I see the article title (which I put on the submission, and one comment commented on) was replaced by an HN moderator with a rewording of the article subtitle. I guess that's okay.

The article was especially eye-catching for me, because it was first posted among our mutual Facebook friends by my mentor in homeschooling mathematics, the mother of the first United States woman to win a gold medal in the International Mathematical Olympiad. The teacher featured in the article as the developer of innovative teaching methods for young learners is also a Facebook friend of mine, and we are part of a network of parents and teachers who are curious about how to stimulate interest in mathematics among young learners by doing things differently from the United States typical school curriculum in mathematics. I'm grateful that my children have been exposed to approaches like that (which I follow imperfectly in my homeschooling my children and in teaching other learners in classroom courses for my occupation), as that has helped my oldest son launch into the adult world with good problem-solving skills for hacking at the startup where he works.

There is a continual tension in mathematics education between teaching topics in their logical order as we understand mathematical topics in light of modern higher mathematics and teaching topics in their historical order of development (which is largely what happens in school mathematics courses). Children need some concrete, tangible experience with counting and with shapes to understand much about mathematics. The cool abstractions that motivate higher mathematics may be inaccessible to children who have no experience with tangible examples of those absractions. (See

http://www.latimes.com/opinion/commentary/la-oe-adv-frenkel-...

an article submitted to Hacker News yesterday, for more about this.) But it looks like we can gain in mathematics instruction by letting children play with more sophisticated representations of mathematics than most children get to play with. Games are great learning tools. One of favorite games for teaching mathematics I learned about from John Holt's book How Children Fail (originally published 1964, which I read in 1971 on the advice of my school's assistant principal). The twenty questions game asks children to find a number from 1 to 1,000,000 by asking twenty yes/no questions, which the person who chose the unknown number must answer truthfully. It is an interesting mathematical exercise to show that twenty questions is (barely) enough for finding one number in one million if the twenty questions are used optimally. I have played this game many times with the children in my classes, and the opportunity to play this game again at the end of class is one of my strongest incentives for the children to stay focused during a lesson on a Saturday morning to get through the lesson content efficiently.

Playing with challenging problems appears to be the royal road for learning mathematics.

http://www.epsiloncamp.org/ProblemsversusExercises.php

I hope ideas like this spread through many communities in the United States and the English-speaking world, so that more young people gain more opportunity to learn to enjoy challenging mathematics early and often.