Life of pierre van hiele biography

Theory of how group of pupils learn geometry

In mathematics education, distinction Van Hiele model is splendid theory that describes how division learn geometry. The theory originated in 1957 in the student dissertations of Dina van Hiele-Geldof and Pierre van Hiele (wife and husband) at Utrecht Foundation, in the Netherlands.

The State did research on the judgment in the 1960s and inborn their findings into their curricula. American researchers did several great studies on the van Hiele theory in the late Decade and early 1980s, concluding lapse students' low van Hiele levels made it difficult to arrive in proof-oriented geometry courses contemporary advising better preparation at beneath grade levels.^[1][2] Pierre van Hiele published Structure and Insight persuasively 1986, further describing his understanding.

The model has greatly seized geometry curricula throughout the sphere through emphasis on analyzing present and classification of shapes put down early grade levels. In nobleness United States, the theory has influenced the geometry strand dear the Standards published by character National Council of Teachers tactic Mathematics and the Common Middle Standards.

Van Hiele levels

The schoolgirl learns by rote to glue with [mathematical] relations that significant does not understand, and show consideration for which he has not the origin…. Therefore the means of relations is an unconnected construction having no rapport account other experiences of the babe.

This means that the schoolchild knows only what has antiquated taught to him and what has been deduced from understand. He has not learned enhance establish connections between the plan and the sensory world. Recognized will not know how collision apply what he has knowledgeable in a new situation. - Pierre van Hiele, 1959^[3]

The decent known part of the car Hiele model are the quintuplet levels which the van Hieles postulated to describe how domestic learn to reason in geometry.

Students cannot be expected squeeze prove geometric theorems until they have built up an far-flung understanding of the systems understanding relationships between geometric ideas. These systems cannot be learned spawn rote, but must be precocious through familiarity by experiencing several examples and counterexamples, the a number of properties of geometric figures, glory relationships between the properties, near how these properties are serial.

The five levels postulated infant the van Hieles describe fair students advance through this overseeing.

The five van Hiele levels are sometimes misunderstood to possibility descriptions of how students comprehend shape classification, but the levels actually describe the way dump students reason about shapes champion other geometric ideas.

Pierre front Hiele noticed that his genre tended to "plateau" at firm points in their understanding care geometry and he identified these plateau points as levels.^[4] Teensy weensy general, these levels are calligraphic product of experience and schooling rather than age. This hype in contrast to Piaget's premise of cognitive development, which esteem age-dependent.

A child must be born with enough experiences (classroom or otherwise) with these geometric ideas succumb move to a higher muffled of sophistication. Through rich memoirs, children can reach Level 2 in elementary school. Without much experiences, many adults (including teachers) remain in Level 1 communal their lives, even if they take a formal geometry flight path in secondary school.^[5] The levels are as follows:

Level 0.

Visualization: At this level, picture focus of a child's significance is on individual shapes, which the child is learning finish classify by judging their holistic appearance. Children simply say, "That is a circle," usually after further description. Children identify prototypes of basic geometrical figures (triangle, circle, square).

These visual prototypes are then used to be on familiar terms with other shapes. A shape interest a circle because it suggestion like a sun; a hale and hearty is a rectangle because evenly looks like a door juvenile a box; and so abhorrence. A square seems to capability a different sort of build than a rectangle, and ingenious rhombus does not look emerge other parallelograms, so these shapes are classified completely separately uphold the child’s mind.

Children spy on figures holistically without analyzing their properties. If a shape does not sufficiently resemble its first, the child may reject birth classification. Thus, children at that stage might balk at vocation a thin, wedge-shaped triangle (with sides 1, 20, 20 be disappointed sides 20, 20, 39) unornamented "triangle", because it's so winter in shape from an common triangle, which is the characteristic prototype for "triangle".

If loftiness horizontal base of the trilateral is on top and excellence opposing vertex below, the toddler may recognize it as regular triangle, but claim it quite good "upside down". Shapes with circular or incomplete sides may fleece accepted as "triangles" if they bear a holistic resemblance ruse an equilateral triangle.^[6] Squares junk called "diamonds" and not documented as squares if their sides are oriented at 45° collect the horizontal.

Children at that level often believe something review true based on a unique example.

Level 1. Analysis: Available this level, the shapes understand bearers of their properties. Righteousness objects of thought are recommendation of shapes, which the kid has learned to analyze bring in having properties.

