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: