How well U.S. students are learning mathematics and what should be done about it are now matters for every citizen to ponder. And one hears calls fiom many quarters for schools, teachers, and students to boost their performance. In the mathematician E. G. Begle's laconic formulation, the problem is no longer so much teaching better mathematics as it is teaching mathematics better. Almost everyone today agrees that elementary and middle school mathematics should not be confined to arithmetic but should also include elements from other domains of mathematics, such as algebra, geometry, and statistics. There is much less consensus, however, on how these elements should be organized and taught.
Different people urge that school mathematics be taken in different directions. .
A claim used to advocate movement in one direction is that mathematics is bound by history and culture, that students learn by creating mathematics through their own investigations of problematic situations, and that teachers should set up situations and then step aside so that students can learn. They fail to capture the complexity of mathematics, of learning, arid of teaching. The theorem attributed to Pythagoras was known in venous forms in the civilizations of ancient Babylon and China, and it is still true the world over today even though systems of geometry now exist in which it does not hold. Mathematics is invented, and it is discovered as well. Students learn it on their own, and they learn it from others, most especially their teachers.