The role of proof in mathematics
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The role of proof in mathematics

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  • http://vimeo.com/3386641
  • http://www.graspr.com/videos/Math-Proof-Negative-Times-Negative-Equals-Positive-3

The role of proof in mathematics The role of proof in mathematics Presentation Transcript

  • The Role of Proof in Mathematics
  • The Role of Proof in Mathematics
  • Proof in Mathematics
    Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of characters.
    Vladimir Arnold
  • Standards on Proof
    Instructional programs that should enable students to:
    • develop and evaluate mathematical arguments and proofs
    • select and use various types of reasoning and methods of proof
    • By the end of middle school, students should be able to understand and produce mathematical proofs
  • http://vimeo.com/3386641
  • Proof
    • Convincing demonstration that a math statement is true.
    • To explain.
    • Informal and formal.
    • Logic
    • No single correct answer
  • Proofs
    Often proofs are constructed by working backwards. For example:
    Starting with the desired conclusion T, you could say, "If I could prove statement A, then using previously proved theorem B, I could conclude that T is true." This reduces your proof to proving statement A, then saying at the end of that proof, "Using Theorem B, T is true."
    Often there are many possibilities for A (and B).
    The trick is to pick one you can prove!
  • Three Forms of Formal Proof
    • Synthetic Geometry
    • Analytic Geometry
    • Transformational Geometry
  • http://www.graspr.com/videos/Math-Proof-Negative-Times-Negative-Equals-Positive-3
  • Synthetic Geometry
    • A system illustrated by proving geometric relationships based on the use of a rational sequence of definitions, postulates, and theorems
    • 19th Century
    • Pure geometry
    • Logical Arguments
  • The most common proof – The Pythagorean Theorem
  • Grade 7 – MathematicsFinding the value of (a-b)2 (Geometrical Proof)
  • Analytic Geometry
    • Also known as coordinate geometry or Cartesian geometry
    • Algebra
    • Graphing Technology
    • Computations
  • Analytic GeometryCartesian Geometry
    Also known as coordinate geometry-graphing
  • Transformational Geometry
    • 20th Century
    • Graphics technology
    • MIRA
    • Plane mirror
    • Is a method for studying geometry that illustrates congruence and similarity by use of transformations
    • Therefore a transformational proof is a proof that employs the use of transformation
  • Transformation Proof
    An isometry is a transformation of the plane that preserves length.  For example, if the sides of an original pre-image triangle measure 3, 4, and 5, and the sides of its image after a transformation measure 3, 4, and 5, the transformation preserved length.               A direct isometry preserves orientation or order - the letters on the diagram go in the same clockwise or counterclockwise direction on the figure and its image.             A non-direct or opposite isometry changes the order (such as clockwise changes to counterclockwise).
  • Transformational Proof
  • http://mathoverflow.net/questions/8846/proofs-without-words
    cut-the-knot.org
    cartoonstock.com
    Bibliography
    www.nctm.org