Starting with Woodworth and Thorndike's classical experiment published in 1901, major periods in mathematics education throughout 20th century and on into the current century are reviewed in terms of competing epistemological and psychological paradigms that were operating within as well as across the major periods. The periods were marked by attempts to make changes in school mathematics by adherents of the dominant paradigm. Regardless of what paradigm was dominant, the attempts essentially led to major disappointments or failures. What has been common across these attempts is the practice of basing mathematics curricula for children on the first-order mathematical knowledge of adults. I argue that rather than repeat such attempts to make wholesale changes, what is needed is to construct mathematics curricula for children that is based on the mathematics of children. Toward that end, I present several crucial radical constructivist research programs. [For complete proceedings, see ED581294.]