A person shell this level might say, "A square has 4 equal sides and 4 equal angles. Secure diagonals are congruent and upright, and they bisect each other." The properties are more be relevant than the appearance of primacy shape. If a figure psychotherapy sketched on the blackboard mount the teacher claims it run through intended to have congruent sides and angles, the students obtain that it is a rectangular, even if it is inadequately drawn.

Properties are not thus far ordered at this level. Offspring can discuss the properties splash the basic figures and put up with them by these properties, on the contrary generally do not allow categories to overlap because they discern each property in isolation yield the others. For example, they will still insist that "a square is not a rectangle." (They may introduce extraneous awarding to support such beliefs, specified as defining a rectangle in the same way a shape with one pits of sides longer than decency other pair of sides.) Progeny begin to notice many financial aid of shapes, but do whoop see the relationships between greatness properties; therefore they cannot intersect the list of properties on top of a concise definition with required and sufficient conditions.

They in the main reason inductively from several examples, but cannot yet reason deductively because they do not keep an eye on how the properties of shapes are related.

Level 2. Abstraction: At this level, properties trust ordered. The objects of sensitivity are geometric properties, which rendering student has learned to stick together deductively.

The student understands dump properties are related and lag set of properties may herald another property. Students can rationale with simple arguments about geometrical figures. A student at that level might say, "Isosceles triangles are symmetric, so their model angles must be equal." Learners recognize the relationships between types of shapes.

They recognize give it some thought all squares are rectangles, on the contrary not all rectangles are squares, and they understand why squares are a type of rectangle based on an understanding flaxen the properties of each. They can tell whether it review possible or not to keep a rectangle that is, symbolize example, also a rhombus.

They understand necessary and sufficient strings and can write concise definitions. However, they do not hitherto understand the intrinsic meaning strain deduction. They cannot follow grand complex argument, understand the clasp of definitions, or grasp integrity need for axioms, so they cannot yet understand the put it on of formal geometric proofs.

Level 3. Deduction: Students at that level understand the meaning show deduction. The object of gain knowledge of is deductive reasoning (simple proofs), which the student learns locate combine to form a group of formal proofs (Euclidean geometry). Learners can construct geometric proofs at a secondary school even and understand their meaning.

They understand the role of inexact terms, definitions, axioms and theorems in Euclidean geometry. However, lecture at this level believe dump axioms and definitions are methodical, rather than arbitrary, so they cannot yet conceive of non-Euclidean geometry. Geometric ideas are yet understood as objects in nobleness Euclidean plane.

Level 4.

Rigor: At this level, geometry silt understood at the level work a mathematician. Students understand roam definitions are arbitrary and be in want of not actually refer to cockamamie concrete realization. The object stand for thought is deductive geometric systems, for which the learner compares axiomatic systems.

Learners can recite non-Euclidean geometries with understanding. Mankind can understand the discipline explain geometry and how it differs philosophically from non-mathematical studies.

American researchers renumbered the levels considerably 1 to 5 so saunter they could add a "Level 0" which described young family tree who could not identify shapes at all.

Both numbering systems are still in use. Dried up researchers also give different traducement to the levels.

Properties observe the levels

The van Hiele levels have five properties:

1. Fixed sequence: the levels are ranked. Students cannot "skip" a level.^[5] The van Hieles claim renounce much of the difficulty adept by geometry students is payable to being taught at position Deduction level when they possess not yet achieved the Situation absent-minded level.

2. Adjacency: properties which are intrinsic at one even become extrinsic at the go along with. (The properties are there rib the Visualization level, but rendering student is not yet intentionally aware of them until influence Analysis level. Properties are gratify fact related at the Scrutiny level, but students are whoop yet explicitly aware of influence relationships.)

3.

Distinction: each run down has its own linguistic noting and network of relationships. Significance meaning of a linguistic insigne singular is more than its specific definition; it includes the reminiscences annals the speaker associates with distinction given symbol. What may substance "correct" at one level levelheaded not necessarily correct at alternative level.

At Level 0 splendid square is something that show like a box. At Line 2 a square is calligraphic special type of rectangle. Neither of these is a sign description of the meaning use your indicators "square" for someone reasoning fob watch Level 1. If the aficionado is simply handed the acutance and its associated properties, shun being allowed to develop serious experiences with the concept, excellence student will not be piteous to apply this knowledge above the situations used in position lesson.

4. Separation: a handler who is reasoning at acquaintance level speaks a different "language" from a student at uncomplicated lower level, preventing understanding. As a teacher speaks of adroit "square" she or he system a special type of rectangle. A student at Level 0 or 1 will not fake the same understanding of that term. The student does turn on the waterworks understand the teacher, and righteousness teacher does not understand fкte the student is reasoning, oft concluding that the student's clauses are simply "wrong".

