American schools do use "manipulatives" to introduce counting and numbers, addition, subtraction, etc. They might use checkers, or popsicle sticks, or anything small and easy to hold/move.

The use of various kinds of mathematical manipulatives and concrete materials is great (including base ten blocks, cuisenaire rods, ten frames, number lines, dice, balance scales with weights, geoboards, pattern blocks, multi-link cubes, etc.). I'm a fan of all of them. But I think the counting board, per se, is a sadly neglected tool, not least because it gives a nice connection to the past.