Transcript Developing Geometric Thinking: The Van Hiele Levels

Developing Geometric
Thinking: The Van Hiele
Levels
Adapted from
Van Hiele, P. M. (1959). Development and learning
process. Acta Paedogogica Ultrajectina (pp. 1-31).
Groningen: J. B. Wolters.
CI 319 Fall 2006
Mara Alagic
1
Van Hiele: Levels of
Geometric Thinking
 Precognition
 Level
0: Visualization/Recognition
 Level 1: Analysis/Descriptive
 Level 2: Informal Deduction
 Level 3:Deduction
 Level 4: Rigor
CI 319 Fall 2006
Mara Alagic
2
Visualization or Recognition
 The
student identifies, names
compares and operates on geometric
figures according to their appearance
 For example, the student recognizes
rectangles by its form but, a rectangle
seems different to her/him then a
square
 At this level rhombus is not recognized
as a parallelogram
CI 319 Fall 2006
Mara Alagic
3
Analysis/Descriptive
 The
student analyzes figures in terms
of their components and relationships
between components and discovers
properties/rules of a class of shapes
empirically by

folding /measuring/ using a grid or diagram, ...
 He/she
is not yet capable of
differentiating these properties into
definitions and propositions
 Logical relations are not yet fit-study
object
CI 319 Fall 2006
Mara Alagic
4
Analysis/Descriptive:
An Example
If a student knows that the
 diagonals
of a rhomb are perpendicular
she must be able to conclude that,
 if two equal circles have two
points in common, the segment
joining these two points is
perpendicular to the segment
joining centers of the circles
CI 319 Fall 2006
Mara Alagic
5
Informal Deduction
 The
student logically interrelates
previously discovered
properties/rules by giving or
following informal arguments
 The intrinsic meaning of deduction
is not understood by the student
 The properties are ordered deduced from one another
CI 319 Fall 2006
Mara Alagic
6
Informal Deduction: Examples
A
square is a rectangle because it has
all the properties of a rectangle.
 The student can conclude the equality
of angles from the parallelism of lines:
In a quadrilateral, opposite sides being
parallel necessitates opposite angles
being equal
CI 319 Fall 2006
Mara Alagic
7
Deduction (1)
 The
student proves theorems
deductively and establishes
interrelationships among networks of
theorems in the Euclidean geometry
 Thinking is concerned with the
meaning of deduction, with the
converse of a theorem, with axioms,
and with necessary and sufficient
conditions
CI 319 Fall 2006
Mara Alagic
8
Deduction (2)
 Student
seeks to prove facts
inductively
 It would be possible to develop an
axiomatic system of geometry, but the
axiomatics themselves belong to the
next (fourth) level
CI 319 Fall 2006
Mara Alagic
9
Rigor
 The
student establishes theorems in
different postulational systems and
analyzes/compares these systems
 Figures are defined only by symbols
bound by relations
 A comparative study of the various
deductive systems can be
accomplished
 The student has acquired a scientific
insight into geometry
CI 319 Fall 2006
Mara Alagic
10
The levels: Differences in
objects of thought
 geometric
figures => classes of
figures & properties of these classes
 students act upon properties,
yielding logical orderings of these
properties => operating on these
ordering relations
 foundations (axiomatic) of ordering
relations
CI 319 Fall 2006
Mara Alagic
11
Major Characteristics of the Levels

the levels are sequential; each level has its own
language, set of symbols, and network of
relations
 what is implicit at one level becomes explicit at
the next level; material taught to students above
their level is subject to reduction of level
 progress from one level to the next is more
dependant on instructional experience than on
age or maturation
 one goes through various “phases” in
proceeding from one level to the next
CI 319 Fall 2006
Mara Alagic
12
References



Van Hiele, P. M. (1959). Development and learning
process. Acta Paedogogica Ultrajectina (pp. 1-31).
Groningen: J. B. Wolters.
Van Hiele, P. M. & Van Hiele-Geldof, D. (1958).
A method of initiation into geometry at secondary
schools. In H. Freudenthal (Ed.). Report on methods
of initiation into geometry (pp.67-80). Groningen: J.
B. Wolters.
Fuys, D., Geddes, D., & Tischler, R. (1988). The van
Hiele model of Thinking in Geometry Among
Adolescents. JRME Monograph Number 3.
CI 319 Fall 2006
Mara Alagic
13