooling math," overcoming math anxiety, and much more...
And a lovely book for preschoolers with some activities that blend math and art:
MathArts: Exploring Math Through Art for 3 to 6 Year Olds by Mary Ann Kohl.