2016/2/11

ICME-2012

On Geometry for Development ofCritical Thinking

Mikio Miyazaki
Shinshu University,Japan

Enhancing Critical Thinking with
Proving in Geometry
Proving in Geometry Critical Thinking
Establishing the universalityof properties

Establishing the truth ofstatements logically

Clarifying implicit theorems ina proof

Revealing implicit evidencesin an explanation

Organizing figural properties
Discovering new properties
based on proofs

Organizing knowledge
Discovering new knowledge
based on explanations

Overcoming counterexamples Making use of proofs andrefutations
Strongly Recommended to Impose Proving in Curriculum for Future Generations

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Importance of Proof & Proving
The teaching and learning of proof isa key component of mathematics andthus of mathematics curricula (Hanna& de Villiers, 2008; 2012).

TIMSS 2011 results

http://timssandpirls.bc.edu/timss2011/52

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Teaching and Learning a Proof in Japan

Grade 4 5 6 789InformalProofa informal
Activity Learning
proof

FormalLearning a formal proof inparagraph modes•Thinking logically,•What is it?
deductively•Why is it needed?
•Expressing precisely •How to construct it?

Ex.

uExplain the sum of uProve the sum of inner
inner angel ofangel of triangle
triangle inductively uProve the properties ofand deductively
triangles
uProve the properties ofquadrilaterals

Gap in the Geometry Curriculum
GradeContents

Geometry: Symmetry, Basic constructions,
7 PlaneCircle and sector
Space Geometry: Solids and spatial figures, Surface
area and volume of solids

to explore figures: Properties of parallel
8 How
lines and angles, Properties of congruent figures,

Proving

InformalConditions of congruent triangles

Proof & Proving :What is it? How to construct?
Figural properties and proof: Triangles,Quadrilaterals

9 Figures and similarity

Inscribed angle and central angle
Pythagoras’ theorem

Formal3

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How do you prove it to establish the universality?

- The sum of the three interior angles of a triangle is 180° -Okamoto etc. (2012). “Gateway to the future math2”, Keirinkan: Osaka.

What kinds of theorems used?

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Why is Properties of parallel lines true?

Prove ∠a = ∠COkamoto etc. (2012). “Gateway to the future math2”, Keirinkan: Osaka.

Proofs in Textbook
Generic Explanation

ProofProperties of parallel lines
Vertical angles are equal(corresponding angles) Okamoto etc. (2012). “Gateway to the future math2”, Keirinkan: Osaka.

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Why are vertical angles equal?

Vertical angles are equal
Okamoto etc. (2012). “Gateway to the future math2”, Keirinkan: Osaka.

Okamoto etc. (2012). “Gateway to the future math2”, Keirinkan: Osaka.

Proofs in Textbook

Generic Explanation

Proof6

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Okamoto etc. (2012). “Gateway to the future math2”, Keirinkan: Osaka.

Why is it true?

Properties of parallel lines(corresponding angles)

Figural Construction

Organizing Figural Properties
Sum of inner angles of triangle

Properties of parallel lines(Alternate interior angles)
Properties of parallel lines(corresponding angles)
Figural Construction

Vertical angles are equal
Straight angle is 180°7

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What can we find from this proof?

Okamoto etc. (2012). “Gateway to the future math2”, Keirinkan: Osaka.

Discovering New Properties Based on Proofs

Okamoto etc. (2012). “Gateway to the future math2”, Keirinkan: Osaka.

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