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![Page 1: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/1.jpg)
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The van Hiele levels of Geometrical thought in an in-service training setting in
South Africa.
Ronél Paulsen South Africa
![Page 2: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/2.jpg)
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Questioning
The importance of questioning Use children’s innate curiosity Probe to become more aware of
thinking processes Develop confidence to question,
challenge and reflect
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Mathematical thinking is a dynamic
process,… enabling the complexity
of ideas we can handle, expands our
understanding (Mason)
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Still drawing on John Mason
What improves Mathematical thinking?
Practice with reflection
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Still drawing on John Mason
What supports Mathematical thinking?
Atmosphere of questioning, challenging, reflection.
This takes time to develop
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Still drawing on John Mason
What provokes Mathematical thinking?
Challenge, surprise, contradiction,
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Japanese lesson plans emphasise what students will think, not what the teacher will say (Stigler 1998)
Teachers anticipate students’ reaction
Because teachers work collaboratively on lessons, each teacher contributes from his or her own experiences
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Rohlen also emphasises that the goal for questioning in Japanese classrooms is to get students to think
They pose questions and ask students to explain their thinking
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Metacognition – knowledge of one’s own thought processes. (Romberg)
Novice vs expert
How do we teach students to become aware of their own metacognitive processes?
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Tom Carpenter and Elizabeth Fennema found that although teachers have a great deal of intuitive knowledge about children’s Mathematical thinking, this knowledge is fragmented(CGI Programme at Wisconsin)
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Teachers should take trouble to understand how children think
Leads to fundamental changes in beliefs and practices
This reflects on students’ learning
(Fennema et al)
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The van Hiele Levels
In 1957, Dutch educators Dina van Hiele-Geldof and Pierre van Hiele proposed that the development of a student's understanding of reasoning and proof progresses through five distinct levels.
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The van Hiele Levels
Students identify and reason about shapes and other geometric configurations based on shapes as visual wholes rather than on geometry properties
Level 0: Visual
For instance, they might identify a rectangle as a "door shape"
They would identify two shapes as congruent because they look the same, not because of shared properties
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The van Hiele Levels
A learner on this level will name the following shapes as rectangle, parallelogram and square just on appearance without knowing any of their properties
Level 0: Visual (continued)
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The van Hiele Levels
They might have problem to give the names of the shapes in the following orientations, “because they don’t look like a rectangle, a parallelogram or a square”
Level 0: Visual (continued)
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The van Hiele Levels
The learners analyse figures in terms of their components and relationships between components, establishes properties of a class of figures empirically, and uses properties to solve problems
Level 1: Descriptive/Analytic
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The van Hiele Levels
Students recognise and characterise shapes by their properties
Level 1: Descriptive/Analytic
For example, they can identify a rectangle as a shape with opposite sides parallel and four right angles
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The van Hiele Levels
Level 1: Descriptive/Analytic
Students at this level still do not see relationships between classes of shapes (e.g., all rectangles are parallelograms), and they tend to name all properties they know to describe a class, instead of a sufficient set
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The van Hiele Levels
Level 1: Descriptive/Analytic (continued)
When learners investigate a certain shape they come to know the specific properties of that figure. For example, they will realise that the sides of a square are equal and that the diagonals are equal. Students discover the properties of a figure but see them in isolation and as having no connection with each other.
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The van Hiele Levels
Level 2: Abstract/Relational Students are able to form abstract
definitions and distinguish between necessary and sufficient sets of conditions for a class of shapes, recognizing that some properties imply others.
Students also first establish a network of logical properties and begin to engage in deductive reasoning, though more for organizing than for proving theorems.
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The van Hiele Levels
Level 2: Abstract/Relational (continued)
When learners reason about and compare the properties of a figure they realise that there are relationships between them.
The relationships being perceived: Exist between the properties of a specific
figure, and Exist between the properties of different
figures.
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The van Hiele Levels
Level 3: Formal Deduction and Proof
Students are able to prove theorems formally within a deductive system.
They are able to understand the roles of postulates, definitions, and proofs in geometry, and they can make conjectures and try to verify them deductively.
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The van Hiele Levels
Level 3: Formal Deduction and Proof (contd) At this level the learner is able to make
deductions. He/she is able to write proofs, understands the role of axioms and definitions, and knows the meaning of necessary and sufficient conditions. The learner reasons formally within the context of a mathematical system, complete with undefined terms, axioms and underlying logical system, definitions and theorems.
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The van Hiele Levels
Level 4: Rigour
The student at this level understands the formal aspects of deduction. Symbols without referents can be manipulated according to the laws of formal logic. The learner can compare systems based on different axioms and can study various geometries in the absence of concrete models.
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The van Tall Hiele Levels
Level zero?
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Level Descriptors
In order to place children on one if the levels of van Hiele, there are some descriptors which can be used to see what children can or cannot do. For the purpose of this talk, I only looked at a few of the descriptors, as set out by van Hiele.
