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