The evolution of math came by way of things like tally marks on sticks and pebbles in pots. The line between symbol and physical system is fuzzy. I think one

The evolution of math came by way of things like tally marks on sticks and pebbles in pots. The line between symbol and physical system is fuzzy. I think one of the reasons that people find math difficult is that the emphasis is on the symbol-manipulation rules and not on the reasons that certain sets of symbol-manipulation rules are useful. Kids are taught to count "1 2 3 4..." as if these symbols were handed down by God rather than being totally arbitrary. They should be taught to count "*, **, ***, ****, ..." and then, when they get to "***********" or so, they should be encouraged to invent shorter ways of writing these unwieldly strings. (Note, BTW, that when you use the "*, **, ***, ****, ..." numeral system, addition and multiplication become a whole lot easier!)