Van Hiele Levelsof understanding shapes in geometry

Van Hiele levels

Visualization/Recognition

Description/Analysis

Informed Deduction

Formal Logic

Rigor/Axiomatic

Van Hiele Properties

Levels are hierarchical--can’t skip a level

Levels are not age dependent

Experience with geometry has greatest influence

Instruction and language higher than the level the student is on could inhibit their learning

teachers need to understand the language and properties of each level

REcognition/visualization

Lack attention to parts/attributes of shapes

Recognize differences between shapes/can compare

Learn names of shapes

Activities:tangramsfind hidden figuresexamples vs. non-examplesmanipulate physical models

Description/analysis

Don’t see relationship between properties

Properties understood independent of each other

i.e. “a square is not a rectangle”

Shapes have properties

Activities:geoboardsfold, measure, cut, look for symmetry, predict shapechange properties and observe, classify

Informed deduction

Properties are related and logically ordered

Follow logical arguments

Relationships between figures

Activities:express relationships verballyopen ended tasks with shapesis converse valid?use deductive language: all, some, none, if-then, what if

Formal logic

Not typically reached until high school or college

Construct deductive arguments

Establish interrelationships among theorems

Activities:drawings and constructionsproofs

Axiomatic/rigor

College level

Highly abstract

Compare deductive systems

Explore geometries based on postulates

Rigorous indirect proof and proof by contrapositive

A problem

Two brothers discover a quadrilateral shaped island. How can they divide the land

fairly between them?

Extension: What if they each wanted the same amount of coastline?

Patty paper geometry

Draw a line segment AB.

Find its midpoint by folding only.

Make a line a parallel to AB.

Make a line b perpendicular to AB.

Patty Paper GEometry

Angle Bisector given angleextension: draw a point on line, what do you know?

Perpendicular Bisector of a line segment given segmentextension: draw point on bisector line, what do you know?

Perpendicular to line through a point given line and point in spaceextension: perpendicular through point on line

- Parallel to line through a point given line and point in spaceextension: create parallelogram

Patty paper geometry

ASA: angle--side--angle

SAS: side--angle--side

AAA: angle--angle--angle

SSS: side--side--side

Draw the appropriate pieces of the triangle on your patty paper. For sides, draw two dots at the

endpoints so you know those are fixed. For angles draw a dot at the vertex of the angle but draw the

sides of the angle at a random length. Cut the paper so you can maniplate each piece for form a triangle. Can you make more than one triangle?

What is your conjecture?

Proof with alice"Then you should say what you mean," the March Hare went on.

"I do," Alice replied; "at least--at least I mean what I say--that's the same thing you know."

"Not the same thing a bit!" said the Hatter. "Why, you might just as well say that 'I see what I eat" is the same thing as 'I eat what I see'!"

"You might just as well say," added the March Hare, "that 'I like what I get' is the same thing as 'I get what I like'!"

"You might just as well say," added the Dormouse, "that 'I breathe when I sleep' is the same thing as 'I sleep when I breathe'!"

Contrapositiveconverse & Inverse

other resources for proof

If you give a moose a muffin

Ad for "A Fish Died"

Sherlock Holmes: if-then statements

Computer Programming (http://beta.appinventor.mit.edu)

Logic Puzzles/LSAT (http://www.logic-puzzles.org/)

Rube Goldberg devices (http://www.rubegoldberg.com/)

Rube Goldberg

House project