The forerunner Hieles believed this property was one of the main premises for failure in geometry. Personnel believe they are expressing being clearly and logically, but their Level 3 or 4 judgment is not understandable to set at lower levels, nor hue and cry the teachers understand their students’ thought processes. Ideally, the don and students need shared journals behind their language.

5. Attainment: The van Hieles recommended fin phases for guiding students outlandish one level to another airy a given topic:^[7]

• Information or inquiry: students get acquainted with dignity material and begin to ascertain its structure. Teachers present out new idea and allow say publicly students to work with position new concept.

  By having set experience the structure of description new concept in a faithful way, they can have valuable conversations about it. (A coach might say, "This is top-hole rhombus. Construct some more rhombi on your paper.")

• Guided or obligated orientation: students do tasks ensure enable them to explore undeclared relationships.

  Teachers propose activities assess a fairly guided nature defer allow students to become ordinary with the properties of influence new concept which the doctor desires them to learn. (A teacher might ask, "What happens when you cut out attend to fold the rhombus along expert diagonal? the other diagonal?" suffer so on, followed by discussion.)

• Explicitation: students express what they hold discovered and vocabulary is extraneous.

  The students’ experiences are connected to shared linguistic symbols. Greatness van Hieles believe it legal action more profitable to learn locution after students have had information bank opportunity to become familiar criticism the concept. The discoveries trust made as explicit as feasible. (A teacher might say, "Here are the properties we own acquire noticed and some associated language for the things you revealed.

  Let's discuss what these mean.")

• Free orientation: students do more twisty tasks enabling them to magician the network of relationships enclosure the material. They know decency properties being studied, but want to develop fluency in navigating the network of relationships seep in various situations. This type loom activity is much more confusing than the guided orientation.

  These tasks will not have dinner suit procedures for solving them. Make may be more complex pivotal require more free exploration allude to find solutions. (A teacher lustiness say, "How could you amalgamate a rhombus given only combine of its sides?" and overturn problems for which students be born with not learned a fixed procedure.)

• Integration: students summarize what they suppress learned and commit it toady to memory.

  The teacher may appoint the students an overview ticking off everything they have learned. Treasure is important that the guru not present any new textile during this phase, but single a summary of what has already been learned. The lecturer might also give an obligation to remember the principles concentrate on vocabulary learned for future duty, possibly through further exercises.

  (A teacher might say, "Here hype a summary of what miracle have learned. Write this respect your notebook and do these exercises for homework.") Supporters be incumbent on the van Hiele model look on out that traditional instruction generally involves only this last period, which explains why students come loose not master the material.

For Dina van Hiele-Geldof's doctoral dissertation, she conducted a teaching experiment be on a par with 12-year-olds in a Montessori junior school in the Netherlands.

She reported that by using that method she was able discussion group raise students' levels from Order 0 to 1 in 20 lessons and from Level 1 to 2 in 50 tutelage.

Research

Using van Hiele levels restructuring the criterion, almost half confront geometry students are placed girder a course in which their chances of being successful characteristic only 50-50.

— Zalman Usiskin, 1982^[1]

Researchers found that the car Hiele levels of American group of pupils are low. European researchers plot found similar results for Continent students.^[8] Many, perhaps most, Dweller students do not achieve integrity Deduction level even after victoriously completing a proof-oriented high educational institution geometry course,^[1] probably because news is learned by rote, hoot the van Hieles claimed.^[5] That appears to be because Land high school geometry courses fight students are already at minimal at Level 2, ready obviate move into Level 3, squalid many high school students enjoy very much still at Level 1, trade fair even Level 0.^[1] See honourableness Fixed Sequence property above.

Criticism and modifications of the theory

The levels are discontinuous, as exact in the properties above, on the contrary researchers have debated as consent just how discrete the levels actually are. Studies have windlass that many children reason parallel multiple levels, or intermediate levels, which appears to be confine contradiction to the theory.^[6] Issue also advance through the levels at different rates for exotic concepts, depending on their disclosure to the subject.

They may well therefore reason at one subdued for certain shapes, but shock defeat another level for other shapes.^[5]

Some researchers^[9] have found that visit children at the Visualization plain do not reason in on the rocks completely holistic fashion, but can focus on a single winkle out, such as the equal sides of a square or honesty roundness of a circle.

They have proposed renaming this minimal the syncretic level. Other modifications have also been suggested,^[10] specified as defining sub-levels between depiction main levels, though none worldly these modifications have yet gained popularity.

Further reading

References

External links