developing geometric thinking: the van hiele levels

Developing Geometric Developing Geometric Thinking: The Van Hiele Thinking: The Van Hiele

LevelsLevels Adapted from Adapted from

Van Hiele, P. M. (1959). Development and learning Van Hiele, P. M. (1959). Development and learning process. process. Acta Paedogogica UltrajectinaActa Paedogogica Ultrajectina (pp. 1-31). (pp. 1-31).

Groningen: J. B. Wolters.Groningen: J. B. Wolters.

20082008 Mara AlagicMara Alagic

Van Hiele: Levels of Van Hiele: Levels of Geometric ThinkingGeometric Thinking

PrecognitionPrecognition Level 0: Level 0:

Visualization/RecognitionVisualization/Recognition Level 1: Analysis/Descriptive Level 1: Analysis/Descriptive Level 2: Informal DeductionLevel 2: Informal Deduction Level 3:DeductionLevel 3:Deduction Level 4: RigorLevel 4: Rigor

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PrecognitionPrecognition Level 0: Visualization/RecognitionLevel 0: Visualization/Recognition Level 1: Analysis/Descriptive Level 1: Analysis/Descriptive Level 2: Informal DeductionLevel 2: Informal Deduction Level 3:DeductionLevel 3:Deduction Level 4: RigorLevel 4: Rigor

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Visualization or Visualization or RecognitionRecognition

The student identifies, names The student identifies, names compares and operates on geometric compares and operates on geometric figures according to their appearance figures according to their appearance

For example, the student recognizes For example, the student recognizes rectangles by its form but, a rectangles by its form but, a rectangle seems different to her/him rectangle seems different to her/him then a squarethen a square

At this level rhombus is not At this level rhombus is not recognized as a parallelogramrecognized as a parallelogram

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Analysis/DescriptiveAnalysis/Descriptive The student analyzes figures in The student analyzes figures in

terms of their terms of their componentscomponents and and relationshipsrelationships between components between components and discovers and discovers properties/rulesproperties/rules of a of a class of shapes empirically byclass of shapes empirically by

folding /measuring/ using a grid or diagram, ...folding /measuring/ using a grid or diagram, ... He/she is not yet capable of He/she is not yet capable of

differentiating these properties into differentiating these properties into definitions and propositionsdefinitions and propositions

Logical relations are not yet fit-study Logical relations are not yet fit-study object object

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Analysis/Descriptive: Analysis/Descriptive: An An ExampleExample

If a student knows that the If a student knows that the diagonals of a rhomb are diagonals of a rhomb are

perpendicularperpendicularshe must be able to conclude that,she must be able to conclude that, if two equal circles have two if two equal circles have two

points in common, the segment points in common, the segment joining these two points is joining these two points is perpendicular to the segment perpendicular to the segment joining centers of the circlesjoining centers of the circles

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Visualization/RecognitionVisualization/Recognition Level 1: Analysis/DescriptiveLevel 1: Analysis/Descriptive Level 2: Informal DeductionLevel 2: Informal Deduction Level 3:DeductionLevel 3:Deduction Level 4: RigorLevel 4: Rigor

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Informal DeductionInformal Deduction The student logically interrelates The student logically interrelates

previously discovered previously discovered properties/rules by giving or properties/rules by giving or following informal argumentsfollowing informal arguments

The intrinsic meaning of deduction The intrinsic meaning of deduction is not understood by the studentis not understood by the student

The properties are ordered - The properties are ordered - deduced from one anotherdeduced from one another

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Informal Deduction: Informal Deduction: ExamplesExamples

A square is a rectangle because it A square is a rectangle because it has all the properties of a rectangle.has all the properties of a rectangle.

The student can conclude the The student can conclude the equality of angles from the equality of angles from the parallelism of lines: In a quadrilateral, parallelism of lines: In a quadrilateral, opposite sides being parallel opposite sides being parallel necessitates opposite angles being necessitates opposite angles being equal equal

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Deduction (1)Deduction (1) The student proves theorems The student proves theorems

deductively and establishes deductively and establishes interrelationships among networks of interrelationships among networks of theorems in the Euclidean geometry theorems in the Euclidean geometry

Thinking is concerned with the Thinking is concerned with the meaning of deduction, with the meaning of deduction, with the converse of a theorem, with axioms, converse of a theorem, with axioms, and with necessary and sufficient and with necessary and sufficient conditions conditions

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Deduction (2)Deduction (2) Student seeks to prove facts Student seeks to prove facts

inductivelyinductively It would be possible to develop an It would be possible to develop an

axiomatic system of geometry, but axiomatic system of geometry, but the axiomatics themselves belong to the axiomatics themselves belong to the next (fourth) levelthe next (fourth) level

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RigorRigor The student establishes theorems in The student establishes theorems in

different postulational systems and different postulational systems and analyzes/compares these systemsanalyzes/compares these systems

Figures are defined only by symbols Figures are defined only by symbols bound by relationsbound by relations

A comparative study of the various A comparative study of the various deductive systems can be accomplisheddeductive systems can be accomplished

The student has acquired a scientific The student has acquired a scientific insight into geometryinsight into geometry

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The levels: Differences in The levels: Differences in objects of thoughtobjects of thought

geometric figures => classes of geometric figures => classes of figures & properties of these classesfigures & properties of these classes

students act upon properties, yielding students act upon properties, yielding logical orderings of these properties logical orderings of these properties => operating on these ordering => operating on these ordering relationsrelations

foundations (axiomatic) of ordering foundations (axiomatic) of ordering relationsrelations

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Major Characteristics of the LevelsMajor Characteristics of the Levels the levels are sequential; each level has its own the levels are sequential; each level has its own

language, set of symbols, and network of language, set of symbols, and network of relationsrelations

what is implicit at one level becomes explicit at what is implicit at one level becomes explicit at the next level; material taught to students the next level; material taught to students above above their level is subject to reduction of leveltheir level is subject to reduction of level

progress from one level to the next is progress from one level to the next is more more dependant on instructional experience than on dependant on instructional experience than on age or maturationage or maturation

one goes through various “phases” in one goes through various “phases” in proceeding from one level to the nextproceeding from one level to the next

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ReferencesReferences Van Hiele, P. M. (1959). Development and Van Hiele, P. M. (1959). Development and

learning process. learning process. Acta Paedogogica UltrajectinaActa Paedogogica Ultrajectina (pp. 1-31). Groningen: J. B. Wolters.(pp. 1-31). Groningen: J. B. Wolters.

Van Hiele, P. M. & Van Hiele-Geldof, D. (1958). Van Hiele, P. M. & Van Hiele-Geldof, D. (1958). A method of initiation into geometry at A method of initiation into geometry at

secondary schools. In H. Freudenthal (Ed.). secondary schools. In H. Freudenthal (Ed.). Report on methods of initiation into geometryReport on methods of initiation into geometry (pp.67-80). Groningen: J. B. Wolters.(pp.67-80). Groningen: J. B. Wolters.

Fuys, D., Geddes, D., & Tischler, R. (1988). Fuys, D., Geddes, D., & Tischler, R. (1988). The The van Hiele model of Thinking in Geometry Among van Hiele model of Thinking in Geometry Among AdolescentsAdolescents. JRME Monograph Number 3.. JRME Monograph Number 3.

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developing geometric thinking: the van hiele levels

Documents

informal deductionlevel

analysisdescriptive

van hiele levels

recognitionthe student

geometric figures

level rhombus

opposite angles

equal circles