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Level 0 Descriptors
The learners identify and operate on
shapes (e.g., squares, triangles) and
other geometric configurations (e.g.,
lines, angles, and grids) according to
their appearance.
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Level 0 Descriptors
The learner identifies instances of a shape by its appearance as a whole in a simple drawing diagram or set of cut-outs.
The learner identifies instances of a shape by its appearance as a whole in different positions
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Three year olds can sort and classify (Level 0)
Flat shapes and space shapes
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Recognizing spheres and circles have the same shape
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The visual effect!
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Level 0 Descriptors
A learner identifies parts of a figure but: Does not analyse a figure in terms of
its components. Does not think of properties as
characterising a class of figures. Does not make generalisations about
shapes or use related language.
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Level 0 Descriptors
Examples Does not analyse a figure in terms of its components.
This is a rectangle This is not a rectangle
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Level 0 Descriptors
Examples Does not analyse a figure in terms of its components.
This is a kite This is not a kite
![Page 35: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/35.jpg)
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Level 0 Descriptors
Examples Does not analyse a figure in terms of its components
This is a kite This is a rhombus This is a square?
![Page 36: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/36.jpg)
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Level 1 Descriptors
The learners analyse figures in terms of their components and relationships between components, establishes properties of a class of figures empirically, and uses properties to solve problems
![Page 37: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/37.jpg)
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Level 1 Descriptors
The learner sorts shapes in different ways according to certain properties, including a sort of all instances of a class from non-instances.
Does not explain subclass relationships beyond specific instances against given list of properties.
![Page 38: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/38.jpg)
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Level 1 Descriptors
Does not explain how certain properties
of a figure are interrelated
Does not formulate and use formal
definitions
![Page 39: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/39.jpg)
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Level 1 Descriptors
ExampleA learner tells what shape a figure is, given certain properties.
Quadrilaterals of which the diagonals bisect each other
![Page 40: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/40.jpg)
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Level 1 Descriptors
ExampleA learner tells what shape a figure is, given certain properties.
Quadrilaterals of which the diagonals bisect each other
![Page 41: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/41.jpg)
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Level 2 Descriptors
Learners formulate and use definition, give
informal arguments that order previously
discovered properties, and follows and
gives deductive arguments.
![Page 42: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/42.jpg)
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Level 2 Descriptors
The learner identifies sets of properties
that characterise a class of figure and
tests that these are sufficient.
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Level 2 Descriptors
Recognises the role of deductive arguments and approaches problems in a deductive manner, but
Does not grasp the meaning of deduction in an axiomatic sense (e.g. does not see the need for definitions and basic assumptions).
![Page 44: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/44.jpg)
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Level 2 Descriptors
Does not formally distinguish between
a statement and its converse
Does not yet establish inter-relationships
between networks of theorems.
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Moving on…
The learner must have completed levels
0, 1 and 2 in order to successfully cope
with the proofs in Euclidean geometry.
Levels are hierarchical
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Level 3 Descriptors
Learners establish, within a postulation
system, theorems and inter-
relationships between networks of
theorems.
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Level 3 Descriptors
Recognises characteristics of a formal definition (e.g. necessary and sufficient conditions) and equivalence of definitions.
Compares and contrasts different
proofs of theorems
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Level 3 Descriptors
ExampleCompares and contrasts different proofs of theorems
Proving the theorem of Pythagoras
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Level 4 Descriptors
Learners rigorously establishes
theorems in different postulational
systems and analyses/compares these
systems
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Examples of teachers involving in in-service training programme
Some video clips
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The video
Taken during in service training. All teachers are enrolled for an Advanced Certificate Programme at the University of South Africa. English second or third language speakers.All experienced primary school teachers Reading to do for preparationWe discussed the van Hiele levelsFlat shapes cut outsSimulate a classroom situationWe concentrated basically on level zero van Hiele levelsTeachers were required to do various activities:Sort the shapes with curvesSort shapes with three sidesFind the squaresShapes with equal sides
![Page 52: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/52.jpg)
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Mathematical thinking?
The person who took the video, also asked
the questions.
Look carefully and you will probably note
many areas where the questioning
techniques can be improved upon.
Impromptu and unedited
![Page 53: 1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa paulsr@unisa.ac.za.](https://reader036.fdocuments.in/reader036/viewer/2022062322/56649e315503460f94b224dd/html5/thumbnails/53.jpg)
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Cut –out shapes
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55DropperMedicine spoon
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Fenemma E, Carpenter TP, Franke, ML, Levi L, Jacobs VR, and Empsen SB. 1996. A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education 27(4): 403-434
Van de Walle, J.A. 2007. Elementary and middle school Mathematics – teaching developmentally. Sixth Edition. New Jersey: Pearson Education.
Rohlen, T and Le Trendre, (1998) Teaching and Learning in Japan. Cambridge University Press.
Bibliography
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Carpenter T, Fennema E, Franke ML, Levi L Empson SB (2000) Research Report, National Centre for improving student learning and achievement in Mathematics and Science. University of Wisconsin − Madison
Mason, J Thinking Mathematically