Van Hiele Levels and Achievement in Secondary School Geometry
Van Hiele Levels of understanding shapes in geometry.
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Transcript of Van Hiele Levels of understanding shapes in geometry.
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