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Fakulti Pendidikan Teknikal UTHM

KOD KURSUS :BBR 23203PEMBELAJARAN GEOMETRI

NAMA AHLI KUMPULAN:

RAJENDRAN A/L DHARMALINGAM ( DB100785 )

NARAYANASAMY A/L ANGAMUTHU (DB100740)

PEERIYA RAJ A/P SUPRAMANIAM ( DB100741 ) THIAGU A/L NARAYANASAMY ( DB100226 )

KANMALAR A/P VIJEYASINGAM (DB100224)

ROHINI DEVI A/P MAHALINGAM ( DB100742 )

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2.0 PEMBELAJARAN GEOMETRI

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2.1

Perspektif Piaget dan

Inhelder

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TEORI PIAGET

* Diperkenalkan oleh Jean Piaget, berasal daripada

Switzerland,merupakan ahli psikologi yang banyak

menyumbang kepada pemahaman bagaimana kanak-kanak

belajar.

TEORI-TEORI PEMBELAJARAN

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PIAGET MENDAPATI BAHAWA

PERKEMBANGAN KOGNITIF KANAK-KANAK

BERBEZA DAN BERUBAH MELALUI EMPAT PERINGKAT:

Peringkat Deria Motor ( 0 – 2 tahun )

Peringkat Pra –

Operasi ( 2 –

7 tahun)

Peringkat Operasi Konkrit ( 7 -11 tahun)

Peringkat Operasi Formal ( 11 tahun ke dewasa)

http://2.bp.blogspot.com/_d7u0FbTf7eU/TDnRObCZTtI/AAAAAAAAAMo/w0yDxAcJrwE/s1600/perkbgn+piaget.jpg

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Kematangan biologi. Proses pemikiran yang dipengaruhi oleh

faktor baka/genetik.

Interaksi individu dengan persekitaran. Individu berinteraksi

dengan persekitaran untuk memahami perkara-perkara yang

berlaku. Mereka akan meneroka, menguji, memerhati dan

menyusun maklumat yang didapati.

Pengalaman sosial. Perkembangan kognitif juga dipengaruhi

oleh pergaulan individu dengan orang-orang di persekitaran :

ibu bapa, adik beradik, rakan, guru dan sebagainya.

Keseimbangan. Proses keseimbangan adalah merupakan salah

satu cara yang digunakan oleh individu untuk mengadaptasi

dengan situasi-situasi yang baru dialami.

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D I DALAM T EORI P EMBELAJARAN

K OGNITIF P IAGET (1970), TERKANDUNG

HURAIAN LIMA PANDANGANNYA YANG

ASAS :

# Skema( schema) # Adaptasi (adaptation)

# Keseimbangan (equilibration)

# Asimilasi (assimilation )

# Akomodasi (accomodation)

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2.2

Perspektif van Hieles

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Van Hieles model

Introduction

Two Dutch educators, Dina and Pierrevan Hiele suggested that children may

learn geometry along the lines of astructure for reasoning that they

developed in the 1950’s.

They work was incorporated into the curriculaof the Soviet Union in the 1960’s but it was not

until the late 1970’s that it began to receive

international attention although not universallyaccepted by all north America educators, the

van Hiele model is mathematically elegant andworthy of investigation.

1) The van Hiele of geometric reasoning are sequential, students must pass through all theprior levels to arrive at any specific level.

2) This levels are not age dependent in the way Piaget described development.

3) Geometric experiences have the greatest influence on advancement through thelevels.

4) Instruction and language at a level higher than the level at the student mayinhibit learning.

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Properties

1) Fixed sequence

The levels are hierarchical, students cannot“skip” a level. Van Hiele claim that much of

the difficulty experienced by geometrystudents is due to being taught at the

Deduction level when they have not yetachieved the abstraction.

5) Attainment

The van Hiele recommended 5 phases forguiding students from one level to another.

2) Adjacency

Properties which are intrinsic at one levelbecome extrinsic at the next. ( the propertiesare there at the level visualization level, butthe students is not yet consciously aware ofthem)

3) Distinction

Each level has its own language and networkof relationship.

4) Separation

A teacher who is reasoning at one levelspeaks a different “language” from a student

at a lower level, preventing understanding.

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Phases ofLearning

The majority of high school geometry courses is taughtat level 3. The van Hieles also identified somecharacteristics of their modal, including the fact that aperson must proceed through the levels in order , thatthe advancement from level to level depends more oncontent and mode of instruction than on age, and thateach level has its own vocabulary and its own system ofrelations. The van Hieles proposed sequencial phasesof learning to help students move from one level toanother.

Phase 1- Inquiry / Information At this initial stage, the teacher andthe students engaged inconversation and activity about theobjects of study for this level.Observation are made, questions

are raised and level specificvocabulary is introduced.

