id: 06145587 dt: j an: 2013b.00625 au: Jiang, Zhonghong; O’Brien, George E. ti: Multiple proof approaches and mathematical connections. so: Math. Teach. (Reston) 105, No. 8, 586-593 (2012). py: 2012 pu: National Council of Teachers of Mathematics (NCTM), Reston, VA la: EN cc: G60 G40 E50 ut: geometric concepts; secondary school mathematics; preservice teachers; mathematical logic; technology; trigonometry ci: li: http://www.nctm.org/publications/article.aspx?id=32488 ab: Summary: One of the most rewarding accomplishments of working with preservice secondary school mathematics teachers is helping them develop conceptually connected knowledge and see mathematics as an integrated whole rather than isolated pieces. To help students see and use the connections among various mathematical topics, the authors have paid close attention to selecting such problems as the Three Altitudes of a Triangle problem. They presented the problem to preservice teachers (referred to here as “students") in their Geometry for Teachers class. Using technology to explore the three altitudes of a triangle problem, students devise many proofs for their conjectures. After all the proof approaches were presented and discussed in class, students were excited about what they had learned and felt ownership of the proofs. They commented that they not only understood how to prove that the three altitude lines of any triangle are concurrent but also saw connections between this problem situation and various mathematical topics. In addition, their explorations of multiple approaches to proofs led beyond proof as verification to more of illumination and systematization in understandable yet deep ways [{\it M. D. de Villiers}, Rethinking proof with the Geometer’s Sketchpad. Emeryville, CA: Key Curriculum Press (1999; ME 2000d.02607)]; expanded their repertoire of problem-solving strategies; and developed their confidence, interest, ability, and flexibility in solving various types of new problems. These benefits, in turn, will be passed on to their own students. (ERIC) rv: