Teaching Math from Logic in the Primary Grades

I recently read a posting entitled "Math Magic" by Lisa VanDamme on the blog of The Objective Standard. What VanDamme describes seems to be a slow re-invention of the elementary mathematics curriculum that I fondly remember from grade school in Poland, the Łukasiewicz curriculum. Łukasiewicz, a philosopher of logic and mathematics who would later write Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic (Oxford University Press, 1957,) was briefly (in 1919) Poland's minister of Education, which prompted him to write a radical, logic-based primary-school mathematics curriculum. The greatest difference between his curriculum and the usual one was the removal of almost all memorization, and its replacement with algorithms based on principles that the student already understood. For example, no memorizing the 9 times N multiplication table: the student learns to take 10 times N (already understood, easy) and subtract N. Of course, after calculating the result a few times the student remembers the result, at least as well as though it had been memorized - but also understands how getting this result was grounded in previously understood principles.

One beautiful thing was really understanding the logic of everyday algorithms. Learning to add and subtract minutes, hours of the day and days of the week - in first grade, I understood not only how to tell time, but also the logical principles of modulo arithmetic and number bases. Another was how quickly one could advance through concepts when no time was wasted on memorization: the Łukasiewicz curriculum, at least in the school I attended, got to analytic geometry and trigonometry in grade 4, calculus in grade 5. I don't know if the Łukasiewicz curriculum was ever translated into English, although it is the standard in Hungary, Japan etc.