Using Technology to Unify Geometric Theorems about the Power of a PointReport as inadecuate

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Mathematics Educator, v21 n1 p11-21 2011

In this article, I describe a classroom investigation in which a group of prospective secondary mathematics teachers discovered theorems related to the power of a point using The Geometer's Sketchpad (GSP). The power of a point is defines as follows: Let P be a fixed point coplanar with a circle. If line PA is a secant line that intersects the circle at points A and B, then PA[middle dot]PB is a constant called the power of P with respect to the circle. In the investigation, the students discovered and unified the four theorems associated with the power of a point: the secant-secant theorem, the secant-tangent theorem, the tangent-tangent theorem, and the chord-chord theorem. In our journey the students and I also discovered two kinds of proofs that can be adapted to prove each of the four theorems. As teacher educators, we need to design learning tasks for future teachers that deepen their understanding of the content they are likely to teach. Having a profound understanding of a mathematical idea involves seeing the connectedness of mathematical ideas. By discovering and unifying the power-of-a-point theorems and proofs, these future teachers experienced what it means to understand a mathematical theorem deeply. GSP was an instrumental pedagogical tool that facilitated and supported the investigation in three main ways: as a management tool, motivational tool, and cognitive tool. (Contains 13 figures.)

Descriptors: Geometric Concepts, Mathematics Teachers, Teacher Education Programs, Secondary School Mathematics, Preservice Teachers, Mathematics Instruction, Mathematics Education, Mathematical Logic, Validity, Educational Technology, Computer Software

Mathematics Education Student Association, University of Georgia. 105 Aderhold Hall, Athens, GA 30602. Tel: 706-542-4194; Fax: 706-542-4551; e-mail: mesaprez[at]; Web site:

Author: Contreras, Jose N.


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