Phase 2-Directed Orientation The students explore topicsthrough materials that the teacherhas carefully sequenced. These

activities should gradually reveal tothe students to structurecharacteristic at this level.

Phase 3-Explication

Building on their previous experiencesstudents express and exchange theiremerging views about the structures thathave been observed. Other than to assist thestudents in using accurate and appropriatevocabulary, the teachers role is minimal. It isduring this phase that the level system ofrelations begins to become apparent.

Phase 4-Free Orientation Students encounter more complextasks – tasks with many steps, tasks

that can be completed more than oneway, and open ended tasks. They gainexperience in resolving problems ontheir own and make explicit manyrelations among the objects of thestructures being studied.

Phase 5- integration Students are able tointernalize and unifyrelations into and new bodyof thought. The teacher canassist in the synthesis by

giving “global surveys” ofwhat students already havelearned.

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Levels

The van Hiele model asserts that the learner moves sequentially through fivelevels of understanding.

Different numbering systems are found in the literature but the van Hielesspoke of level 0 through 4.

Level 0 : Visualization •Students recognize figures as total entities (triangles,square) but do not recognize properties of these figure(right angles in a square)•Sorting, identifying and describing shapes.•Manipulating physical model.•Seeing different sizes and orientations of the same shape as to distinguish characteristics of a shape and the futures that are

not relevant.•Building, drawing, making, putting together, and taking apart shapes.

Level 1 : Analysis •Students analyse component parts of the figures (opposite angles of parallelograms are congruent), but interrelationshipsbetween figures and properties cannot be explained.•Shifting from simple identification to properties, by using concrete or virtual models to define measure, observe and changeproperties.•Using models or technology to focus on defining properties, making property lists and discussing sufficient conditions todefine a shape.•Doing problem solving. Including tasks in which properties of shapes are important components.•Classifying using properties of shapes.

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Level 2 : informal Deduction / Abstraction

•Students can establish interrelationships of properties within figures(in a quadrilateral opposite sides being parallel necessitates opposite angles being congruent) among figures(a square is a rectangle because it has all the properties of a rectangle). Informal proofs can be followed but students do not seehow the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises.

•Doing problem solving, including tasks in which properties of shapes are important component.

•Using models and property lists and discussing which group of properties constitute a necessary and sufficient condition fora specific shape.•Using informal deductive language (“all”, “some”, “none”, “if-then”, “what if”)

•Investigating certain relationships among polygons to establish if the converse is also valid(eg : if a quadrilateral is a rectangle,it must have four right angles. If a quadrilateral has four right angles, must it also be a rectangle.)

•Using models or drawings (including dynamic geometric) as tools to look for generalizations and counter example.

•Making and testing hypotheses.

•Using properties to define a shape or determine if a particular shape is included in a given set.

Level 3 : Deduction •At this level the significance of deduction as a way of establishing geometric theory within an axiom system is understood.•The interrelationship and role of undefined terms, axioms, definitions, theorems and formal proof is seen.•Possibility of developing a proof a proof in more than one way is seen.

Level 4 : Rigor •Student at this level can compare different axiom systems (non-Euclidean geometry can be studied).•Geometry is seen in the abstract with a high degree of rigor, even without concrete examples.

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2.3

Perspektif Hoffer

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Hoffer represent geometrical thinking in two dimensional matrix.

The first dimension consists of five geometrical skills (Visual,

Descriptive, Drawing, Logical and Applied).

The second one deals with the levels of geometric thinking(Recognition, Observation, Analysis, Ordering, Deduction and

Abstraction).

(Hoffer, 1981, p15). This paper restricted with three geometric skills(Visual, Descriptive and Logical) and three levels of geometricalthinking (Recognition, Analysis and Deduction).

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Table (1): the levels of geometric thinking distributed according togeometric skills.

Visual

Recognize geometric shapes by its' picture without knowing the shapes

prosperities Recognize the relationship between different kinds of geometric shapes

Uses information about a geometric shape to deduce more information

Descriptive

Naming a geometric shape. Explain statements that describe geometricshape

Describes the relationships between geometric shapes. Defines geometricconcepts clearly.

Understand the difference between the definition, postulate and theorem

Logical

Understand the meaning of shape reservation in different situations.

Uses the prosperities of geometric shapes to identify the subset relation

Uses logic to prove and being able to deduce new knowledge from givenfacts

The previous slide can be translated into a pyramid of three levels

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The previous slide can be translated into a pyramid of three levels.

Pyramid of three levels of geometric thinking.

Level (3)Logical

Level (2)

Descriptive

Level (1) Visual

Recognize

Shapes Recognize relationbetween Shapes Deduce moreinformation

DescribeShapes

Describe relation

between Shapes Deduce more

information

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