If one topic is likely to be stressed by algebra and geometry teachers, it is reasoning. In algebra classes, students are constantly being asked to show their work and justify their simplifications, often without formal connection to proof concepts or the proof process. In geometry classes, students are expected to learn how to write simple proofs. However, evidence shows that students are not learning these reasoning skills. In the 1985–86 National Assessment of Educational Progress, Silver and Carpenter (1989, 18) found that “many eleventhgrade students are confused about the fundamental distinctions among mathematical demonstrations, assumptions, and proofs.” Most students thought a theorem was a demonstration or an assumption. Senk (1985) found that only about 30 percent of students mastered proof wTiting in geometry, despite being enrolled in a year-long course emphasizing proof. Thompson (1992) found that roughly 60 percent of precalculus students were successful at trigonometric-identity proofs, more than 30 percent could complete number-theory proofs dealing with divisibility, and less than 20 percent could handle indirect arguments or proof by mathematical induction.
On Baica’s trigonometric identity