Mathematics starts with counting. A one-year old child can count to 1, a two-year to 2, a three-year to 3 and then it goes quickly, although learning all the decimal digits from 0 to 9 fills time in pre-school, while two-digit decimal numbers have to wait to early school.

Birds can count, typically the number of eggs in the crest by sight, and maybe even more primitive creatures such as salamanders and fish.

The binary system of counting with only two digits 0 and 1, is simpler than the decimal with ten digits, and so can be mastered by younger children.

The app Counting leads young children into a playful world of counting in binary form towards a gradual grasp and control of the world during many school years.

Counting starts with counting apples or more generally things of the same kind. The child is led to binary representations in different forms such as color, shape, type, size where two of one kind is replaced by one of the next higher kind, ultimately represented by the two digits 0 and 1 in a position system where the kind is the position. Children are invited to invent individual binary representations as symbols to compare and find a common on to use together with 0 and 1 in positions. An abacus with just one bean can be used instead of 0 and 1 and can be fabricated in pre-school using simple elements:

The next step is start counting just about anything: things. age, time, size, length, width, volume, weight, and to express the number in the chosen binary representation. 

Next step is to add binary numbers by successive replacements of two of one kind with one of the next kind. And then go to subtraction, multiplication and even division by one digit binary numbers. 

With this system a pre-school child can learn to master basic arithmetics with binary numbers (with a home fabricated abacus as computing machine), and even write codes in symbolic form expressing basic operations (as expressed in the operations of the abacus).  This is much more than given to children in traditional math education, as an expression of the power of the tools of the digital society.