the role of proof in mathematics
TRANSCRIPT
The Role of Proof in Mathematics
The Role of 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
Proof in Mathematics
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
Find the value of (a-b)2
Suppose AC = AF = A unitsAB = AG = b units\ BC = AC - AB = (a - b) unitsGF = AF - AG = (a - b) unitsHI = BC = (a - b) units and HE = GE = (a - b) units\ Area of the square HIDE = HI * HE= (a - b) * (a - b) = (a - b)2 sq. units ..........IArea of square HIDE = Area of square ACDF - Area of rectangle ACIG - Area of rectangle GHEF= AC * AF - AC * AG - GH * GF= a.a - a.b - (a-b).b= a2 - ab - b(a-b)= a2 - ab - ab + b2
= a2 - 2ab + b2 ..........II\ From I and II Þ (a - b)2 = a2 - 2ab + b2
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
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).
Transformation Proof
Transformational Proof
http://mathoverflow.net/questions/8846/proofs-without-words
cut-the-knot.org
cartoonstock.com
Bibliography
www.nctm.org