against human nature.rnUse of manipulatives. These are basicallyrntoys of various kinds—counters,rncubes, sticks, marbles, grids, and so on.rnEven the NCTM admits in its Standardsrnthat manipulation of manipulatives is incapablernof proving any mathematicalrnproposition, yet it advocates such arnhands-on approach well into the fifthrngrade.rnThis approach is closely . . .
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